quantum mechanic
TRANSCRIPT
CHAPTER 11
QUANTUM MECHANICS
11.1. In conformity with the scope of this book, the emphasis of the present
chapter is on the mathematics of quantum mechanics, the physical ideas entering
the discussion only in a secondary way. Limitation of space further demands that
only the important, and this happily implies the more elementary, portions of the
wide field be presented. Complete exclusion of physical ideas would, however,
leave its subject matter so poorly joined and so incomprehensible to the student who
has no prior knowledge of quantum mechanics that the value of an entirely formal
treatment appears questionable. It is also true that no part of applied mathematics
exacts from its student a more radical change from his customary habits of thought,
a greater tolerance for new methods of inquiry, than does this latest branch. In order
to provide the proper attitude of mind, we preface the later mathematical
developments by a few qualitative remarks whose relevance to the present book is
but auxiliary.
The central notion of classical mechanics is the mass point, or particle.
Classical theory therefore presupposes, tacitly, that a physical system can in
principle be recognized as a particle, or a set of particles. Until the advent of
quantum physics this dogma has never been questioned; in fact scientific
philosophers have frequently inflated it to the dimensions of a universal proposition
claiming that all physical systems are composed of particles. The method of
physical description in best accord with this fundamental attitude is clearly this: To
correlate instantaneous positions of a given particle with instants of time, assuming
motion to be continuous in space and time. Thus, if a particle moves along the X-
axis, the complete description of its motion would appear in the form ๐ฅ = ๐(๐ก).
Now it is conceivable that such a correlation becomes impossible, and the
question then arises whether this fundamental mode of description should be
abandoned in such circumstances. The answer which has often been given and
which the modern physicist emphatically rejects is the flatly negative one, the
answer alleging that classical description is intrinsically evident and that the relation
๐ฅ = ๐(๐ก) has meaning even when the functional relation cannot be established. On
the other hand, one would not like to discard this successful description lightly, for
instance because of certain practical and accidental difficulties in the procedure of
measuring ๐ฅ as a function of ๐ก. The criterion which has ultimately produced clarity
is this: A method of description must be abandoned when it becomes impossible,
not because of experimental difficulty, but because its use contradicts known laws
of science. Classical description has become impossible for the latter reason, as the
following simple example will show.
Imagine an oscillating mass point, e.g., the bob of a pendulum. As long as
the eye can follow the bob, correlations between x and t can certainly be made. But
suppose the mass point is made to increase its frequency of vibration. The eye will
soon be unable to perceive instantaneous positions, but the camera can still establish
them. when the camera fails, oscillographic methods may be available, and after
that, ingenious devices perhaps not yet invented may serve. But ultimately, a barrier
of an essential kind will be encountered. Let us assume that the bob oscillates 1010
times per second. It is a fact of atomic physics that visible light requires about 10โ8
seconds to be emitted (or reflected). Thus if it were used as the medium of report,
the light-emitting mass would have to remain in a given position for approximately
that length of time. In the present instance, however, the bob executes 100
vibrations within this period. A similar argument can finally be used to invalidate
every other means for establishing the classical correspondence. The latter has to
be ultimately abandoned because its use contradicts the laws of optics.
What, then, can be done? Perhaps the example suggests an answer. While a
snapshot can in principle no longer be taken of the rapidly oscillating bob, a time
exposure would reveal some features of its dynamical b behavior. It would give
essentially a correlation between the time the bob spends within a given interval dx
and the location of that interval, in other words between x and the probability wdx
of encountering it in dx. This leads to a less pretentious description of the physical
system called a mass point, of the form w= p (x), and this description is characteristic
of quantum mechanics. It is to be noted that p (x) can be inferred from the classical
relation ๐ฅ = ๐(๐ก), but not ,๐(๐ก) from ๐ค = ๐(๐ฅ).
Quantum mechanics provides the means for deducing probability relations
of the type described, and it does so in a logically consistent fashion. But before
turning to this central issue, let us see what has become of the concept: particle. Our
time exposure has left it very ill defined. Indeed if the system called a mass point
were invisibly small or never sufficiently stationary to permit the classical
description, the customary properties of particles would never be exhibited. By the
criterion of essential observability, the concept would lose its physical significance.
From a misunderstanding of this situation there has arisen a claim that quantum
mechanics leads to a dualism, to the monstrous conception that ultimate entities of
physics like electrons are both particles and waves; the correct statement is that they
are neither particles nor waves, but more abstract entities for the description of
which quantum mechanics gives most simple and successful rules. The question as
to the particle or wave nature of an electron must be put in the same class as that
regarding its color or, to use a lighter metaphor due to the philosopher Dingle, as
the question concerning the color of an elephantโs egg if an elephant laid eggs.
Despite this fundamental situation we shall place no ban upon the use of the
terms particle, wave, etc.; we shall even adhere to universal practice in calling the
electron one of the elementary particles of nature; we do this only, of course, as a
concession to usage. But whenever a paradox arises, the reader should endeavor to
resolve it by recalling that the โclassical languageโ when applied to atomic entities
is in fact metaphoric.
AXIOMATIC FOUNDATION
11.2. Definitions.
For the sake of brevity all historical considerations are omitted here. Nor
will any attempt be made to โdeduceโ quantum mechanics either from classical
physics or from outstanding experimental facts, for in a strict logical sense this
cannot be done. We shall, however, present the framework of the theory with utmost
economy of thought and space, committing the reader to the tacit understanding that
all experimental consequences of the theory outlined have been verified as far as
they could hitherto be tested.
On a physical system, by which is meant any object of interest to physic or
chemistry, numerous observation or measurements can be made. The quantities so
observed or measured, such as size, energy, position and momentum, are called
observables. It is well to think of these observables without ascribing to them the
intuitive qualities they possess in classical mechanics. Position, or energy, is not so
much possessed by a system as it is characteristic of a certain measuring process
which can be carried out upon it. The measurement of an observable upon a system
yield a number.
In defining the state of a physical system considerable caution must be
exercised, for we wish to remain in keeping with the requirements outlined in the
introductory paragraphs. First it is well to notice that by state the scientist never
means anything not subject to arbitrary fixation; indeed the definition of state is
made to conform to the needs of each particular subject. It is quite different, for
instance, in classical mechanics from what it is in thermodynamics or in
electrodynamics. Hence we need not feel ill at ease when in quantum mechanics a
new choice is made. Leaving elucidation until later: a state is1 a function of certain
variables, a function from which by the rules of quantum theory significant
1 The reader who dislike this phrase may substitute โis represented byโ for the simple โisโ. We
wish to warn, however, that the spirit of quantum mechanics permits no distinction in meaning
between these two expressions.
information can be obtained. The variables may be chosen in several ways, each
giving rise to a consistent description equivalent to all others; here they will be
taken to be space coordinates, for this gives rise to the form of quantum mechanics
most commonly used, namely Schrodingerโs. By state function, we thus mean a
mathematical construct, โ (๐ฅ1, ๐ฆ1, ๐ง1; ๐ฅ2, ๐ฆ2, ๐ง2; โฆ ๐ฅ๐, ๐ฆ๐, ๐ง๐). It is possible, as we
shall later see, to associate the variables ๐ฅ1 . . . ๐ง๐ with the dimensions of
configuration space of the classical analogue of the system in question. In particular,
the number of variables needed in โ for a complete description of its behavior (at
a given instant of time) has always been found to be equal to the number of its
classical degrees of freedom. This must indeed be the case in order that large scale
bodies be consistently described both by quantum mechanics and by classical
mechanics. States may change with time; hence a state in its widest meaning may
be written
โ (๐ฅ1๐ฆ1๐ง1โฆ๐ง๐, ๐ก)
Certain restrictions are to be placed upon state functions, restrictions which
will take on greater plausibility in view of the postulates of the next section. Most
important among them are two: first โ , which may be a complex function, must
possess an integrable square2 in the sense that
โซโ โ โ ๐๐ < โ (11-1)
Where ๐๐ is the โVolume of configuration space,โ i.e., in rectangular
coordinates
๐๐ โก ๐๐ฅ1๐๐ฆ1๐๐ง . . . ๐1๐๐ฅ๐๐๐ฆ๐๐๐ง๐
Second,
2 This statement requires modification in some cases. See remarks concerning โcontinuous
spectrum,โ sec. 11.9c. Condition (1) must be rigorously maintained without exception when โซ ๐๐ is
finite. It seems best to present the foundations of the theory with this restriction., leaving necessary
generalizations for later
โ is single โ valued (11-2)
The function โ may of course be expressed in any other system of space
coordinates by the ordinary geometric transformations of Chapter 5. Condition (2)
is particularly important when one of the variables is an angle, say ๐ผ, for it then
requires that
โ (๐ผ) = โ (๐ผ + 2๐๐) (11-3)
๐ being an integer
Finally we must include in our list of definitions another mathematical
construct, that of an operator. Every specific mathematical operation, like adding
6, or multiplying by c, or extracting the third root, etc., can be represented by a
characteristic symbol which is then called an operator. Operators are:
6+, ๐. , โ3
,๐
๐๐ฅ , โซ ๐๐ก
๐
๐, ๐ด
๐2
๐๐ฅ2+ ๐ต
๐
๐๐ฅ+ ๐ถ, and so forth. In general they act on
functions. They can be applied in succession. When they are so applied, the order
in which the operators occur is important. For convenience, let us use more general
symbols for operators, such as P and Q. If P stands for + and Q for c., then PQF
means ๐ผ + ๐๐ ๐คโ๐๐๐ ๐ is a function; however QPf means c ( + f ). Thus
QPf = PQf + ( c โ 1 ) (11-4)
Such an equation is said to be an operator equation. The reader will at once
verify that, if P stands for / x and Q for x., the operator equation
PQf - QPf = f (11-5)
There is an important difference between eqs. (4) and (5); the second is
homogeneous in f, the first is not. From the second, f may be canceled symbolically
so that it reads
PQ โ QP = 1 (11-5)
Only homogeneous operator equations of this kind, usually written in the
latter form without explicit insertion of the operand f, are of interest in quantum
mechanics.
The formalism of operator is convenient also in other ways. It is possible,
for instance, to define a periodic function โ (๐ฅ) by writing
๐โ๐ทโ (๐ฅ) = โ (๐ฅ)
D being d/dx; for the left-hand side is, on expansion, simply the Taylor
series for โ (๐ฅ + โ).
Two operators, P and Q, are said to commute when PQ โ QP is zero. Thus
c and d/dx; commute if c is a constant. Other examples of commuting operators
are: x and d/dx; d/dx and โซ ๐๐ฅ๐
๐ if and b are constant; + and ( b ). Clearly,
every operator commutes with itself or any power of itself, provided that by the n-
th power we mean the n-fold iteration of the operator.
11.3. Postulates.3
a. The fundamental postulates of quantum mechanics are three in number.
The first concerns the use of observables.
Brief reflection will show that classical physics associates with observables
certain definite function of suitable variables: x, y, z with position, mv with linear
momentum, 1
2๐๐ฃ2 with kinetic energy, and so forth. These function are chosen to
describe experience most adequately. There is no logical reason which would
exclude the use of more abstract mathematical entities in this association. It has
3 Henceforth in the present section, and in all subsequent sections up to 11.25, states will be
supposed to be independent of the time; i.e., โ does not contain I, Such states are known as
stationary ones, and the part of quantum mechanics dealing with them will be called quantum
statics. In quantum dynamics, introduced in sec. 11.25, a new postulate (Schrodingerโs โtimeโ
equation) will be needed. This postulate is not included in the present list. Nor do we include the
Pauli principle, which is also of axiomatic status, and which will be presented in sec. 11.33. The
presented limitation is made for pedagogical reasons
indeed been found that, for the description of atomic phenomena, certain operators
should replace the functions which in classical mechanics represent observables.
The first postulate may be stated as follows:
To every observable there corresponds an operator.
The correct operator to be associated with a given observable must be found
by trial. In the following table we give a brief summary of the four most important
operators of quantum mechanics; the observable in question are understood to refer
to systems classically described as groups of mass points having 3n degrees of
freedom (j = 1, 2, .. .,n), subject to no external forces ( total energy constant ) and
not requiring relativity treatment. The first column gives the name of the
observable, the second its classical representation, the third its quantum mechanical
representation.
Cartesian coordinate ๐ฅ๐ ๐ฅ๐
Cartesian component
of linear momentum of
j-th particles
๐๐ฅ๐ = ๐๐๏ฟฝ๏ฟฝ๐ โ
๐ ๐
๐๐ฅ๐
X-component of
angular momentum of
j-th particles
๐๐(๐ฆ๐๏ฟฝ๏ฟฝ๐ โ ๐ง๐๏ฟฝ๏ฟฝ๐) โ
๐( ๐ฆ๐
๐
๐๐ง๐โ ๐ง๐
๐
๐๐ฆ๐
Total energy 1
2โ
1
๐๐(
๐
๐๐ฅ๐2 + ๐๐ฆ๐
2
+ ๐๐ง๐2 + ๐ (๐ฅ1 . . . ๐ง๐ )
โ โ2
2โ
1
๐๐( ๐2
๐๐ฅ๐2 +
๐๐=1
๐2
๐๐ฆ๐2 +
๐2
๐๐ง๐2) + ๐
(๐ฅ1 . . . ๐ง๐ )
๐๐ is the mass of the j-th particle; ั is an abbreviation for Planckโs constant, h,
divided by 2๐.
The operator form of the Cartesian coordinate ๐ฅ, is identical with its
classical representation and has been included only for formal reasons. Linear
momentum, a differential operator, is basic in the construction of the last two entries
in the table.
When the operator corresponding to the linear momentum p of a single
particle is written in the vector form iัโ, those corresponding to angular
momentum and energy of this particle may be constructed according to classical
formulas: Angular momentum = ๐ ร ๐ = โ๐ั๐ ร โ, and Energy = (1
2๐) ๐2 +
๐ = โ(ั2
2๐) โ2 + ๐. These vector form are valid in all other system of coordinates
and should be used as the basis for transformation.
In view of the table, the reader will easily verify the following operator
equations:
Let ๐๐ stand for the operator โ k-th Cartesian coordinate,โ ๐๐ for the k-th
component of linear momentum. Then
๐๐๐๐ โ ๐๐๐๐ = โ๐ั๐ฟ๐๐
(11-6)
Also, if ๐ฟ๐ฅ, ๐ฟ๐ฆ, and ๐ฟ๐ง denote the components of the angular momentum
operator for a single particle,4
๐ฟ๐ฅ๐ฟ๐ฆ โ ๐ฟ๐ฆ๐ฟ๐ฅ = ๐ั๐ฟ๐ง
๐ฟ๐ฆ๐ฟ๐ง โ ๐ฟ๐ง๐ฟ๐ฆ = ๐ั๐ฟ๐ฅ (11-7)
4 ๐ฟ๐ฆ and ๐ฟ๐ง may be obtained from ๐ฟ๐ง in the table by cyclical permutation of coordinates
๐ฟ๐ง๐ฟ๐ฅ โ ๐ฟ๐ฅ๐ฟ๐ง = ๐ั๐ฟ๐ฆ
Commutation rules, like (6) and (7), are often sufficient to define the
operators involved without recourse to their explicit form, but the latter is usually
helpful.
b. The second postulate states:
The only possible values which a measurement of the observable whose
operator is P can yield are the eigenvalues pฮป of the equation
๐๐๐ = ๐๐๐๐ (11-8)
Provided ๐๐ obeys conditions (1) and (2),namely: โซ๐๐โ ๐๐๐๐ <
โ ๐๐๐ ๐๐ is single-valued.
The range of integration depends on the particular problem under
consideration, as will be seen later.
We illustrate the meaning of this postulate by a few examples. Let us find
the measurable values of the linear momentum of a particle, known to be
somewhere on the X-axis between the finite points ๐ฅ = ๐ผ and ๐ฅ = ๐. The operator
P is โ๐ั (๐
๐๐ฅ). Eq. (8) therefore becomes a first-order differential equation which
can obviously be satisfied if ๐๐ is assumed to be a function of x only. It reads
โ๐ั๐๐๐
๐๐ฅ= ๐๐๐๐ (11-9)
And has the solution
๐๐ = ๐๐(๐ั)๐๐๐ฅ
is this solution satisfactory from the point of view of eqs. (1) and (2) ? It is certainly
single-valued; moreover, โซัฑ๐โ ัฑ๐๐๐ฅ = (๐ โ ๐)๐
โ๐ is finite for every finite c.
Hence bo restriction upon ๐๐ results; ๐๐๐ values of the linear momentum may be
found upon measurement. The eigenvalues of the linear momentum from a
continuous spectrum ( ฮป is not a discrete index) and every function of the form
๐๐(๐
ั)๐๐ฅ
with constant ๐ is an eigenfunction. As far as measurable values of linear
momentum are concerned, quantum mechanics leads to the same result as classical
physics.
This in not true for the ๐๐๐๐ข๐๐๐ momentum of a single particle. Here eq.
(8) reads
โ๐ั (๐ฅ๐
๐๐ฆโ ๐ฆ
๐
๐๐ฅ)ัฑ๐ = ๐๐๐๐ (11-10)
Provided we consider the ๐ง-component and write ัฑ๐ for the eigenvalues.
Obviously, ัฑ๐ must be a fuction of both x and y. But a simple transformation of
coordinates reduces the equantion to a simpler form. On putting ๐ฅ = ๐ cos ๐ and
๐ฆ = ๐ sin ๐ , we have
๐
๐๐= โ๐ sin ๐
๐
๐๐ฅ+ ๐ cos ๐
๐
๐๐ฆ= ๐ฅ
๐
๐๐ฆโ ๐ฆ
๐
๐๐ฆ
Therefore aq. (10) becomes
โ๐ั ๐ัฑ๐๐๐
= ๐๐ัฑ๐
And ัฑ๐ is seen to be a funcition of ๐ alone. The solution is
ัฑ๐ = ๐๐(๐ั)๐๐๐
It certainly has an integrable square, because the range of ๐ extends from 0
to 2๐, or more exactly, from 2๐๐ to 2๐(๐ + 1), where ๐ is an integer. But ัฑ๐
violates the condition of single-validness which must be imposed in the from (3).
To satisfy it we must require that
ัฑ๐(๐) = ัฑ๐(๐ + 2๐)
And this implies ๐(2๐๐
ั)๐๐ = 1. This is true only if
๐๐ = ๐ั , ๐ an integer (11-11)
Hence the only observable values of the angular momentum are given by
(11), and the eigenfunctions are ๐๐(๐
๐). This result is identical with the postulate of
the older Bohr theory concerning angular momentum.
Next we consider the possible values of the total energy of a single mass
point. The energy operator appearing in the table is often referred to as the
Hamiltonian operator and is denoted by the symbol H. let us use ๐ธ๐ for the
eigenvalues. The operator equation then becomes
๐ปัฑ๐ โก โั2
2๐โ2ัฑ๐ + ๐(๐ฅ, ๐ฆ, ๐ง)ัฑ๐ = ๐ธ๐ัฑ๐ (11-12)
This equation, written perhaps more frequently in the form
โ2ัฑ๐ + 2๐
ั2 (๐ธ๐ โ ๐)ัฑ๐ = 0 (11-12)
Was found by ๐๐โ๐๏ฟฝ๏ฟฝ๐๐๐๐๐๐ and bears his name. Its solutions and eigen-values
clearly depend on the functional nature of ๐(๐ฅ, ๐ฆ, ๐ง); they will be reserved for
detailed consideration in secs. 9 et seq.
A rather peculiar result is obtained when (8) is applied to the coordinate
โoperatorโ. The eigenvalues of โ๐ฅโ are the values ฮพ๐for which the equation
๐ฅ. ัฑ๐ = ฮพ๐ัฑ๐
an ordinary algebraic one, possesses solutions. On writing it in the form
(๐ฅ โ ฮพ๐)ัฑ๐ = 0
It is evident that either ๐ฅ = ฮพ๐ or ัฑ๐ = 0. In plainer language, ัฑ๐ as a function
of ๐ฅ vanishes everywhere except at ๐ฅ = ฮพ๐, a constant. From a rigorous
mathematical point of view such a function is a monstrosity, but it is useful for
certain purposes to introduce it, as Dirac5 has done. It is called ๐ฟ(๐ฅ โ ฮพ๐), the
symbol being fashioned after the Kronecker ๐ฟ, and is best visualized as something
like lim๐โ0
๐๐โ(๐ฅโฮพ๐)2/๐
. For later use the constant ๐(๐) will be so chose that
โซ ๐ฟ(๐ฅ โ ๐)๐๐ฅ = 1,โ
โโ so that
โซ ๐(๐ฅ)๐ฟ(๐ฅ โ ๐)๐๐ฅ = ๐(๐)โ
โโ (11-13)
now it is clear that such a โfunctionโ can be formed for every value ฮพ๐ , hence every
point of the X-axis is an eigenvalue of the ๐ฅ-coordinate. 6
The significance of the second postulate is best grasped when it is regarded
as furnishing a catalogue of the measurable values of all observables for which
operators are known. It implies no information concerning the meaning of the
eigenfunction ัฑ๐. These are, of course, states of the system in the sense explained.
Their nature will unfold itself when the third postulate has been set forth. For the
present we only note that every ๐๐ is indeterminate with respect to a constant
multiplier; eq.(8) will also be satisfied by constant ๐๐ On the other hand,
โซ๐๐โ๐๐ ๐๐ exists. We may require, therefore, that ๐๐ is normalized after the manner
of sec.8.2. Henceforth this will be assumed unless a statement to the contrary is
made. In this connection it may be recalled, however, that normalization may fail
intrinsically when the eigenvalues ๐๐ form a continuous spectrum. In Chapter 8 this
was shown to be the case in instances where the range of the fundamental variable
became infinite. These require special treatment.
The ๐๐ will be orthogonal if operator and boundary conditions conform to
the circumstances of the Sturm- Liouville theory (sec.8.5). this theory, as will later
5 Dirac, P.A.M., โPrinciples of Quantum Mechanics,โ Third Edition; Clarendon Press, Oxford, 1947.
6 The operation ๐ฅ. has continuous spectrum. Correspondingly , the integral โซ ๐ฟ2(๐ฅ โ ๐)๐๐ฅ does not
exist ! See ses. 11.9c.
be seen, covers most of the cases occurring in quantum mechanics, but must be
generalized somewhat to be applicable to complex operators.
c. We turn to the third postulate which states:
when a given system is in a state โ , the expected means of a sequence of
measurements on the observable whose operator is P is given by
๏ฟฝ๏ฟฝ = โซ โ โ ๐โ ๐๐ (11-14)
The expected means is defined as in statistics : If a large number of
measurements is made on the system, and the measured values are ๐1,๐2,โฆโฆ..
๐๐ ,then ๏ฟฝ๏ฟฝ โก1
๐โ ๐๐๐๐=1 . Note that eq. (14) does not predict the outcome of a single
measurement.
In writing (14) we are again supposing that โ is normalizes. This can be
brought about all physical problems by โconfiningโ the system in configuration
space, that is, by taking the volume in which it moves to be finite, so that โซ๐๐
exists. Even if the volume is infinite, โซโ โ โ ๐๐ may still exist, but in general the
situation then calls for special treatment involving the use eigendifferentials instead
of eigenfunctions. * A more general form of eq. (14), which often works when the
volume of configuration space is infinite, is the following
๏ฟฝ๏ฟฝ = ๐๐๐๐ โ โ
โซ๐ ๐โ ๐๐
โซ๐โ โโ ๐๐
(11-14โ)
*See Morse, P.M., and Feshbach, H., โMethods of Theoretical Physics,โMeGraw
Hill Book Co., Inc., 1953. We illustrate the meaning of (14) by a few examples.
Let a system having one degree of freedom be in a state described byโ =
(๐/๐)1
4๐โ (๐
2)(๐ฅโ ๐)2
. Then the mean value of its position will be:
๏ฟฝ๏ฟฝ = โซ โ 2๐ฅ๐๐ฅ = ๐โ
โโ
Its mean momentum:
๏ฟฝ๏ฟฝ๐ฅ = โ๐ั โซโ โ โฒ๐๐ฅ = 0
Its mean kinetic energy:
๏ฟฝ๏ฟฝ๐๐๐ = โั2
2๐โซโ โ โฒโฒ๐๐ฅ =
ั2
2๐โซ(โ โฒ)2 ๐๐ฅ =
๐
2 โ ั2
2๐
It is interesting to note that, the more concentrated the function โ (the
greater b) the larger will be the mean kinetic energy. To calculate the mean total
energy we should have to know the form of ๐ (๐ฅ).
Let us take โ = ๐๐๐๐ฅ / (b โ a)1/2 . We then find
๏ฟฝ๏ฟฝ = โซ โ โ๐ฅโ ๐๐ฅ = ๐ + ๐
2
๐
๐
๏ฟฝ๏ฟฝ๐ฅ = โ๐ัโซ โ โโ โฒ๐๐ฅ = ๐โ๐
๐
๏ฟฝ๏ฟฝ๐๐๐ = โโ2
2๐ โซ โ โโ โฒโฒ๐๐ฅ =
๐2โ2
2๐
๐
๐
If in this example the range is extended to infinity, let us say in such a way
that โ ๐ = ๐ โ โ, the function ๐๐๐๐ฅ can clearly not be normalized One just then
eq. (14โ) in the form
๏ฟฝ๏ฟฝ = limโโ
โซ โ โ๐โ ๐๐ฅ๐
โ๐
โซ โ โโ ๐๐ฅ๐
โ๐
Which gives the same result as those obtained above.
The three postulates here stated and exemplified do not reveal an intuitive
meaning of the state function โ . It is therefore not unusual in textbooks on quantum
mechanics to add another postulate stating that โ โ(๐ฅ)โ (๐ฅ) signifies the probability
that the โparticleโ whose state is โ be found at the point ๐ฅ of configuration space (
with suitable generalization for more than one degree of freedom). This is indeed
true, and it may be well for the reader to form this basic conception; but this
statement is not a further postulate since it may be deduced from those already
given.( Cf. sec.6. )
DEDUCTIONS FROM THE POSTULATES
11.4. Orthogonality and Completeness of Eigenfunctions.
In Chapter 8, orthogonality and completeness of the eigenfunctions belonging to
the Sturm-Liouville operator L have been discussed. The proofs there given need to
be generalized if they are to be applied to quantum mechanics, for the operators
occurring there are not all af the same structure as L. (one of the mist important
equations encountered, the one-dimensional Schr๏ฟฝ๏ฟฝdinger equantion (12), is of the
Sturm-Liouville type.) They often involve many variables, they may be differential
operators of the first order, they may be complex; in fact they may not be differential
operators at all. To simplify the theory we shall assume that the eigenvalues ๐๐ of
eq. (8) are discrete, and that the boundary conditions on acceptable state functions
are of the form 1 and 2. Whenever convenient we shall even assume that โ vanishes
at the boundary of configuration space, over which integrations are to be carried
out, in a manner suitable to our needs. Unless these restrictions are made the
arguments become involved and in some respects problematic. It would the be
necessary to conduct a separate proof for every problem of interest; thus elegance
would fall prey to rigor.
We first define what is meant by an Hermitian operator. Let u and v be two
โacceptableโ functions, defined over a certain range of configuration space ๐. We
then say that the operator P in Hermituan if
โซ๐๐ขโ โ ๐๐ฃ๐๐ = โซ
๐๐ฃ โ ๐โ๐ขโ๐๐ (11-15)
All operators of interest in quantum mechanics have property. As a sample
proof The hermitian property of ๐ฅ . is obvious. To prove it for the Hamiltanian H,
two partial integrations are necessary; the details may be left as an exercise for the
reader.
Hermitian operators real eigenvalues. The fact follows at once from eq. (15)
the eigenvalues of P are defined by the equation.
๐ัฑ๐ = ๐๐ ัฑ๐ (11 โ 16)
This also implies the validity of the equantion
๐โัฑ๐โ = ๐๐
โัฑ๐ โ (11-17)
Now multiply (16) by ัฑ๐ โ and (17) by ัฑ๐ , and integrate over ๐๐ obtaining
โซัฑ๐โ ๐ัฑ๐๐๐ = ๐๐ โซัฑ๐
โ ๐ัฑ๐๐๐ โซัฑ๐๐โัฑ๐
โ๐๐ = ๐๐โโซัฑ๐
โ ๐ัฑ๐๐๐
By (15) the left-hand sides of these two equation are equal, for ัฑ๐ is certainly an
acceptable function in the sense outlined before. Hence ๐๐โ = ๐๐ ; i.e., ๐๐ is real.
Since the eigenvalues of operators are measurable values of observables, which
must of necessity be real, the physical significance of an operator is assured when
it has the Hermitian property.
Let us again consider eq. (16). If ัฑ๐ is some other eigenfunction, it is
evident that
โซัฑ๐โ ๐ัฑ๐๐๐ = ๐๐ โซัฑ๐
โ ๐ัฑ๐๐๐ (11-18)
But if we start with the equation
๐โัฑ๐โ = ๐๐ัฑ๐
โ
Which in true because ๐๐ is real, we also conclude that
โซัฑ๐๐โัฑ๐
โ ๐๐ = ๐๐ โซัฑ๐โ ๐ัฑ๐๐๐ (11-19)
Combining (18) and (19) we find
โซัฑ๐โ ๐ัฑ๐๐๐ โ โซัฑ๐๐
โัฑ๐โ ๐๐ = ( ๐๐ โ ๐๐) โซัฑ๐
โ ๐ัฑ๐๐๐
If P is Hermitian the left-hand side vanishes. Hence either ๐๐ = ๐๐ or
โซัฑ๐โ ๐ัฑ๐๐๐ = 0 . we see that eigenfunctions of Hermition operators, belonging to
different eigenvalues, are orthogonal.
The completeness of the eigenfunctions of all operators employed in
quantum mechanics is usually assumed. To the authorsโ knowledge, Rigorous proof
has not been given. Since, however, our main interest will be in the Schrodinger
equation which is of the Sturm-Liou vile type, this point need not detain us further.
In the following we shall assume completeness of all ฯฮป whenever this property is
needed.
Problem. Show that the angular momentum operator ๐ฟ๐ง = โ๐ั ( ๐/๐๐ ) is
Hermitian.
11.5. Relative Frequencies of Measured Values.
Important consequent can now be deduced from the third postulate, eq. (14). We
first note that, if P is Hermitian, every power of P is Hermitian. Moreover, if (14)
is true for every operator P, it must certainly hold for the operator๐๐. It implies,
therefore,
๐๐ = โซโ โ ๐๐โ ๐๐, ๐ = 1, 2, โฆ (11-20)
The left-hand side stands, of course, foe the r-th moment of the statistical
aggregate or the measured values, i.e.,
๐๐ = โ ๐๐๐ ๐๐๐ (11-21)
Provided ๐๐ is the relative frequency of the occurrence of the i-th eigenvalue ๐๐ in
the sett of measurement. In accordance with eq. (20), the state function โ predict
not only the mean, but all moments of the aggregate of measurements.7 Now eq.
(20) may be transformed as follows. Let the eigenfunctions of P be denoted by ัฑฮป,
7 for terminology., see sec. 12.3.
so that ๐ัฑ๐ = ๐๐ัฑ๐. On allowing P to operate on both side of this equations, there
result ๐2ัฑ๐ = ๐๐๐ัฑ๐ = ๐๐2ัฑ๐. By continuing this process, the relation
๐๐ัฑ๐ = ๐๐๐ัฑ๐ (11-22)
Is established. If the function โ appearing in (20) is expanded in terms of the ัฑ๐,
โ = โ๐ัฑ๐๐
And this series is substituted, we find
๐๐ = โซโ๐โ๐
๐๐
ัฑ๐โ๐๐ ัฑ๐๐๐ = โ๐
โ๐๐๐๐
๐๐
โซัฑ๐โัฑ๐ ๐๐๐
= โ๐โ๐๐๐
๐
๐
By virtue of (22) and the orthogonality of the ัฑ๐. Comparing this with (21)
it is clear that
โ๐๐๐๐๐
๐
= โ|๐|2๐๐
๐
๐
For every integer r. But this can be true only if
๐๐ = |๐|2 (11-23)
In Words: when the system is in the state โ , a measurement of the
observable corresponding to P will yield the value ๐๐ with a probability (relative
frequency) |๐|2,๐ being the coefficient of ัฑi in the expansion โ ๐ัฑ๐,๐ and ัฑ๐
is one of the eigenfunctions of P. The coefficients ๐ are called probability
amplitudes.
They may be expressed on terms of โ and ัฑ๐ by the relation
โซัฑ๐โโ ๐๐ = โ ัฑ๐
โ๐ ๐ัฑ๐๐๐ = ๐ (11-24)
Consequently, eq. (23) may also be written
๐๐ = | โซัฑ๐โโ ๐๐|2 (11-25)
An interesting result is obtained when, in this equation, we let โ be one of
the eigenfunctions belonging to the operator P itself, e.g., ัฑj. It then reads
๐๐ = | โซัฑ๐โโ ๐๐|2 = ๐ฟ๐๐
All relative frequencies are zero expect the one measuring the occurrence of
the eigenvalue๐๐, which is unity. Thus we conclude that an Eigen state ัฑ๐ of an
operator P is a state in which the system yields with certainly t5he value ๐๐ when
the observable corresponding to P is measured. Eigen functions are simply state
functions of this determinate character.
11.6. Intuitive Meaning of a State Function.
Consider now a system, like a simple mass point with one degree of
freedom, whose state function is โ (๐ฅ). We wish to know the probability that a
measurement of its position will give the value ๐ฅ = ๐. The eigenfunction
corresponding to the operator ๐ฅ for the value ฮพ has been shown to be
ัฑ๐ = ๐ฟ (๐ฅ โ ๐)
Eq. (25) now reads
๐๐ = | โซ ๐ฟ (๐ฅ โ ๐) โ (๐ฅ)๐๐ฅ |2 = |โ (๐)|2 (11-26)
By virtue of (13). The probability (destiny) of finding the system at ฮพ is
given by the square of its state function. This fact provides a simple intuitive
meaning for the state function. It can be generalized to several dimensions Let
๐1, ๐2, . . . , ๐๐ be the coordinates on which โ depends. Using the former argument,
the eigenfunction corresponding to t5he composite coordinate operator ๐1 โ ๐2 โ โ โ โ
๐๐ may be shown to be
ัฑ๐1๐2โโโ๐๐ = ๐ฟ(๐1 โ ๐1)๐ฟ(๐2 โ ๐2)๐ฟ(๐๐ โ ๐๐) (11-27)
If, therefore, we wish to find the probability ๐๐1๐2โโโ๐๐ of finding the system at the
point (๐1๐2 ๐๐) of configuration space, we must use eq. (25) with ัฑ๐ replaced by
(27). Hence
๐๐1โโโ๐๐ =
|โฌโโโ โซ ๐ฟ (๐1 โ ๐1) โโโ ๐ฟ (๐๐ โ ๐๐)โ (๐1๐2 ๐๐)๐๐1๐๐2 โโโ ๐๐๐|2 =
|โ (๐1๐2 ๐๐)|2
11.7. Commuting Operators.
Let P and R be two operators satisfying the relation PR โ RP = 0, and let
their eigenfunctions be ัฑ๐ and๐ฅ๐, that is
๐ัฑ๐ = ๐๐ัฑ๐, ๐ ๐ฅ๐ = ๐๐๐ฅ๐ (11-28)
We assume the state function to be ัฑ๐ so that, when P is measured, there result with
certainty the value๐๐. But
๐ ๐ัฑ๐ = ๐ ๐ัฑ๐ = ๐๐๐ ัฑ๐
Considering only the last two members of this equation, we may say that
(๐ ัฑ๐) is an eigenfunction of P, namely that belonging to the eigenvalue๐๐. But this
is possible only if ๐ ัฑ๐ = const. ัฑ๐. Comparison with the second equation (28)
shows the constant to be one of the๐๐, and ัฑ๐ to be one of the eigenfunction๐ฅ๐. We
conclude that commuting operators have simultaneous eigenstates; i.e.,
measurements on their observable yield definite values for both; they do not
โspread.โ
The fact that, when P and Q are non-commuting operators and the state of
the system is an eigenstate of P, measurement on Q will give a statistical aggregate
of values and not a single one with certainty, is usually attributed to the interference
of measuring devices. For instance, the measurement of a particleโs position
disturbs its momentum, and vice versa, so that when one is ascertained with
precision, the other quantity loses it. From this point of view, measurements on the
observables associated with commuting operators are said to be compatible, the
procedures of measurements do not conflict do not conflict with each other.
11.8. Uncertainty Relation.
The proof of the famous Heisenberg uncertainty principle which will now
be given requires the use of an inequality, similar to a well known relation due to
Schwarz, though not identical with it. (Cf. eq. 3-112.)
Functions in the sense specified in connection with the definition of Hermitian
operators (sec. 11.4), then
โซ๐ขโ๐ข๐๐ โ โซ ๐ฃโ๐ฃ๐๐ โง1
4[โซ(๐ขโ๐ฃ + ๐ฃโ๐ข)๐๐]2 (11-29)
We assume a system to be in a stateโ , which need not be an eigenstate of
any particular operator, and we are interested in the result of measurements on the
observables belonging to two operators, P and Q, at present unspecified. Introduce
into eq. (29) the following functions
๐ข = (๐ + ๏ฟฝ๏ฟฝ)โ ๐๐๐ ๐ฃ = ๐(๐ โ ๏ฟฝ๏ฟฝ)โ
Where ๏ฟฝ๏ฟฝ and ๏ฟฝ๏ฟฝ are mean values associated with P and Q through the relation (14).
Eq. (29) then reads
โซ( ๐ โ ๏ฟฝ๏ฟฝ )โโ โ(๐ โ ๏ฟฝ๏ฟฝ )โ ๐๐ โ โซ(๐ โ ๏ฟฝ๏ฟฝ)โโ โ(๐ โ ๏ฟฝ๏ฟฝ)โ ๐๐ โง
1
4[ ๐ โซ(๐ โ ๏ฟฝ๏ฟฝ)โโ โ(๐ โ ๏ฟฝ๏ฟฝ)โ ๐๐ โ ๐ โซ(๐ โ ๏ฟฝ๏ฟฝ)โ โ โ(๐ โ ๏ฟฝ๏ฟฝ)โ ๐๐]2
Now P and Q are Hermitian and satisfy eq. (15); ๏ฟฝ๏ฟฝ and ๏ฟฝ๏ฟฝ are constants. Therefore
the inequality reduce to
โซโ โ ( ๐ โ ๏ฟฝ๏ฟฝ )2โ ๐๐ โ โซ โ โ(๐ โ ๏ฟฝ๏ฟฝ)2 โ ๐๐ โง โ1
4[โซโ โ(๐๐ โ ๐๐)โ ๐๐]2 (11-30)
Let us consider the meaning of the quantity โซโ โ ( ๐ โ ๏ฟฝ๏ฟฝ )2โ ๐๐. When โ is
expanded in eigenfunctions ัฑฮป of P, โ = โ ๐ัฑ๐๐ , and the expansion is introduced
in the integral, the result is โ |๐|2(๐๐ โ ๐ ๏ฟฝ๏ฟฝ)2, and this, in view of eq. (23),is
nothing other than the dispersion8 of the statistical aggregate of p-measurements
about their mean. For this quantity we may introduce the more familiar symbolโ๐2 .
A similar identification is to be made forโซโ โ(๐ โ ๏ฟฝ๏ฟฝ)2 โ ๐๐. Inequality (30) then
takes the more interesting form
โ๐2 . โ๐2 โง โ1
4[โซโ โ(๐๐ โ ๐๐)โ ๐๐]2 (11-31)
Now if P and Q commute, the right-hand side is zero, and it is possible for
โ๐2 ๐๐ โ๐2 to be zero, or even for both to vanish. This state of affairs recalls the
result of sec. 7, which was that both p- and q-measurements could yield single
values without spread.
When P and Q do not commute, relation (31) sets a lower limit for the
product of the dispersions, often called uncertainties. Suppose, for instance, that P
is the operatorโ ๐ั (๐
๐๐), the linear momentum associated with q, and Q stands for
the coordinate q. We then have
๐๐ โ ๐๐ = ๐ั (11-32)
When this is put into (31) the result is โ๐2 โ โ๐2 โงั2
4 , or , written in terms of
standard deviations, ๐ฟ๐ and ๐ฟ๐
๐ฟ๐ โ ๐ฟ๐ โง ั/2 (11-33)
This is Heisenbergโs uncertainly relation.
8 The โdispersionโ is the square of the so-called โstandard deviation.โ It is an index of the โ spreadโ
of the measurements. See chapter 12.
Our result need not be east in the form of an inequality. It is indeed quite
possible to calculate both ๐ฟ๐ and ๐ฟ๐ separately and exactly when the state function
โ is given, as the postulates show.
A slight generalization of the present conclusions is also possible. There
are other operators, such as ๐ฟ๐ง and ๐ (ef. Eq. 10 et seq.) which also obey eq. (32).
In fact all quantities which are called canonically conjugate in classical physics9
have operators which satisfy it. (Later we shall see that energy and time belong to
this class.) For all these, the uncertainty relation in the form (33) is valid.
Problem. Show that, if the state function โ is an eigenfunctions of the
angular momentum operator ๐ฟ๐ง corresponding to the eigenvalue ๐ฟ๐ง the product of
๐ฟ๐๐ฅ and ๐ฟ๐๐ฆ is at least as great as (ั/2) ๐๐ง
SCHR๏ฟฝ๏ฟฝDINGER EQUATIONS
Attention will now be given to the eigenvalues and eigenfaunctions of
the energy operator, that is, to the solutions of the various forms of the Schrแฝdinger
equations, eq. (12)
11.9. free Mass Point.โ The simplest example of a physical system is
the free mass point for which the potential energy V may be taken to be zero. In that
case eq. (12) reads
โ2ัฑ + ๐2ัฑ = 0 (11-34)
Provided we omit the subscript ฮป and write๐2 โก 2๐๐ธ/ั2. This quantity
๐2 has a rather simple classical significance which it is well to recognize at once.
For if E is the total energy of the particle, which is in this case purely kinetic,
then๐ธ =1
2๐๐ฃ2 = ๐2/2๐. Hence ๐ โก
๐
ั, ๐ being the classical momentum of the
particle. Note also that k has the dimension opf a reciprocal length.
Eq. (34) has already been solved in Chapter 7 (cf. eq. 7-33), where it appeared as
the space form of the wave equation. To select the proper solution, we must consider
the fundamental domain, ๐, of our problem. Here, a great number of possibilities
present themselves.
a. Enclosure is a Parallelepiped. If the particle is known to be within a
parallelepiped of side lengths ๐1, ๐2 and๐3 , then ๐ is this volume of space.
Moreover, since |ัฑ(๐ฅ๐ฆ๐ง)|2 has already been identified as the probability of finding
the particle at the point ๐ฅ, ๐ฆ, ๐ง this quantity must certainly be zero everywhere
outside ๐. For reasons of continuity (which can, by more expanded arguments, be
shown to result from our axioms) we require that|ัฑ|2, and hence ัฑ itself, shall
vanish on the boundaries of ๐ also. In view of this boundary condition, the solution
of (34) in rectangular coordinates, namely eq. 7-36, must be chosen, in more
explicit form it reads
ัฑ = (๐ด1๐๐๐1๐ฅ + ๐ต1๐
โ๐๐1๐ฅ) (๐ด2๐๐๐2๐ฆ + ๐ต2๐
โ๐๐2๐ฆ)(๐ด3๐๐๐2๐ฆ + ๐ต3๐
โ๐๐2๐ฆ,
๐2 = ๐12 + ๐2
2 + ๐32
The origin of the parallelepiped may be taken in one corner. Vanishing
of ัฑ at the boundary then requires:
๐ด๐ + ๐ต๐ = 0, ๐ด๐ ๐๐๐๐ ๐๐ + ๐ต๐ ๐
โ๐๐๐ ๐๐ = 0, ๐ โ 1 ,2 ,3
The first condition makes each parenthesis of ัฑ a sine-function; the second implies
๐๐ = ๐๐ ๐
๐๐
Where ๐๐ is an integer. Hence
ัฑ = ๐ sin(๐1๐
๐1๐ฅ) sin(
๐2๐
๐2๐ฆ) sin(
๐3๐
๐3๐ง) (11-35)
and
๐2 = (๐12
๐12 +
๐22
๐22 +
๐32
๐32 )๐
2
So that
๐ธ = ๐2ั2
2๐[(๐1
๐1)2 + (
๐2
๐2)2 + (
๐3
๐3)2] (11-36)
If ฯ is to be normalized, โซฯ โ ฯdxdydz = 1, and the constant c has the value
๐ = (8
๐1๐2๐3)
12โ
= (8
๐)
12โ
The permitted energy values form a denumerably infinite set. Their arrangement
is best represented by constructing a lattice of points filling all space, with the
โreciprocalโ parallelepiped of sides 1 ๐1โ , 1 ๐2
โ , 1 ๐1โ as crystallographyc unit. If
from a given point lines are drawn to all other points, the squares of the lengths of
these lines (multiplied by ๐2โ2/2m) are the energies of our problem. However, not
all these lines represent different states. The function ฯ changes only it sign when
one of the integers ๐1, ๐2 ๐๐ ๐3 changes sign; it is not therebly converted into a new,
linearly independent function. Hence only the lines lying in one octant of the lattice,
with the origin of the lines at one corner, will represent different states. If some of
the๐โฒ๐ are equal there will be degeneracy (cf. Sec. 8. 6), for then an interchange of
the corresponding ๐โฒ๐ will not produce a different E, while ฯ will be changed into
a function which is nearly independent from the original one.
b. Enclosure is s sphere. Eq. (34) must now be solved in sphericak
coordinates. But this has already been done in sec. 8.4 (cf. Eq. 8-25), for an
acoustical problem. The eigenfunctions are, aside from normalizing factorฯ =
๐๐(๐, ๐)๐โ1
2โ ๐ฝ๐+1
2
(๐๐). The permitted energies are determined by the condition
๐ฝ๐+
1
2
(๐๐) = 0 where ๐ is the radius of the enclosure. For any integer๐, there will be
an infinie set of roots of ๐ฝ๐+
1
2
which we shall label ๐๐๐, ๐ = 1, 2, โฆ ,โ. the permitted
kโs are therefore
๐๐๐ =๐๐๐๐
And hence E, which will also depend on two indices (quantum numbers) is given
by
๐ธ๐๐ =ั2
2๐๐2(๐๐๐)
2
The simple model treated here is called the โinfinite potential holeโ. It form the
basisfor many nuclear quantum mechanical calculations and is one of the favored
starting points for considerations leading to nuclear shell structure. * A solution of
the potential-hole problem with finite walls ฯฎ requires the use of bessel functions
inside, Hankel functions outside the hole. The sequence of the energy values is
unaltered, but all levels are depressed9
c. No enclosure. When the particle is allowed to exist anywhere in
space, the former boundary conditions need not be applied. The simplest way to
treat this case is to return to case (a) and permit ๐1, ๐2, ๐๐๐ ๐3 to become infinite.
Let us first consider the eigenvalues. The lattice of points will condense as the๐โs
increase, unti finally it forms a continuum; the energy states (length of the
connecting lines squared) will also move closer and closer together untill finally all
(positives) energies are permitted. A similar effect may be brought about by
increasing the mass of the particle, as a glance et eq. (36) will show. Quantum
mechanics indicates no quantization of the energy for particles which are not
restricted in their motion, or which have an infinite mass.
What happens to the ฯ-function, (35), as the ๐โฒ๐ increase? Clearly, the
normalizing constant c tends to zero, causing ฯ also to vanish. The meaning of this
is quite simple: As the space in which the mass point moves increases indefinitely,
the chance of finding it at a given point, |ฯ(x, y, z)|10
* Mayer, M.G. and Jensen, J.H.D., โElementary Theory of Nuclear Shell Structure.,โ John Wiley
and Sons, Inc., New York, 1955.
ฯฎ Margenau, H., Phys. Rev. 46, 613 (1934) 10 Another procedure is dicussed for instance in Sommerfeld, A., โAtombau und Spektrallinen,โ
Vol. II.
Approaches zero. The failure of the normalization rule is therefore not merely
a mathematical phenomenon, but physically reasonable. To circumvent it, several
procedures may be employed. One is to suppose that there is an infinite number of
particles in all space, N per unit volume, and accordingly to putโซ|ฯ|2 ๐๐, taken over
a unit of volume, equal to N. This leaves c finite.11
When there are no boundary conditions the ฯ-function need not be written as a
product of sines. In fact in the absence of an enclosure sine, cosine and exponential
functions are equally acceptable. Hence we may, if we desire, write
ฯ๐ธ = ๐(๐)๐๐๐โ๐ , ๐ธ =
ั2
2๐๐2
Using the notation explained in connection with eq. (38) of the Chapter 7.
Problem. Calculate eigenfunctions and eigenvalues of a free particle enclosed
in a cylinder of ๐ radius and length๐, obtaining
ฯ = c๐๐[(๐๐๐)๐ง+๐๐]๐ฝ๐(๐ผ๐)
where โ ๐ผ is a root of ๐ฝ๐,
๐ธ๐ =ั2
2๐(๐2๐2
๐2+โ2 ๐ผ2)
11.10. One-Dimensional Barrier Problems.- For a one-dimensional problem the
Schrแฝdinger equation is
๐2ฯ
๐๐ฅ2+2๐
ั2[๐ธ โ ๐(๐ฅ)]ฯ = 0
Let us take V to be the step function given by the solid line in Fig. 1, that is: ๐ =
0 ๐๐ ๐ฅ < 0, ๐ = ๐ = ๐๐๐๐ ๐ก๐๐๐ก ๐๐ ๐ฅ > 0. The solutions for the two regions are
easily written down:
11 Another procedure is discussed for instance in Sommerfeld. A., โAtombau und Spektrallinien,โ
Vol. II.
ฯ๐ = ๐ด๐๐๐๐๐๐ฅ + ๐ต๐๐
โ๐๐๐๐ฅ, ๐ฅ < 0 (๐๐๐๐ก ๐๐ 0)
ฯ๐ = ๐ด๐๐๐๐๐๐ฅ + ๐ต๐๐
โ๐๐๐๐ฅ, ๐ฅ > 0 (๐๐๐โ๐ก ๐๐ 0)
With
๐๐ =โ2๐๐ธ
ั ๐๐๐ ๐๐ =
โ2๐(๐ธ โ ๐)
ั
But how are they to be joined? The differential equation tells us that ฯโฒโฒ
suffers a finite discontinuity as we pass across the discontinuity in V. The increases
in ฯโฒ in crossing the origin will be
lim๐โ0
โซ ฯโฒโฒ๐๐ฅ = ๐
โ๐
lim๐โ0
๐( ฯ๐โฒโฒ + ฯ๐
โฒโฒ) = 0
Hence ฯโฒ (and a fortioriฯ) remains continuous at the origin. The constants
A and B must therefore be fixed by requiring
ฯ๐(0) = ฯ๐(0); ฯ๐โฒ(0) = ฯ๐โฒ(0)
In addition to these two we have an equation expressing normalization, three
relations in all. However, there are four constant (๐ด๐ , ๐ด๐ , ๐ต๐, ๐ต๐) to be determined.
The mathematical situation is therefore such that one of them may be chosen at will.
Let us then put ๐ต๐ equal to zero. The physical meaning of this will at once be clear.
On applying the continuity conditions we have
A๐ + B๐ = ๐ด๐; ๐๐(A๐ โ B๐) = ๐๐๐ด๐
Whence
Bl =๐๐ โ ๐๐๐๐ + ๐๐
๐ด๐
The coefficient A and B have a simple significance. Let us analyze from our
fundamental point of view a state function of the formฯ = ๐ด๐๐๐๐๐ฅ + ๐ต๐โ๐๐๐๐ฅ. In
view of the third postulate (eq. 14โ) it represents a mean momentum
๏ฟฝ๏ฟฝ = โ๐ ัโซฯ โ ฯโฒdx
โซฯ โ ฯdx
And a mean square momentum
๐2 = โั2โซฯ โ ฯโฒโฒdx
โซฯ โ ฯdx
We have intentionally left the limits of integration indefinite. In evaluating
the integrals occuring here we assume that the range of integration is very much
larger than the wave length of the particles, 2๐/๐. The integral over the last two
terms of ฯ โ ฯ = A โ A + B โ B + AB โ ๐2๐๐๐ฅ + ๐ด โ ๐ต๐โ2๐๐๐ฅ will then vanish, and
โซฯ โ ฯdx = (|๐ด|2 + |๐ต|2)๐
๐ being the range of integration. By the similar procedure,
โซฯ โ ฯโฒdx = ik(|๐ด|2 + |๐ต|2)๐ ๐๐๐ โซฯ โ ฯโฒโฒdx = โik(|๐ด|2 + |๐ต|2)๐
Hence
๏ฟฝ๏ฟฝ = ๐ั|๐ด|2 โ |๐ต|2
|๐ด|2 + |๐ต|2, ๐คโ๐๐๐ ๐2 = ๐2ั2
It will also be observed that ฯ is an ๐๐๐๐๐๐ ๐ก๐๐ก๐๐ of the operator (โ๐ั๐
๐๐ฅ)2, but
not of โ๐ั๐
๐๐ฅ.
Translated into particle language, this state of affairs must be expressed as
follows. Since all particles have a root mean square momentum along x is smaller
than๐ั, some of them must be traveling to the right, others to the left, with
momentum ๐ั. If a fraction ๐ผ travels to the right and ๐ฝ to the left,
(๐ผ โ ๐ฝ)๐ั = ๏ฟฝ๏ฟฝ, (๐ผ + ๐ฝ)๐ั = โ๐2
Whence
๐ฝ
๐ผ= (1 โ
๏ฟฝ๏ฟฝ
๐ั)/(1 +
๏ฟฝ๏ฟฝ
๐ั) =
|๐ต|2
|๐ด|2
In our problem ๐ฝ
๐ผ is the reflection coefficient of the barrier of potential energy V. In
view of eq. (37) it is given by
๐ =|๐๐ โ ๐๐|
2
|๐๐ + ๐๐|2
Two case of interest may be distinguished, (a) E < V, (b) E > V. In classical
mechanics, a particle would certainly be reflected in case a, (R=1), certainly
transmitted in case b, (R=0). The matter is not quite simple in quantum mechanics.
In case a, ๐๐ is real but ๐๐ is imaginary. R is thus always 1 in agreement with the
classical prediction. But in case b both ๐๐ and ๐๐ are real, and R < 1 but not zero.
Hence every potential barrier reflect, particles, even though classically one would
expect them to be only retarded.
Before leaving this matter, we must justify the procedure of setting ๐ต๐ equal
to zero. This is now. This is now seen to mean omission of a beam of particles
travelling to the left in the region to the right of the origin. Had such a beam been
included, the physical condition corresponding to ฯ would have implied the
incidence of two beams of particls upon the origin, one from the left and one from
the right. In that case, ๐ฝ
๐ผ is not the reflection coefficient of the barrier. The ฯ-
function we have chosen permits that interpretation, for it corresponds to one beam
incident from the left, one reflected and one transmitted beam.
Problem. Prove that ๏ฟฝ๏ฟฝ is the same whether it is computed to the left or to the right
of the origin. [use condition (37)].
A study of more complicated barriers, such as that depicted in fig. 2, reveals a new
and striking feature: the โtunnel effectโ. The energy E of the incident particles is
assumed to be greater than ๐1and๐3, but smaller than ๐2, so that from the classical
point of view every particle would certainly be reflected. If we define
๐12 =
2๐
ั2(๐ธ โ ๐1); ๐
2 = โ๐22 =
2๐
ั2(๐2 โ ๐ธ); ๐3
2 =2๐
ั2(๐ธ โ ๐3)
The ฯ-function for the three regios are
ฯ1 = ๐ด1๐๐๐1๐ฅ + ๐ต1๐
๐๐1๐ฅ, ๐ฅ < 0
ฯ2 = ๐ด2๐๐๐ฅ + ๐ต2๐
โ๐๐ฅ, 0 โค ๐ฅ โค ๐ผ
ฯ3 = ๐ด3๐๐๐3๐ฅ, ๐ฅ > ๐ผ
The continuity conditions for ฯ and ฯโฒ at both x=0 and c= ๐ผ are seen to be:
๐ด1 + ๐ต1 = ๐ด2 + ๐ต2
๐๐1(๐ด1 โ ๐ต1 = ๐(๐ด2 โ ๐ต2)
๐ด2๐๐๐ผ + ๐ต2๐
โ๐๐ผ = ๐ด3๐๐โ๐3๐ผ
๐(๐ด2๐๐๐ผ โ ๐ต2๐
โ๐๐ผ) = ๐๐3๐ด3๐๐๐3๐ผ
From these , ๐ต1, ๐ด2, and ๐ต2 may be eliminated. When this is done we obtain the
relation
๐ด1 =1
2๐ด3๐
๐๐3๐ผ {(1 +๐3๐1) cosh ๐๐ผ + ๐ (
๐
๐1โ๐3๐) sinh ๐๐ผ}
An argument similar to that which led us to identify the reflection coefficient R with
|๐ต|2/|๐ด|2, shows the transmissions coefficient of the present barrier to be
๐ =|๐ด3|
2๐3|๐ด1|2๐1
This may be computed from (38). In doing so we assume that ๐๐ผ โซ 1 so that both
cosh ๐๐ผ and sinh ๐๐ผ become 1
2๐๐๐ผ. Then
๐ = 16๐1๐3
(๐1 + ๐3)2 + (๐ โ๐1๐3๐)2โ ๐โ2๐๐ผ
As the width of the barrier increases, the factor ๐โ2๐๐ผ(sometimes called the
โtransparency factorโ) rapidly diminishes.
The surprising fact is that particles are able to โtunnelโ through the barrier although
their kinetic energy is not great enough to allow them to pass it. Clasically speaking,
the kinetic energy of a particle would be
Negative while it is in region 2. Quantum mechanically, this statement is devoid of
meaning, since it is improper to compute ๐ธ โ ๐ for this region alone.12
Fig. 3 gives a qualitative plot of the (real part of the) ฯ-function in the three regions
here considered. It is seen that the barrier attenuates the wave coming from the left,
permitting a fraction of its amplitude to pass out at๐ผ. The situation is quite
analogous t the passage of a wave through an absorbing layer.
11.11. Simple Harmonic Oscillator.-The potential energy, usually expressed in the
form 1
2๐๐ฅ2, is
1
2๐๐2๐ฅ2 when written in terms of the mass m and the classical
frequency ๐ = 2๐๐ฃ of the oscillator. The meaning of ๐ is
12 More complicated barriers are discussed by Condon, E. U., Rev. Mod. Phys. 3, 43 (1931),
Eckart, C., Phys. Rev. 35, 1303 (1930)
Simply that of a parameter appearing in V; we must no longer expect the oscillator
to go back and forth ๐/2๐ times per second. The Schrแฝdinger equation is
๐2๐
๐๐ฅ2+ (๐ โ ๐ฝ2๐ฅ2)๐ = 0
If we use the abbreviations
๐ =2๐๐ธ
ั2, ๐ฝ =
๐๐
ั
The substitution ๐ = โ๐ฝ๐ฅ reduces (39) to the form of the differential equation for
โHermitโs orthogonal functionsโ
๐2๐
๐๐2+ [1 โ ๐2 + (
๐
๐ฝโ 1)]๐ = 0
Which was studied in chapter 2 (cf. Eq. 2-66). It was there found that its solution is
of the form ๐โ๐2/2๐ป(๐),๐ป(๐) being solution of Hermiteโs equation (2-62). Now
๐ป(๐) is a polynominal if the quantity๐ผ, which corresponds to the present 1
2(๐
๐ฝโ 1),
as an integer. Unless this is true, H is a superposition of the infinite sequence (2-
63) and (2-64). But both of these approach infinity like๐๐2, as closer inspection will
show. If they are multiplied by๐โ๐2/2, they will not yield a ฯ-function which has
an integrable square between the limits โโ ๐๐๐ +โ, which we are here assuming
to exist. Hence ๐ป(๐) must be chosen in its polynominal form, ๐ป๐(๐). Also,
1
2(๐/๐ฝ โ 1 ) = ๐, and this leads to
๐ธ๐ = (๐ +1
2) ั๐ = (๐ +
1
2) โ๐ฃ
๐๐ = ๐๐โ(๐ฝ/2)๐ฅ2๐ป๐(โ๐ฝ๐ฅ)
If theoscillator has three degrees of freedom, the Schrแฝdinger equuation is
โ2๐ + (๐ โ ๐ฝ2๐2)๐ = 0
When the same abbreviations as above are used. The method of separation of
variables (chapter 7) which involves the substitution of ๐(๐ฅ), ๐(๐ฆ), ๐(๐ง) for ฯ at
once reduces this partial differential equation to three ordinary ones
๐โฒโฒ + (๐1 โ ๐ฝ2๐ฅ2)๐ = 0, ๐โฒโฒ + (๐2 โ ๐ฝ
2๐ฅ2)๐ = 0
๐โฒโฒ + (๐3 โ ๐ฝ2๐ฅ2)๐ = 0
Provide that ๐1+๐2 + ๐3 = ๐. Each of these has a solution of the form (41), so that
๐๐1๐2๐3=๐๐
โ(๐ฝ/2)๐2๐ป๐1(โ๐ฝ๐ฅ)โ๐ป๐2(โ๐ฝ๐ฆ)โ๐ป๐3(โ๐ฝ๐ง)
๐ธ๐1๐2๐2 = (๐1 + ๐2 + ๐3 +3
2)ั๐
The orthogonality of the functions (41) has been proved in eq. 3-92. From this
formula, the normalizing constant c may also be computed. For if
โซ ๐2๐โ๐ฝ๐ฅ2
โ
โโ
๐ป๐2(โ๐ฝ๐ฅ)๐๐ฅ = ๐ฝโ
12โซ ๐2๐โ๐
2โ
โโ
๐ป๐2(๐)๐๐
= ๐2 โ 2๐๐!โ๐
๐ฝ= 1
Then
๐ = (๐ฝ
๐)1/4
(๐! 2๐)โ1/2
A similar computation, which involves three integrations, yields for the constant c
of eq. (42) the value
(๐ฝ
๐)1/4
(๐1! ๐2! ๐3! 2๐1+๐2+๐3)โ1/2
Further mathematical details concerning the functions here encountered, as well as
table of the ๐ป๐-polynomials, are given in sec. 3.10.
Problem. The treatment above implide that the 3- dimensional oscillator was
istropic ; bound with equal force in all directions. Calculate eigenvalues and
eigenfunctions for an anisotropic oscillator with potential energy
๐ =1
2๐(๐1
2๐ฅ2 + ๐22๐ฅ2 + ๐3
2๐ฅ2)
11.12. Rigid Rotator, Eigenvalues Eigenfunctions of ๐ฟ2 . โA rigid rotator is a pair
of point masses held together by a rigid, inflexible and inextensible (massless)
bond. A diatomic molecule is a fiar approximation to a rigid rotator. Before
attempting to solve the Schrแฝdinger equation for such a system it is well to digress
briefly and considen the eigenvalue equation for an operator which so far we have
not introduced, but which is easily constructed. We have seen that the operators
corresponding to the components of angular momentum of a particle are
๐ฟ๐ฅ = โ๐ั (๐ฆ๐
๐๐งโ ๐ง
๐
๐๐ง)
๐ฟ๐ฆ = โ๐ั (๐ง๐
๐๐ฅโ ๐ฅ
๐
๐๐ง)
๐ฟ๐ง = โ๐ั (๐ฅ๐
๐๐ฆโ ๐ฆ
๐
๐๐ฅ)
From these, we wish to construct the operator
๐ฟ2 = ๐ฟ๐ฅ2 + ๐ฟ๐ฆ
2 + ๐ฟ๐ง2
It is advantageous to do this in polar (spherical) coordinates*13 putting ๐ฅ =
๐ sin ๐ cos๐, ๐ฆ = ๐ sin ๐ sin๐ ๐ง = ๐ cos ๐, ๐ค๐ โ๐๐ฃ๐
๐
๐๐ฅ= ๐ ๐๐๐๐๐๐ ๐
๐
๐๐+1
๐๐๐๐ ๐๐๐๐ ๐
๐
๐๐โ1
๐
๐ ๐๐๐
๐ ๐๐๐
๐
๐๐
๐
๐๐ฆ= ๐ ๐๐๐๐ ๐๐๐
๐
๐๐+1
๐๐๐๐ ๐๐ ๐๐๐
๐
๐๐+1
๐
๐๐๐ ๐
๐ ๐๐๐
๐
๐๐
๐
๐๐ง= ๐๐๐ ๐
๐
๐๐โ1
๐๐ ๐๐๐
๐
๐๐
When these results are introduced in (44) and (45) is formed, there results
๐ฟ2 = โั2 {1
sin ๐
๐
๐๐(sin ๐
๐
๐๐) +
1
๐ ๐๐2๐
๐2
๐๐2}
The observable value which the square of the angular momentum may assume are
the eigenvalues p of the equation
๐ฟ2๐ = ๐๐
This equatioan is easily solved by the metode of separation of variables (ef. Chapter
7 ). Clearly, ฯ is a function of ๐ and ๐. Put ฯ = ำจ (๐) . ๐ท(๐) into (47). This equation
will then break up into two ordinary equations (the process is analogous to the
constructions of eqs. 7-42a and 7-42b):
ั2 {1
sin ๐
๐
๐๐(sin ๐ำจโฒ) โ
๐2
๐ ๐๐2๐ำจ +
๐
ั2ำจ} = 0
๐ทโฒโฒ = โ๐2๐ท
13 See also the problem at the end of this section
This equation therefore has the solution ๐ท= const.๐๐๐๐, m an integar. The equation
for associated Legendre functions, (eq. 7.45b), except that the constant ๐(๐ + 1)
appearing there is here replaced by๐/ั2. The solution previously obtained is
ำจ = ๐ ๐๐๐๐๐๐
๐(๐๐๐ ๐)๐๐๐(cos ๐)
Now the legendre function ๐๐ was shown to behave singulary at cos ๐ = ยฑ1 unless
๐ is an integer, in fact it would countain unlimited powers of ๐ฅ(= cos ๐). The same
would be true for ำจ if ๐ were arbitrary. But in that caseโซฯ โ ฯdr, which contains
the factor
โซ ำจ2 sin ๐๐๐ = โซ ำจ2๐๐ฅ1
โ1
๐
0
Would centainly not exist. We conclude, therefore, that ๐ must be an integar, and
that the eigenfunction of ๐ฟ2 are
๐ = ๐(๐ + 1)ั2
On other hand, the eigenfunctions of ๐ฟ2 are of the form
๐ ๐๐๐๐๐๐
๐๐๐๐๐(๐๐๐ ๐)๐
๐๐๐ = ๐๐๐(cos ๐) ๐๐๐๐
In the notation adopted in chapter 3 (ef. Eq. 3-43). Since the eigenvalue ๐ does not
depend on ๐ but only on, functions like (48) with different ๐ will satisfy eq. (47)
. The most general solution of that equation is therefore, 14
14 We define here and elsewhere: ๐๐
โ๐ = ๐๐๐, ๐๐ ๐๐ (3 โ 62)
๐ = โ ๐๐๐๐๐(cos ๐)๐๐๐๐
๐
๐โโ๐
In chapter 7 this function has already been encountered; it is called a spherical
harmonic and denoted by ๐๐(๐, ๐) (ef. Eq. 7-43 et seq.). Hence
ฯ = ๐๐(๐, ๐)
Since ๐๐ = sin ๐๐๐๐๐, normalization requires that
โซ ๐๐๐ โ ๐ = 1๐
0
Whan (49) is inserted the integral becomes
2๐โ๐๐ โ ๐๐โซ [๐๐๐(๐ฅ)]2๐๐ฅ
1
โ1
=4๐
2๐ + 1โ|๐๐|
2(๐ + ๐)!
(๐ โ ๐)!
๐
โ๐
๐
โ๐
(cf. Eq. 3-62). Hence, for normalization, the constants ๐๐ appearing in (49) must
satisfy the relation
โ |๐๐|2
๐
๐=โ๐
(๐ + ๐)!
(๐ โ ๐)!=2๐ + 1
4๐
And are otherwise arbitrary.
We are now ready to return to the problem of the rigid rotator. In the first place we
shall assume it proper to replace it by a single mass, rigidly tied to center of rotation,
and having the same moment of inertia as the original system. The condination upon
the state function in accord with this assumption-aside simple ๐ = ๐,a constant. The
best procedure is therefore to write down the Schrรdinger equation for a particle
moving in three dimensions, and then to put ๐ = ๐, ๐ ฯ /dr = 0. This requares the
use of polar (spherical) coordinates. The potential energy, in this case, is cleary
constant and may be taken to be zero.
Schrรdingerโs equation reads15 (ef. Chapter 5 for transformation of โ2)
1
๐2๐
๐๐(๐2
๐๐
๐๐) +
1
๐2 sin ๐
๐
๐๐(sin ๐
๐๐
๐๐) +
1
๐2๐ ๐๐2๐
๐2๐
๐๐2+2๐
ั2๐ธ๐ = 0
When ๐ is put equal to a the first term on the left vanisehes, and the remainder
becomes very similar to๐ฟ2๐. Indeed if we introduce, a new operator โ2 definde
as(1/ั2)๐ฟ2, eq. (51) may be written
โ2๐ =2๐๐2
ั2๐ธ๐
But the eigenvalues of a โ2 are obviously๐(๐ + 1), and its eigenfunctions are the
same as those of ๐ฟ2. The constant(2๐๐2/ั2)๐ธ, must be identified with๐(๐ + 1).
Hence the eigenvalues and eigenfunctions are
๐ธ =ั2
2๐๐2๐(๐ + 1); ๐๐,๐ = ๐1(๐, ๐)
Problem. Show by vector algebra that
โโ2 = (๐ ๐ฅ โ)2 = โ๐2โ2 + 2๐๐
๐๐+ ๐2
๐2
๐๐2
Hint: Note that(๐ ๐ฅ โ)2 = ๐ โ [โ ๐ฅ (๐ ๐ฅ โ)]. Then use (4-26) forโ ๐ฅ ๐ ๐ฅ ๐.
11.13. Motion in a Central Field. โBy central field is meant a field of force in which
the potential energy is a function of r only; V is independent of ๐ and ๐. The
isotropic three-dimentional oscillator treated in sec. 11 is an example of motion in
a central field. Another is the motion of a particle in a coulomb field. It is to this
last example, an electron attracted by a positive point charge (hydrogen atom), that
we shall chiefly direct our attention. But before considering this spesific case a few
general features of the central field problem will be exposed.
15 To avoid confusion, we write M for the electron mass in this section, returning to the symbol m
in the next.
It is now clear that the Laplacian, โ2, in spherical polar coordinates has the form.
โ2=1
๐2{๐
๐๐(๐2
๐
๐๐) + โ2}
Where โ2 is given by (by) divided by โั2. The eigenvalues of the โ2 are ๐(๐ + 1).
The Schrแฝdinger equation therefore reads
1
๐2{๐
๐๐(๐2
๐๐
๐๐) โ โ2๐} +
2๐
ั2[๐ธ โ ๐(๐)]๐ = 0
We write ๐ as a product of a function R(r) and another.๐ด(๐, ๐), which depends
only on the angles. The operator โ2 acts only on A. Eq. (55), after multiplication
by ๐2 and subsequent division by๐ โ ๐ด, has the form
๐๐๐(๐2
๐๐ ๐๐)
๐ +2๐๐2
ั2[๐ธ โ ๐(๐)] =
โ2๐ด
๐ด
The left-hand side of this equation is a function of ๐ alone, the right a function of
๐and๐. By the argument which is familiar from chapter 7, each side must be a
constant, say๐. Thus
โ2๐ด = ๐๐ด
But this is simply the eigenvalue equation forโ2. We see, then, that
๐ = ๐(๐ + 1), ๐๐๐ ๐ด = ๐1(๐, ๐)
The left-hand side of (56) becomes
๐
๐๐[๐2
๐๐
๐] +
2๐๐2
ั2[๐ธ โ ๐(๐) โ
๐(๐ + 1)
2๐๐2ั2] ๐ = 0
And the substitution ๐(๐) = ๐๐ (๐) reduce this to
๐โฒโฒ +2๐
ั2[๐ธ โ ๐(๐) โ
๐(๐ + 1)ั2
2๐๐2]๐ = 0
The depelovement so far has been totally independent of the form of V, except in
assuming it to be a function of ๐ alone. The result obtained are therefore valid for
any central field. Summarizing them, we may say:
The energy states of a particle in a central field are always of the form
๐ =1
๐๐๐(๐)๐๐(๐, ๐)
And the function ๐๐ is determined by eq. (57b). It was necessary to add a subscript
๐ ๐ก๐ ๐ because the differential equation contains ๐ as a parameter. The energies E
are obtained solely from eq. (57b)
That equation looks very much like the one-dimensional Schrแฝdinger equation,
๐โฒโฒ +2๐
ั2[๐ธ โ ๐(๐ฅ)]๐ = 0
But with the term ๐(๐ + 1)ั2/2๐๐2 added to the normal potential
energy. What is the meaning of that term? In classical mechanics, the energy of a
particle moving in three dimensions differs from that of a one-dimensional particle
by the kinetic energy of a rotation1
2๐๐2๐2. This is precisely the quantity๐(๐ +
1)ั2/2๐๐2, for we have seen that ๐(๐ + 1)ั2 is the certain value of the square of
the angular momentum for the state๐๐, in classical language(๐๐2๐)2, which when
divided by2๐๐2, gives exactly the kinetic energy of rotation.
There is, however, one further difference between (57b) and (58). The
fundamental range of ๐ in (57b) starts at ๐ = 0 and is limited to positive values,
whereas the range of ๐ฅ in (58) may include negative values. This fact often has a
more important effect on the eigenvalues than the addition of the terms just
mentioned.
Let us now solve eq. (57b), assuming a Coulomb field, e.g., ๐(๐) = โ๐2/๐.
The energies E will then be the energy levels of the hydrogen atom16. For
sufficiently large ๐ the solution is determined by
๐โฒโฒ โ (๐ผ
2)2
๐ = 0
Provided we define
(๐ผ
2)2
= โ2๐๐ธ
ั2
The solution of (59) is๐โ = ๐1๐(๐ผ/2)๐ + ๐2๐
(๐ผ/2)๐, and this represents the behavior
of the correct๐ ๐๐ก โ. Let us first suppose that ๐ผ is real, which means that the energy
of the particle is negative. U will then certainly not have an integrable square (note
that the radial integral has then the form โซ ๐ 2๐2๐๐ = โซ๐2๐๐โ
0 if the coefficient ๐1
fails to vanish. But we cannot simply put it equal to zero because we have boundary
conditions to full fill! Without going further in our analysis at the moment we
expect, therefore, that only special values of ๐ผ will produce accetable solutions
when ๐ผ is real. If the total energy of the particle is negative (classically speaking,
the particle is bound to the attracting center), the energy is expected to be quantized.
The following analysis will bear this out.
If ๐ผ is imaginary, which means that E is positive, ๐โ shows sinusoidal behaviot. It
has, in fact, the typical form of the state function for a free particle, and the failure
of normalization occurs in the milder manner which we have previously found
associated with the presence of a continuous spectrum of eigenvalues. There is
indeed no way of choosing ๐1 ๐๐ ๐2
Or ๐ผ which would make one ๐โ more acceptable than another. We conclude that,
when E is positive, the energy spectrum is continuous.
16 If ๐2 is replaced by ๐๐2, ๐ = 2 represents ionized helium, Z=3 doubly ionized lithium, etc.
From the point of view of classical physics this result is welcome, for when E is
positive the particle is ionized and moves through the space, its energy being
unrestricted.
We now discuss the bound states in a more rigorous manner. Put๐ธ = โ๐, so that
๐ is positive. Our interest will now return to eq. (57a) which forms a more suitable
basis for the present discussion. Let๐ = ๐ฅ/๐ผ, where ๐ผ is defined by (60). Eq. (57a)
then reads, after some cancellation,
๐ฅ๐2๐
๐๐ฅ2+ 2
๐๐
๐๐ฅ+ [2๐๐2
ั2๐ผโ๐ฅ
4โ๐(๐ + 1)
๐ฅ] ๐ = 0
But this is precisely the differential equation for associated Laguere functions,
which was studied in chapter 2 (cf.eq.71). for our immadiatepurpose we shall write
that equation with n* in place of n, since otherwise our nation would be in conflict
with physical convention. To summarize the result of sec. 2.16:
The equation
๐ฅ๐ฆโฒโฒ + 2๐ฆโฒ + [๐ โ โ๐ โ 1
2โ๐ฅ
4โ๐2 โ 1
4๐ฅ] ๐ฆ = 0
Has solution possessing an integrable square17 of the form
๐ฆ = ๐โ๐ฅ/2๐ฅ(๐โ1)/2๐ฟ๐โ๐ (๐ฅ)
Provided n* and k are positive integers. Moreover, ๐ โ โ๐ โฅ 0 sinc otherwise ๐ฟ๐โ๐
would vanish.
On comparing (61) and (62) we find, in the first place, that(๐2 โ 1)/4 = ๐(๐ + 1),
hence
๐ = 2๐ + 1
Secondly,
17 The reader should convince himself of this fact by going back to see sec. 2.16.
๐ โ โ๐ โ 1
2= ๐ โ โ๐ =
2๐๐2
ั2๐ผ
When the value of ๐ผ is inserted here and the relation is solved for W, we find
๐ =1
2
๐๐4
(๐ โ โ๐)2ั2
Because of the conditions on n* and k, the quantity ๐ โ โ๐ cannot be zero. It is
usually denoted by n and called the total quantm number (after the role it played in
the Bohr theory). Our conclusion, then, is this: The energy states of the hydrogen
atom are
๐๐ = โ๐ธ๐ =1
2 2๐๐2
๐2ั2
And the corresponding eigenfnctions are, in accordance with (63),
๐ ๐,๐ = ๐๐,๐๐โ๐ฅ2๐ฅ๐๐ฟ๐+๐
2๐+1(๐ฅ)
The variable x being defined by
๐ฅ = ๐ผ๐ =โ8๐๐
ั๐ =
2๐๐2
๐ั2๐
In the Bohr theory of hydrogen, the first orbit has a radius
๐0 =ั2
๐๐2= 0,53 ๐ฅ 10โ8 ๐๐
It sometimes convient to express x in terms of it. Thus๐ผ = 2/๐๐0, and
๐ฅ =2
๐ ๐
๐0
It is to be noticed that x represents a different variable for each energy state; the
quantum number n determining W appears as a scale factor in the dimensionless
variable x.
Some integrals involving๐ ๐,๐, which occur frequently in physical and chemical
problem, have been evaluated in sec. 3.11., see also the example at the end of sec.
3.11, which is of interest in this connection.
For later use, we write down in explicit form the state function for the normal
hydrogen atom. It is
๐ 1,0 = ๐1,0๐โ๐/๐0๐ฟ1
1 = 2๐0โ3/2๐โ๐/๐0
For this state ๐1 = ๐๐๐๐ ๐ก๐๐๐ก = (4๐)โ1/2 when the function is normalized. Hence
the total ground state function is
๐0 = (๐๐03)โ1/2๐โ๐/๐0
ฮจ-functions for the higher states are listed in explicit form in Pauling and Wilson18.
When the charge on the nucleus is not e but๐๐, ๐0 must be replaced by๐0/๐, so that
๐0 = (๐3
๐๐03)
1/2
๐โ๐๐/๐0
Problem a., using the result of chapter 3, show that the normalizing factor in (65)
is
๐๐,๐ = (2
๐๐0)3/2
{(๐ โ ๐ โ 1)!
2๐[(๐ + ๐)!]3}
1/2
Problem b. Work out the problem of the isotropic oscillator using spherical
coordinates, and show that the result agree with those obtained in (42) and (43).
11.14 Symmetrical Top. โIn dealing with the problem f the rotating rigid body
attention must be given to the kinetic energy operator. To obtain it we first observe
18 Pauling, L., and Wilson, E. B., Jr., โIntroduction to Quantum Mechanicsโ McGraw-Hill Book
Co., 1935.
that its form in rectangular coordinates, for the n particle problem (cf. Sec. 11.31)
is ๐๐ = โ ั2
2โ
โ๐2
๐๐๐๐
๐โ1
The position of a rigid body is best expressed in terms of the Eulerian angles,
introduced in sec. 9.5. it was there shown that the classical kinetic energy is given
by
๐๐ =1
2โ๐๐(๐ฅ๐
2 + ๐ฆ๐2 + ๐ง๐
2)
๐
๐=1
=1
2๐ด๐ฝ2 +
1
2๐ด๐ผ2๐ ๐๐2๐ฝ +
1
2๐ถ(๐พ + ๐ผ cos ๐ฝ)2
Let us define a line element constructed from the Cartesian coordinates
๐๐ = โ๐๐๐ฅ๐ , ฦ๐ = โ๐๐๐ฆ๐ , ๐๐ = โ๐๐๐ง๐
As follows:
๐๐ 2 =โ(๐
๐
๐=1
๐๐2 + ๐ฦ๐
2 + ๐๐๐2)
This is clearly identical with2๐๐๐๐ก2. From the form of ๐๐ in Eulerian coordinates it
is seen that ๐๐ 2 in these coordinates is given by
๐๐ 2 = ๐ด๐๐ฝ2 + ๐ด ๐ ๐๐2๐ฝ๐๐ผ2 + ๐ถ(๐๐พ + cos ๐ฝ๐๐ผ)2
Now the quantum mechanical form of T is the Laplacian operator corresponding to
the line element๐๐ 2, multiplied by โั2/2. The problem is therefore to transform
the Laplacian operator from a set of coordinates in terms of which the line element
is given by (68), to a new set in terms of which the lines element is (69). This
problem has been discussed in sec. 5.17. if
๐๐ 2 =โ๐๐,๐๐๐๐๐๐๐๐,๐
Then
โ๐2๐ =
1
โ๐โ
๐
๐๐๐[โ๐ ๐๐,๐
๐
๐๐๐๐]
๐,๐
On indentifying the ๐๐,๐ from (69) we find (putting ๐1 = ๐ฝ, ๐2 = ๐ผ, ๐3 = ๐พ)
(๐๐,๐) = (
๐ด 0 00 ๐ด๐ ๐๐2๐ฝ + ๐ถ๐๐๐ 2๐ฝ ๐ถ cos ๐ฝ0 ๐ถ cos ๐ฝ ๐ถ
)
And hence
(๐๐,๐) =
(
1
๐ด0 0
01
๐ด๐ ๐๐2๐ฝโ
cos๐ฝ
๐ด๐ ๐๐2๐ฝ
0 โcos ๐ฝ
๐ด๐ ๐๐2๐ฝ
1
๐ถ+๐๐๐ 2๐ฝ
๐ด๐ ๐๐2๐ฝ)
, ๐ = ๐ด2๐ถ ๐ ๐๐2๐ฝ
When these results are substitued in the expression for โ๐2๐ we have
๐๐ = โั2
2โ๐2= โ
ั2
2 sin ๐ฝ{๐๐
๐๐ฝ(๐ ๐๐๐ฝ
๐ด
๐๐
๐๐ฝ) +
๐
๐๐ผ[๐ ๐๐๐ฝ
๐ด๐ ๐๐2๐ฝ
๐๐
๐๐ผโ๐ ๐๐๐ฝ๐๐๐ ๐ฝ
๐ด๐ ๐๐2๐ฝ
๐๐
๐๐พ]
+๐
๐๐พ[โ๐ ๐๐๐ฝ๐๐๐ ๐ฝ
๐ด๐ ๐๐2๐ฝ
๐๐
๐๐ผ+ (
๐ ๐๐๐ฝ
๐ถ+๐ ๐๐๐ฝ๐๐๐ 2๐ฝ
๐ด ๐ ๐๐2๐ฝ)๐๐
๐๐พ]}
= โั2
2๐ด{๐2๐
๐๐ฝ2+ cot ๐ฝ
๐๐
๐๐ฝ+
1
๐ ๐๐2๐ฝ
๐2๐
๐๐ผ2+ (๐๐๐ก2๐ฝ +
๐ด
๐ถ)๐2๐
๐๐พ2
โ2 cos ๐ฝ
๐ ๐๐2๐ฝ
๐2๐
๐๐ผ๐๐พ}
Since the potential energy in this problem is zero, the Schrแฝdinger equation
becomes
๐๐ = ๐ธ๐
It is separable; for if we put
๐ = ๐ข(๐ผ) โ ๐ฃ(๐พ) โ ๐ค(๐ฝ)
The function ๐ข ๐๐๐ ๐ฃ are seen to satisfy equations of the form
๐2๐2๐ข
๐๐ผ2+ ๐1
๐๐ข
๐๐ผ+ ๐0๐ข = 0, ๐2
๐2๐ข
๐๐พ2+ ๐1
๐๐ข
๐๐พ+ ๐0๐ข = 0
Where the coefficient ๐0, ๐1, ๐2 are not function of๐ผ, and the coefficients ๐0, ๐1, ๐2
are not function of ๐พ. Such equations have solutions
๐ข = ๐๐๐๐ผ, ๐ฃ = ๐๐๐๐พ
m and k being roots of algebraic quadratic equations involving the coefficients
๐ ๐๐๐ ๐. However, these need not be solved here, since the condition of single-
valuedness dictates that m and k be integers. We therefore put
๐ข = ๐๐๐๐ผ, ๐ฃ = ๐๐๐พ๐พ
๐,๐พ = 0, ยฑ1, ยฑ2, ๐๐ก๐
The Schrแฝdinger equation now reduces to the following ordinary differential
equation in the independent variable๐ฝ:
๐คโฒโฒ + cot ๐ฝ๐คโฒ
โ[๐2
๐ ๐๐2๐ฝ+ (๐๐๐ก 2๐ฝ +
๐ด
๐ถ)๐พ2 โ 2
cos๐ฝ
๐ ๐๐2๐ฝ๐พ๐ โ
2๐ด
ั2๐ธ]๐ค = 0
The substutions
1
2(1 โ ๐๐๐ ๐ฝ) = ๐ฅ
๐ค(๐ฝ) = ๐ฅ|๐พโ๐|2 (1 โ ๐ฅ)
|๐พ+๐|2 ๐น(๐ฅ)
Which are suggested when this equation is examined for its singularities along the
lines of chapter 2, transform it to
(๐ฅ2 โ ๐ฅ)๐2๐น
๐๐ฅ2+ [(1 + ๐)๐ฅ โ ๐]
๐๐น
๐๐ฅโ ๐(๐ + ๐)๐น = 0
The new parameters being defined as follows:
๐ = 1 + |๐พ โ๐| + |๐พ +๐|
๐ = 1 + |๐พ โ๐|
๐(๐ + ๐) = ๐ด(2๐ธ
ั2โ๐พ2
๐ถ) + ๐พ2 โ
1
4(๐ โ 1)2 โ
1
2(๐ โ 1)
This last relation, when rearranged, may be written
๐ธ =ั2
2๐ด[(๐ +
๐ + 1
2) (๐ +
๐ โ 1
2) + (
๐ด
๐ถโ 1)๐พ2]
Reference to chapter to chapter 2, eq. 56 will show at once that the differential
equation for F is none other than the familiar hypergeometric equation defining the
Jacobi polynomials, provided n is an integer. Unless this condition is satisfied, F
will diverge for๐ฅ = 1, i.e., for๐ฝ = ๐.
Eq. (70) takes a simpler form when we introduce the new quantum number
๐ฝ = ๐ +๐ โ 1
2= ๐ +
1
2|๐พ โ๐| +
1
2|๐พ +๐|
Which is evidently a positive integer or zero. We then obtain
๐ธ =ั2
2๐ด[๐ฝ(๐ฝ + 1) + (
๐ด
๐ถโ 1)๐พ2]
An equation which determines the energy levels of the symmetrical top. Note that
the quantity 1
2|๐พ โ ๐| +
1
2|๐พ + ๐| is equal to the larger of the two integers K and
M; in consequence of this neither |๐พ| nor |๐| can be greater than J.
The energy levels of the spherical top (A=C) are those already obtained in sec. 11.12
(cf. Eq. 11-53).
MATRIX MECHANICS
11.15 General Remarks and procedure. โThe formulation of quantum mechanics
we have given in the foregoing sections was historically precede by Heisenbergโs
matrix theory. The latter, while it appears at first glance to be an altogether different
mathematical structure, strikingly produced the same result as the former. But when
the initial amazement subsided both formulations were recognized as equivalent. In
the present text the Schrแฝdinger -Dirac theory was discussed first because its axioms
seem perhaps less strange, and because its point of view has been more widely
adopted. The terminology of matrix mechanics, however, enjoys great popularity
and its often conducive to clarity of expression.
Its possible, and perhaps pedagogically worthwhile, to derive Heisenbergโs theory
from the postulates of part of this chapter. But when this is done, the impressive
element of uniqueness which attaches to matrix mechanics is completly lost. To
preserve it we proceed to state the basic facts of the theory first, to give an example
of its application, and then to exhibit its relation to the preceding developments. We
can afford to be brief, for when the equivalence of the theories is once established,
no new insight is likely to be gained by deducing former result over again in a
different manner. As before, attention will be limited to what we have called
quantum statics. The principal facts of chapter 10 will be used.
Heisenberg associates with every observable a square Hermitian matrix. As in the
Schrแฝdinger theory, one of the chief concerns of matrix mechanics is the
determination of the measureable values of an observable. Let it be desired to find
the observable values of a quantity H, which, classically, is a function of the
cartesian coordinates ๐๐ and momenta๐๐,๐ป = ๐ป(๐๐. . . ๐๐; ๐๐. . . ๐๐) in our
example we shall specify H to be the energy, but this restriction is not necessary.
Heisenbergโs directions are the find a set of matrices๐ธ๐, ๐ธ๐, ..., ๐ธ๐; ๐ท๐, ๐ท๐, โฆ , ๐ท๐
which (a) satisfy the commutation rules
๐ธ๐๐ธ๐ โ ๐ธ๐๐ธ๐ = ๐ถ; ๐ท๐๐ท๐ โ ๐ท๐๐ท๐ = ๐ถ; ๐ท๐๐ธ๐ โ ๐ธ๐๐ท๐ = โ๐โ๐น๐๐๐ฌ
(11โ71)
where E is the unit matrix; (b) render the matrix
H (๐ธ๐, โฆ๐ธ๐; ๐ท๐โฆ ๐ท๐) diagonal (11โ72)
By H (๐ธ๐โฆ ๐ธ๐; ๐ท๐โฆ ๐ท๐) is meant, of course, the matrix which is the
same funcition of matrices ๐ธ๐โฆ ๐ท๐ that the ordinary function H is of ๐1โฆ ๐๐.
The existence of the matrix H and its uniqueness will be assumed. when such a set
of matrices has been found, the diagonal elements of H will be the measurable
values in question. (It is also true that the squares of the absolute values of the
elements (๐๐)๐๐ are simply related to spectroscopic transition probabilities, as will
be show later; but this does not concern us here). We illustrate the power of the
method by an example.
11.16. Simple Harmonic Oscillator.
The Hamiltonian function is (cf. sec. 11)
H = ๐2
2๐ +
1
2๐๐2๐2
Hence, if P and ๐ธ are matrices,
๐ฏ = (๐ธ,๐ท) = 1
2๐ (๐ท๐ + ๐2๐๐๐ธ2)
The straightforward way of working this problem would be to select a set
of matrices such as, e.g.,
๐๐๐ = ๐ฟ๐,๐โ1, ๐๐๐ = โ๐โ๐๐ฟ๐,๐+1 (11-73)
which satisfy the commutation rule (71)
(๐๐)๐๐ (๐๐)๐๐ = โ๐โ๐๐ฟ๐๐ (11-71a)
as the reader may verify. These โ must then be subjected to a similarity
transformation with some other matrix, say S, until the new matrices
๐ธโฒ = ๐บโ๐ ๐ธ๐บ,๐ทโฒ = ๐บโ๐๐ท๐บ
when substituted in H, make H a diagonal matrix. (Cf. Chap. 10.) This procedure,
however, is usually very cumbersome and is rarely used. The success of the
matrix method depends frequently on fortunate guesses or on specific properties
of the Hamiltonian. In the present instance the following consideration lead most
directly to a solution of the problem.
Supposes that the matrices ๐ท ๐๐๐ ๐ธ, which satisfy (71a) and make H
diagonal, have already been found. Then
๐ป๐๐ = 1
2๐ (๐2 + ๐2๐2๐2)๐๐ = ๐ธ๐๐ฟ๐๐ (11-74)
provider we write ๐ธ๐ for the diagonal elements of H.
Now let ๐จ ๐ท โ ๐๐๐๐ธ
And
๐ฉ ๐ท + ๐๐๐๐ธ.
Then, because of eq. (71a),
๐จ๐ฉ = 2๐๐ฏ +๐๐โ๐ (11โ7 5)
and
๐ฉ๐จ = 2๐๐ฏ โ๐๐โ๐ (11โ76)
Now form ๐จ๐ฉ๐จ from (75) and (76):
A(2mHโ ๐๐โ๐) = (2๐๐ฏ+๐๐โ๐) A
โ๐ด๐๐๐
(๐ธ๐๐ฟ๐๐ โ 1
2๐โ๐ฟ๐๐) = โ(๐ธ๐๐ฟ๐๐ โ
1
2๐โ๐ฟ๐๐)๐ด๐๐
๐
๐ด๐๐ (๐ธ๐ โ ๐ธ๐ + ๐โ) = 0.
Hence ๐ด๐๐ vanishes unless
๐ธ๐ โ ๐ธ๐ = โ๐
Next, form BAB from (75) and (76):
B(2mH + ๐๐โ๐) = (2๐๐ฏโ๐๐โ๐)B
โ๐ต๐๐๐
(๐ธ๐๐ฟ๐๐ + 1
2๐โ๐ฟ๐๐) = โ(๐ธ๐๐ฟ๐๐ โ
1
2๐โ๐ฟ๐๐)๐ต๐๐
๐
๐ต๐๐ (๐ธ๐ โ ๐ธ๐ + ๐โ) = 0 or by changing the subscripts,
๐ต๐๐ (๐ธ๐ โ ๐ธ๐ + ๐โ) = 0
Hence ๐ต๐๐ vanishes unless. Now take a diagonal element of eq. (76):
(BA)๐๐ = 2m (๐ธ๐ โ 1
2โ๐) (11โ77)
But
(๐ต๐ด)๐๐ =โ๐ต๐๐๐ด๐๐ .
๐
Each term in the summation over ๐ vanishes except the one for which ๐ธ๐ = ๐ธ๐ โ
๐โ. Suppose ๐ธ๐ is given. Then either
๐ธ๐ = ๐ธ๐ โ ๐โ Is another eigenvalue, in which case the right side of eq. (77) is
finite. Or there is no eigenvalue which is less than ๐ธ๐ by โ๐. Then the right side of
(77) is zero and
๐ธ๐ = 1
2 โ๐
This must be the lowest eigenvalue. From his analysis we may conclude that the
sequence of eigenvalue is
1
2 โ๐,
3
2 โ๐,
5
2 โ๐ ๐๐ก๐.
In agreement with the results of sec. 11.
11.17. Equivalence of Operator and Matrix Methods.
We first establish a theorem of great importance in quantum mechanics. Consider
a differential, Hermitian operator L of the kind discussed in sec 4, which generates,
through the eigenvalue equation.
๐ฟโ ๐ = ๐๐โ ๐
A complete set of orthonormal functions ๐๐. Whether ๐๐ is a function of one or
many coordinates is unimportant in this connection. If we introduce other operator
M, N which act on the same variables as L we can clearly form two square arrays
of numbers, i.e., matrices, by the rule:
๐๐๐ ๐๐ โ M ๐๐ , ๐๐๐
๐ N
๐d (11-78)
d being the element of configuration space of the variables of . The theorem
asserts that equations which hold between the operators M and N, also hold between
the matrices formed by the rule (78). To prove this it is necessary only to establish
this parallelism for the two fundamental operations, addition and multiplication.
(M+N)๐๐ = ๐๐๐ + ๐๐๐ (11-79)
(๐๐)๐๐ =โ๐๐๐
๐
๐๐๐
(11โ80)
The first of these is at once evident from (78). To prove the second, let us expand
the function ๐๐ in terms of the ๐ themselves:
๐๐๐ โ ๐ผ๐๐๐๐ (11-81)
By the general procedure of finding the expansion coefficients,
๐๐ = ๐ N
๐๐ = ๐๐๐ (11-82)
The left side of (80) is, by definition , ๐MN
๐๐๐. On using (81)
And (82), this becomes (MN) ๐๐ = ๐ M
โ๐๐๐
=โ ๐
๐๐๐๐
=โ๐๐๐๐,
In accord with eq. (80).
If, then, we wish to form matrices satisfying relations like (71) or (71a), we need
only find operators which conform to them, select an orthonormal set of functions
and construct the matrices by means of the rule (78).
Problem a. The operator Q = ๐๐๐ฅ, and P = -โ๐โ๐๐ฅ(d/dx) stisfy
PQ โ QP = ๐โ1
Use the functions ๐ =
1
2 ๐๐๐๐ฅ, k = 0 , ยฑ1, 2, .... to construct the matrices ๐๐๐,
๐๐๐. They we bill found to be identical with those given in eq. (73).
Problem b. Construct the matrices ๐๐๐ and ๐๐๐, using X = x, P = -
๐โ(d/dx), and taking as the orthonormal set the normalized Hermite orthogonal
functions discussed in chapter 3. Note that ๐ and ๐ช can only be 0 or positive.
Ans. ๐๐๐ = โ(๐ + 1)2๐ฝ๐ฟ๐,๐+1 + โ๐/2๐ฝ๐ฟ๐,๐โ1;
๐๐๐ = ๐โ๐ฝ(๐ โ๐)๐๐๐. ฮฒ is defined after eq. (39).
Show that these matrices satisfy (71a)
It is interesting to note here that a Hermitian operator, defined by eq. (15),
generates a Hermitian matrix (cf. Sec. 11.10). for
๐ ๐
๐๐ =
๐๐
๐d
Simply means
๐๐๐ = ๐๐๐
In our present notation.
The success of Heisenbergโs directions is now easily understood. The differential
operators which obey relations analogous to those prescribed for Heisenbergโs
matrices (71) are
๐๐ = ๐๐, ๐๐ = -๐โ๐
๐๐๐
In other words precisely the former, schrรถdinger operators. Suppose we select an
orthonormal set of functions ,๐, belonging to the operator L, and construct
(๐๐)๐๐ = ๐๐๐๐d
The fact that there are also others, like the ones considered in problem a, need not
disturb us here. The Schrรถdinger equation which results when they are used appears
different, to be sure, but reduces to its familiar form when a change of variable is
made.
When these matrices are substituted into the functional from H the result is the same
as if we had at once formed
๐ป๐๐ = โซ๐๐โ๐ป๐๐ ๐๐
As follow from the theorem we have proved. But the only condition under which
this matrix can be diagonal is
๐ป๐๐ = ๐๐๐๐ ๐ก. ๐๐ (11โ83)
That is to say, the ๐โfunctional must be chosen to be eigen functions of the
Hamiltonian H. The problem of making the matrix H diagonal is equivalent to
selecting the proper ๐๐, I. e., to solving the Schr๏ฟฝ๏ฟฝdinger equation. To see that the
diagonal elements of H are the permissible energies ๐ธ๐ of the former theory, we
need only substitute ๐ป๐๐ = ๐ธ๐๐๐ into (83), obtaining
๐ป๐๐ = ๐ธ๐๐ฟ๐๐
It is easy to extend the Heisenberg theory beyond the limits of the present
development. The second postulate, eq. (8), is valid if P is interpreted as a matrix
and ๐๐ as a vector. In the terminology of chapter 10, the ๐๐ are then the
eigenvectors of the matrix P, and the ๐๐ are its eigenvalues. The relation of the
eigenvectors to the state functions is not difficult to see. Suppose we choose a basic
orthonormal set of functions, ๐๐. Expand the eigenfunctions ฯ๐ appearing in the
operator equation
๐๐๐= ๐๐๐๐ (11โ84)
In terms of them, viz.,
๐๐ =โ๐๐๐๐๐๐
.
Now multiply (84) by ๐๐โ and integrate. We find immediately
โ๐๐๐ ๐
๐๐๐ = ๐๐ ๐๐๐
And conclude that the eigenvector ฮจ๐ has as component the coefficients which
appear in its expansion in terms of the basic ๐. More explicitly,
๐๐ =
[ ๐๐1๐๐2๐๐3... ]
The last equation then reads (๐๐๐)๐ = ๐๐(๐๐)๐. If the basic set is identical with the
eigenfunctions of the operator P, eigenvector has only one non-vanishing
component.
Finally, even the third postulate, (14), may be retained in the Heisenberg
theory if its form is suitably changed. We interpret ๐ as a vector ๐ with components
๐๐, the ๐๐, being the coefficients in the expansion of the functions ๐ = in terms
of our basic ๐๐ (๐ without subscript here denotes an arbitrary state function, not
necessarily one of the set ๐๐ ), but ๐โ not as the complex conjugate, but the
associate vector:
ฯโ = ( ๐1โ ๐2
โ ๐3โ . . .)
P represents the matrix ๐๐๐ = โซ๐๐โ ๐๐๐ ๐๐ Eq. (14) must then be modified to
๏ฟฝ๏ฟฝ = ฯโ ๐ฯ
Which reads, when written more explicitly,
๏ฟฝ๏ฟฝ = โ๐๐โ
๐๐
๐๐๐๐๐
When the ๐๐ are taken to be the aigenstates of the operator P, the matrix P becomes
diagonal, and
๏ฟฝ๏ฟฝ =โ๐๐โ
๐
๐๐๐๐
which is the same relation as was found the Schr๏ฟฝ๏ฟฝdinger theory under these
conditions.
Problem. Calculate the integral
โซ ๐๐โ
โโ
๐โ๐2๐ป๐(๐ฅ)๐ป๐(๐ฅ)๐๐ฅ
By the methods of matrix mechanics. Let ๐๐ = ๐๐๐
โ๐2/2 ๐ป๐ (๐), ๐๐ = ๐๐
๐โ๐2/2๐ป๐ (๐) , where ๐๐, ๐๐ are normalizing factors, and note that, aside from
normalizing factors, the integral is the matrix element ( ๐ฅ๐)๐๐. Now ๐ฅ๐๐ is given
by eq. (3-39); this may be used in calculating
(๐๐)๐๐ = โ ๐๐๐๐๐๐๐๐๐โฏ๐๐๐๐,๐,๐,โฏ๐
APPROXIMATION METHODS FOR SOLVING EIGENVALUE
PROBLEMS
11.18. Variantional (Ritz) Method.
In chapter 8 we showed that the differential equation ๐ฟ(๐ข) + ๐๐ค๐ข = (๐๐ขโฒ)โฒ โ
๐๐ข + ๐๐ค๐ข = 0 is the necessary (though not sufficient!) condition upon u if it is to
minimize the integral ษ (๐ข) = โซ(๐๐ขโฒ2 + ๐๐ข2) ๐๐ฅ . Futhermore, it was seen that
ษ (๐) could be transformed (cf. eq. 8โ37) by simple steps to โโซ๐ข๐ฟ(๐ข)d ๐. The
theory in this simple form is applicable to every one-dimensional Schr๏ฟฝ๏ฟฝdinger
equation, for in that case the Hamiltonian operator H = -(โ2/2m) (๐2/๐๐ฅ2) + V (๐ฅ)
is of the form โL if only we identify p with โ2/2m and q with V. Hence we may at
once say that the Schrodinger equation is the necessary condition upon ๐ so that
the integral
โซ๐๐ป๐d
Shall be a minimum. The one-dimensional variation theory may also be applied,
though in a somewhat more cumbersome manner, to every ordinary differential
equation to which the multi-dimensional Schrรถdinger equation gives rise on
separation of variables. It is possible, however, to prove a far more general theorem
which is of utmost utility in numerous problems of applied mathematics, a theorem
of which the former statement is a special case.
Let P be a Hermitian operator. We wish to find the normalized function ๐
which will make the integral
โซ ๐๐๐d
A minimum. The integration extends, as usual over configuration space, and we
shall assume for the sake of definiteness that is a finite portion of configuration
space. Certainly, the necessary condition upon ๐ is that
๐ฟ {โซ๐๐๐๐ โ โซ๐๐๐}
Shall vanish; is an undetermined ( Lagrangian) multiplier (cf. Sec. 6.5). Now the
variation symbol and the integral sign are commutable in this expression because
the limits of the integration are supposed finite and fixed. Hence we have
โซ๐ฟ๐. ๐๐๐ + โซ๐. ๐ฟ(๐๐)๐ โ ๐ โซ ๐ฟ๐. ๐๐ โ ๐ โซ๐๐ฟ๐๐ = 0 (11-85)
The second integral in this expression may be transformed in two steps.
Firts, ๐ฟ(๐๐) may be replaced by ๐(๐ฟ๐) since the operator P suffers no variation.
Second because P is Hermitian and both ๐ ๐๐๐ ๐ฟ๐ are acceptable functions,
โซ๐๐(๐ฟ๐)d = โซ๐ฟ๐. ๐๐d. Eq. (85) therefore reads
โซ๐ฟ๐(๐๐ โ ๐๐)d +โซ ๐ฟ๐(๐๐โ ๐๐)d =0 (11-86)
Here ๐ฟ๐ is an entirely arbitrary functions. Let us take it to be real, so that ๐ฟ๐
= ๐ฟ๐. ๐ธ๐. (86) can be satisfied only if
P๐ โ ๐๐ + ๐๐โ ๐๐ = 0
On the other hand, if we take ๐ฟ๐ to be imaginary, so that ๐ฟ๐ = โ๐ฟ๐, we conclude
๐๐ โ ๐๐ โ ๐๐ + ๐๐ = 0
Addition of the last two equations yields
๐๐ = ๐๐
Subtraction gives
๐๐ = ๐๐
We have shown that, if
๐ฟ โซ๐๐๐๐ = 0 (11-87)
For normalized ๐, this functionmust satisfy the eigenvalue equation
๐๐ = ๐๐ (11-88)
Which also automatically determines ๐. Whether, when (88) is satisfied, the
minimum of, or indeed the integral, โซ๐๐๐๐, actually exists, is a point we have
not investigated. It is customary in physics not to worry about these eventualities,
for they are difficult to discuss. The mathematical equivalence of the minimal
property of the integral and eq. (88) is usually taken as a matter of faith.
If ๐ satisfies eq. (88), thenโซ๐๐๐๐, = ๐ . From what has been said it follows,
therefore, that the integral โซ๐๐๐๐ computed with a function different from the
minimizing ๐, cannot be smaller than๐. But here a slight complication arises, for
there are many eigenvalues๐. All that we can really say is that for a function ๐ in
the โneighborhoodโ of ๐๐, the integral will not be greater than ๐๐. Certainly,
however,
โซ๐๐๐๐ =โฅ ๐0 (11-89)
If ๐ is any analytic and continuous function19 and ๐0 the lowest eigenvalue.
19 Restriction to function with a certain number of derivatives is necessary because P is in general a differential operator, and P ๐ must have meaning
The Ritz method, 20 named after its inventor, is a systematic procedure,
based upon the foregoing variational considerations, for solving the eigenvalue
equation (88) by substituting into the integral in (87) a suitable sequence of function
which causes the integral to converge upon the value ๐. Instead of presenting the
method in its original form, we shall here work out some of its features in a manner
more directly adapted to the needs of quantum mechanics, and with a slight loss of
the lowest (normal state) energy of physical energy of physical or chemical system,
hence with identify at once the operator P in (89) with the hamiltionan H.
The simples way of finding an approximation to the lowest wnwrgy of the system
in to use (89) directly. Sometimes a good guess can be made as to the general form
of the true state function , a from which may allow the inclution of one or more
arbitrary parameters. The integral in (89) is then computed with this function, and
the result is minimized with respect to the parameters. An example with the clarify
the methode.
11.19. example: normal state of the helium atom. The helium atom consists
of two electron moving in the field of the nucleus of charge 2e and at the same time
repelling each other. We consider the nucleus as stationary and denote the distances
of the two electrons from it by r1 and r2 respectively; r12 is the interelectronic
distance. The potential energy is โ 2๐2(1/๐1 + 1/๐2) + ๐2/๐12, and the scrodinger
equation.
๐ป = {โโ2
2๐(โ1
2 + โ22) โ 2๐2(1/๐1 + 1/๐2) + ๐2/๐12} = ๐ธ (11-90)
A subscript on the symbol โ indicates that the laplacian is to be taken with respect
to the respect to the coordinates labeled by the subscript. If the term ๐2/๐12 where
absent eq. (90) would be sparable, for then the operator H would be to the sum of
two helium โ ion Hamiltonians, ๐ป = ๐ป1 + ๐ป2, the first acting on the coordinates of
electron 1, the second on those of electrons 2. But the equation.
20 Ritz, W ., J . f. Reine und angew. Math .135, 1 (1909); Courant-Hilbert, p. 150.
(๐ป1 + ๐ป2) = ๐ธ
May be separated of substitution of = ๐ข(1)๐ฃ(2), where ๐ข(1) stand for a function
of the space coordinates of electron 1, the second on those of electron 2. But the
equation.
๐ป1๐ข(1)
๐ข(1)+๐ป2๐ข(2)
๐ข(2)= ๐ธ, a constant
Which indicates that ๐ป1๐ข(1) = ๐1๐ข(1); ๐ป2๐ฃ(2) = ๐ธ2๐ฃ(2); ๐ธ1 + ๐ธ2 = ๐ธ. But the
first two of these are simply Schrแฝdinger equations for the singly charged helium
ion, whose solutions we already know. (Cf. eq. 67a) since we wish to find the lowest
energy of our system, we identify the functions as follow:
๐ข(1) = (๐3
๐๐03)
1/2 ๐โ๐1/๐0. ๐ฃ(2) = (๐3
๐๐03)
1/2 ๐โ๐1/๐0
And is the product of these.
The correct solution of eq. (90) is certainly not of this exact from because
of the โinteraction termโ ๐2/๐12, whose effect on one would expext to be very
complicated indeed. Aside from other changes, it will couse to depend or r12
expicitl. But from a physical point of view, the repulsion between the electrons will
cause both of them to be, on the average, farther away from the nucleus then if the
repulsion were absent. This would mean that the function ๐ข and ๐ฃ are in error with
respect to the scale factor ๐/๐0. If this were smaller, a more extended probability
distribution would result. (for the helium ion Z = 2) it woud seen expendient,
therefore, that we take as our โtrialโ function in the variational procedure teh
function = ๐ข(1)๐ฃ(2) but with an undertermined Z.
In calculating
โซ ๐ป๐๐ (11.90)
It is well to have available the differential equation whose solution are ๐ข and ๐ฃ:
โโ2
2๐โ12๐ข(1) =
๐๐2
๐1๐ข(1) + ๐2๐ธ๐ป๐ข(1), ๐ฃ(2) = ๐ข(2) (11-91)
Here ๐ธ๐ป is the energy is the normal hydrogen atom, ๐ธ๐ป = โ๐2/2๐0(=
โ13.53 ๐. ๐ฃ๐๐๐ก๐ ). the differential ๐๐ in (91) represent, of course, the product of the
volume element for then two electrons. When H is taken from (90), we find, using
(92) and the fact that u is normalized,
โซ ๐ป๐๐ = 2๐2๐ธ๐ป + (๐ โ 2)๐2 โซ (
1
๐1+
1
๐2) 2๐๐ + ๐2 โซ
2
๐12๐๐ (11 โ 92)
The integral
โซ2
๐1๐๐ = โซ
๐ข2(1)
๐1๐๐1. โซ ๐ข
2(2)๐๐2 = โซ๐ข2(1)
๐1. ๐12๐๐1 sin ๐๐๐๐๐
Is easily computed directly. It has, in fact, already been evaluated (cf. sec. 3.11,
example) and found to be ๐/๐0. The other integral, โซ2
๐1๐๐, has the same value.
We leave the evaluation of ๐2 โซ2
๐1๐๐ for later; its value is โ
5
4๐๐ธ๐ป . Hence, eq.
(93) becomes
โซ ๐ป๐๐ = 2๐๐ธ๐ป + (๐ โ 2). 2๐๐2
๐0โ5
4๐๐ธ๐ป (11 โ 93)
= ๐ [2๐ โ 4(๐ โ 2) โ5
4] ๐ธ๐ป
The symbol โ๐๐ in place of Hi j is to remind the reader of the fact that the matrix โ
dose not possess the simple properties of H because the former is not constructed
with an orthonormal, complete set of functions. The denominator in the expression
for E is needed to normalize the function โ . According to the variantional principle,
E โฅ E0, the lowest energy state of the system.
We wish to find the condition that E shall be a minimum, and the minimum value
of E. insertion of (94) gives
๐ธ = โ ๐ผ๐โ๐ผยตโ๐๐
๐๐,ยต=1 / โ ๐ผ๐
โ๐ผยตโ๐๐๐๐ (11 โ 9)
This expression will be an extremum, and we hope a minimum, if E is so adjusted
that ๐๐ธ
๐๐ผ๐ โ and
๐๐ธ
๐๐ผ๐ are zero for every k from 1 to n. let us take the derivative with
respect to ๐ผ๐ โ on both sides of last equation after is it written in form
๐ธ โ๐ผ๐โ๐ผยตโ๐๐
๐๐
= โ๐ผ๐โ๐ผยตโ๐๐
๐๐
The result is
๐๐ธ
๐๐ผ๐ โ โ ๐ผ๐
โ๐ผ๐โ๐๐ + ๐ธ โ ๐ผ๐ฮ๐ ๐ = โ ๐ผ๐โ๐ รฌ, ๐ = 1,2, โฆ , ๐๐๐๐๐ (11 โ 95)
When first term is omitted (๐๐ธ
๐๐ผ๐ โ = 0) the reminder of the equation represent the
condition that E shall be a minimum. Differentiation of (95) with respect to ๐ผ๐ leads
in a similar way to
๐ธ โ๐ผ๐โโ๐๐
๐
= โ๐ผ๐โโ๐๐
๐
An equation which is simply the conjugate of the former. Both may conveniently
be written
โ ๐ผ๐(โ๐ ๐ โ ฮ๐ ๐๐ธ) = 0, ๐ = 1,2, . . . , ๐๐ (11-96)
If this system of equation is to have a solution different from the trivial one,
every ๐ผ๐ = 0, then determinant constructed from the coefficients of the ๐ผ๐ must
vanish. Thus
||
โ11 โ ฮ11๐ธ โ12 โ ฮ12๐ธ โฆ โ1๐ โ ฮ1๐๐ธโ21 โ ฮ21๐ธ โ22 โ ฮ22๐ธ โฆ โ2๐ โ ฮ2๐๐ธโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ .โฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ .โ๐1 โ ฮ๐1๐ธ โ๐2 โ ฮ๐2๐ธ โฆ โ๐๐ โ ฮ๐๐๐ธ
|| = 0 (11-97)
+โฏ +โฏ
๐๐ด ๐๐ต
R
A
This is an equation of the n-th degree in E and therefore has n roots. The lowest of
these will be an approximation to the lowest energy of the system
The other roots approximate, though in general much more poorly, to the n โ 1
higher of the system.
11.21 Example: the hydrogen molecular ion problem. โ the ๐ป2+ โ ๐๐๐
consists of two positive charges + โฏ, which we shall consider stationary and a
distance R apart, and one electron whose distance from the protons will be denoted
by ๐๐ด and ๐๐ต. See Fig. 5.
The Hamiltonian operator is
H = โ2
2๐โ2 โ
โฏ2
๐๐ดโ
โฏ2
๐๐ต+
โฏ2
R
If the term โฏ2
R - โฏ2
๐๐ต were missing, H would be the Hamiltonian of a hydrogen atom
with its proton at A, whose normal state function is (ef.eq.67)
๐ข๐ด = (๐๐ผ๐3)โ1/2โฏโ๐๐ด/๐ผ๐
On the other hand, if the terms โฏ2
R - โฏ2
๐๐ด were missing, the normal state fuction would
be
๐ข๐ต = (๐๐ผ๐3)โ1/2โฏโ๐๐ต/๐ผ๐
B
-โฏ
From a physical point of view one of these solutions is as good as the other: ๐ข๐ด
implies that the electron is entirely attached to proton A, ๐ข๐ต that it is attached to
proton B. Neither is the ease. Let us see what happens if we take for variation
function โ a linear combination of ๐ข๐ด and ๐ข๐ต. We put
โ = ๐ผ๐ด๐ข๐ด + ๐ผ๐ต๐ข๐ต
Using letters as subscripts rather than the number indices which appear in (94).21
The lowest energy is at once obtained as the lowest root of (97) which takes the
simple form
|โ๐ด๐ด โ โ๐ด๐ด๐ธ โ๐ด๐ต โ โ๐ด๐ต๐ธโ๐ต๐ด โ โ๐ต๐ด๐ธ โ๐ต๐ต โ โ๐ต๐ต๐ธ
| = 0 (11-98)
in more complicated molecules it is well to label electron by numbers, nuclei letters.
We here follow this convention. Now โ๐ด๐ด= โซ๐ข๐ดโ๐ข๐ด๐๐ = ฮ๐ต๐ต = 1, because ๐ข๐ด and
๐ข๐ต are normalized. They are not orthogonal, but โ๐ด๐ต = โ๐ต๐ด. Similar, โ๐ด๐ต =
โซ๐ข๐ด๐ป๐ข๐ต๐๐ = โ๐ต๐ด ๐๐๐ โ๐ต๐ต = โ๐ด๐ด since H is insensitive to an interchange of
I A and B. with these simplifications the two roots of (98) are found to be
๐ธ1 =โ๐ด๐ด+โ๐ด๐ต
1+โ๐ด๐ต ๐ธ2 =
โ๐ด๐ดโโ๐ด๐ต
1โโ๐ด๐ต (11-99)
The โ -functions corresponding to these energies are obtained from (96):
๐ผ๐ด(โ๐ด๐ด โ ๐ธ) + ๐ผ๐ต(โ๐ด๐ต โ โ๐ด๐ต๐ธ) = 0
๐ผ๐ด(โ๐ต๐ด โ โ๐ต๐ด๐ธ) + ๐ผ๐ต(โ๐ต๐ต โ ๐ธ) = 0
On inserting E = E1 we get ๐ผ๐ด = ๐ผ๐ต ; hence the corresponding
โ 1 = ๐1(๐ข๐ด + ๐ข๐ต)
If โ 1 is to normalized, ๐1 = [2(1 + โ๐ด๐ต)]-1/2 . if E2 is inserted in (99), we find ๐ผ๐ต
= - ๐ผ๐ด, so that
โ 2 = ๐2(๐ข๐ด โ ๐ข๐ต)
The normalizing factor is in this case ๐2 = [2(1 โ โ๐ด๐ต)]-1/2.
The remainder of the work is the computation of the three quantities โ๐ด๐ต, โ๐ด๐ด and
โ๐ด๐ต. It involves nothing new and will be left to the reader. The integrals are most
easily evaluated in spheroidal coordinates (cf.eq. 5-40). ๐ = ๐๐ดโ ๐๐ต
๐ , ๐ =
๐๐ดโ ๐๐ต
๐ and
๐, the latter measured around R. In terms of these
๐๐ = ๐ 3
8(๐2 โ ๐2)๐๐๐๐๐๐, ๐๐๐ ๐ข๐ด๐ข๐ต = (๐๐ผ0
3)โ1โฏโ(๐ /๐ผ๐)๐
1 โค ๐ < โ; โ1 โค ๐ โค 1
the following results will be found:
โ๐ด๐ต= โฏโ๐(1 + ๐ +
๐2
3
โ๐ด๐ด = ๐ธ๐ป +โฏ2
๐ + ๐ฝ
๐ฝ = โโซ๐ข๐ดโฏ2
๐๐ต๐ข๐ด ๐๐ = โ
โฏ2
๐ [1 โ โฏโ2๐(1 + ๐)]
โ๐ด๐ต = (๐ธ๐ป +โฏ2
๐ )โ๐ด๐ต + ๐พ,
๐พ = โ โซ ๐ข๐ดโฏ2
๐๐ด๐ข๐ต ๐๐ = โ
โฏ2
๐ โฏโ๐(๐ + ๐2) (11-100 )
The parameter ๐ โก๐
๐ผ0; ๐ธ๐ป is defined as in sec. 19. The quantities J and K are of
interest. According to its definitions, J represent the coulomb attraction energy
between a negative charge of density ๐ข๐ด2 and the proton B. the integral K has no
such simple interpretation; it is called an exchange integral. Its importance is best
appreciated if E1 and E2 are written more explicitly with the use of (100):
๐ธ1 = ๐ธ๐ป + โฏ2
๐ +
๐ฝ + ๐พ
1 + โ๐ด๐ต
๐ธ2 = ๐ธ๐ป + โฏ2
๐ +
๐ฝ โ ๐พ
1 โ โ๐ด๐ต
Because K is negative, ๐ธ1 is the lower root, had we omitted the function ๐ข๐ต from
our trial function โ , the variantional result would have been
๐ธ = ๐ธ๐ป + โฏ2
๐ + ๐ฝ
๐ธ1 is lower than this by virtue of the presence of K (and of course โ๐ด๐ต). But in
classical parlance, a lower energy must be regarded as due to the presence of
additional attractive forces between the constituents of the system, i.e., a hydrogen
atom and a proton. These forces would be given by ๐๐พ
๐๐ ; they are commonly called
exchange forces. They possess no classical interpretation; their significance is
rooted entirely in the variational method through which they arise.
Of course ๐ธ1 is only an approximation to the true energy, which is lower for every
R. Its most important feature is that it possesses a minimum, which explains the
stability of the ๐ป2+ ion. Classical mechanics would yield no minimum and is
therefore incompetent to account for the existence of this ion. A detailed
comparison of ๐ธ1 with the experimental energy is given in pauling and Wilson.21
Problem Let ๐ข0, ๐ข1, ๐ข2, be the three lowest energy states of the simple harmonic
oscillator, ๐ป0 its Hamiltonian. The Hamiltonian for an oscillator in an electric field
is H = ๐ป0 + ๐๐ฅ, where k is a constant. Calculate by the variational method the
lowest energy of this system, using as trial functions (a) ๐ข0, (b) ๐0 ๐ข0 +
๐1 ๐ข1 (๐) ๐0 ๐ข0 + ๐1 ๐ข1 + ๐2 ๐ข2 .
21 pauling and Wilson, loc. cit.
Ans. (a) 1
2โ๐ฃ, (๐)โ๐ฃ โ โ๐2๐ฅ01
2 + 1
4(โ๐ฃ)2 โ
1
2โ๐ฃ โ ๐2๐ฅ01
2 /โ๐ฃ (๐) 1
2โ๐ฃ โ
โ๐ฃ๐2๐ฅ01/[(โ๐ฃ)2 โ ๐2๐ฅ12
2 ] (๐๐๐๐๐๐ฅ๐๐๐๐ ๐). Here xii is defined as โซ๐ข๐๐ฅ๐ข๐ ๐๐, as
usual.
11.22 perturbation theory- the following problem is frequently met in
quantum mechanics. We know the energy states of a given system, say an atom,
and also its eigenfunctions. A small perturbation, such as an
This expression is to made as small as possible by choosing Z properly, i.e., the
coefficient must take its maximum value because EH < 0. Putting the derivative
with respect to Z equal to zero, we find for the minimizing Z the value 27/16, which
is somewhat less than 2 as we expected. Hence the best energy value attainable by
adjusting Z in our function is Z (27/4 โ 2Z) EH = 5.695E H. the difference between
these two values is to be ascribed to the defects of the simple trial function here
chosen.
Fig. 11-4
A very interesting summary of the results of the present method as applied to helium
is given by Pauling and Wilson.22 Their table show how the value of the integral
approaches the experimental energy as increasingly refined trial function are used.
22 pauling and Wilson, p. 224
๐1
๐1
๐2
๐12
๐1
To complete the analysis we indicate how the integral
I = โฏ2 โซ๐2
๐12 ๐๐
May be computed. The method is typical of the evaluation of โdouble volumeโ
integrals involving the variable ๐12, and hence perhaps of same interest. The volume
element
๐๐ = ๐12๐๐1๐ ๐๐๐1๐๐1๐๐1. ๐2
2๐๐2๐ ๐๐๐2๐๐2๐๐2
May also be expressed as follows :(see Fig. 4)
๐๐ = ๐12๐๐1๐ ๐๐๐1๐๐1๐๐1. ๐12
2 ๐๐12๐ ๐๐๐๐๐๐๐
Now ๐22 = ๐1
2 + ๐122 โ 2๐1๐12 ๐๐๐ ๐, whence ๐2๐๐2 = ๐1๐12๐ ๐๐ ๐๐๐ provided
๐1and ๐12 are held fixed. By means of this relation sin ๐๐๐ may be eliminated from
the last expression for ๐๐, and we obtain
๐๐ = ๐1๐๐1๐2๐๐2๐12๐๐12๐ ๐๐๐1๐๐1d๐d๐
Substitute this volume element into I, and integrate at once over the angles, thus
introducing the factors 2.2๐ .2๐ . on using the abbreviation ๐ผ = 2Z / ๐ผ0, we obtain
I = ๐ผ6โฏ2
8โญโฏโ๐ผ(๐1+๐2)๐1๐๐2๐2๐๐2๐๐12
The ranges of integration are 0 โค ๐1 โค โ, 0 โค ๐2 โค โ; |๐2 โ ๐1| โค ๐12 โค ๐1 + ๐2.
The absolute value sign on the limit for ๐12 forces us to split the lower limit of ๐12
is ๐2 โ ๐1, in case (b) it is ๐1 - ๐2. Thus
I = ๐ผ6โฏ2
8{โซ โฏโ๐ผ๐1๐1๐๐1
โ
0 โซ โฏโ๐ผ๐2๐2๐๐2 โซ ๐๐12 +
๐2+๐1
๐2โ๐1
โ
๐1
โซ โฏโ๐ผ๐2๐2๐๐2 โซ โฏโ๐ผ๐1๐1๐๐1โ
๐2
โ
0 โซ ๐๐12๐1+๐2
๐1โ๐2}
Inspection show that the two triple integrals are equal, the calculation is now
perfectly straightforward; it makes use of the formula
โซ โฏโ๐๐ฅ๐ฅ๐๐๐ฅโ
0
= ๐โ(๐+1)๐!
And leads to the result
I = 5
8 ๐
โฏ2
๐ผ0=
5
4๐๐ธ๐ป
Which was used above.
11.20 the method of linier variation functions.
It is often convenient to use as the trial function ๐ in โซ๐โ๐ป๐ ๐๐ a linier
combination of definite function ๐ข๐ which are judged suitable for the problem at
hand. The coefficients appearing in the linier combination may then be tread as
variable parameters thus, assume
๐ = โ๐ผ๐๐ข๐
๐
๐=1
Where the uโs need not form an orthonormal set. We define
โซ๐ข๐โ๐ข๐ ๐๐ = ฮ๐๐, โซ๐ข๐
โ๐ป๐ข๐ ๐๐ = โ๐๐ , E โซ๐โ๐ป๐๐๐
โซ๐โ๐๐๐
Electric or magnetic field, is now imposed; this changes, presumably by slight
amounts, both energy and state function. Mathematically, the situation is described
in this way. We know the solutions and eigenvalues of
๐ป0๐๐ = ๐ธ๐0๐๐ (11โ101)
Where ๐ป0 is the โunperturbedโ Hamiltonian. We wish to find solution and
eigenvalues of
๐ป๐๐ = ๐ธ๐๐๐, ๐ป0 + ๐ปโฒ (11โ102)
๐ปโฒ being considered as a โsmallโ addition to ๐ป0. (By a small operator we mean one
whose matrix elements, formed with the function ๐๐, are all small compared with
the diagonal elements of ๐ป0.)
To solve the problem we use the method of linear variation functions, using as our
trial function
๐ = โ ๐๐๐๐๐ (11 โ 103)
If we allow an infinite number of terms in this summation and choose the
coefficients properly, we expect ๐ to be the correct solution of (102), for the๐๐ of
(101) form a complete set. But the since ๐๐ are orthonormal, the energies are given
as roots of (97) with every ฮ๐๐ replaced by a Kronecker ๐ฟ๐๐, so that E appears only
in the principal diagonal. Moreover,
๐ฆ๐๐ = ๐ป๐๐ = (๐ป0)๐๐ + ๐ป๐๐โฒ = ๐ธ๐
0๐ฟ๐๐ + ๐ป๐๐โฒ
๐ป๐๐โฒ = โซ๐๐
โ๐ปโฒ๐๐๐๐
Hence the determinant reads
|
|
๐ป11โฒ โ (๐ธ โ ๐ธ1
0)
๐ป21โฒ
๐ป31โฒ
๐ป41 โฒ
๐ป12โฒ
๐ป22โฒ โ (๐ธ โ ๐ธ2
0)
๐ป32 โฒ
๐ป42โฒ
๐ป13โฒ
๐ป23โฒ
๐ป33โฒ โ (๐ธ โ ๐ธ3
0)
๐ป43โฒ
๐ป14โฒ
๐ป24โฒ
๐ป34 โฒ
๐ป44 โฒ โ (๐ธ โ ๐ธ4
0)
โฏโฏโฏโฏ
โฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏ
|
|
= 0
If all its roots could actually be found they would indeed be the exact energies of
our problem. But in the case we are visualizing certain simplifying approximations
are in order. Suppose we are interested in the energy ๐ธ1, that is, the energy to which
๐ธ10 is changed by the perturbation. (๐ธ1 need not to be the lowest energy of our
system, for the states may be labeled in an arbitrary order.) If ๐ธ10 is a non-
degenarate level, then ๐ธ1will lie much closer to ๐ธ10 than to any other unperturbed
๐ธ๐0. This suggests the following approximations:
a. Put E = ๐ธ10 in all diagonal elements except the first.
b. Since every difference ๐ธ10 โ ๐ธ๐
0 for ๐ โ 1 is large compared to ๐ป๐๐โฒ , the
latter may be omitted in all diagonal elements except first.
c. Neglect all non-diagonal elements except those in the first row and the
first column, since they affect ๐ธ1 only in a secondary way.
When this is done, the determinant reads (we now write ฮ๐ธ1 for the
perturbation ๐ธ โ ๐ธ10 we are seeking)
||
๐ป11โฒ โ ฮ๐ธ1๐ป21โฒ
๐ป31โฒ
๐ป41 โฒ
๐ป12โฒ
๐ธ20 โ ๐ธ1
0
00
๐ป13โฒ
0๐ธ30 โ ๐ธ1
0
0
๐ป14โฒ
00
๐ธ40 โ ๐ธ1
0
โฏโฏโฏโฏ
โฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏ
|| = 0 (11โ105)
It may be evaluated by the usual process of adding multiples of rows or column. In
this instance, multiply the second row by ๐ป12โฒ /(๐ธ2
0 โ ๐ธ10), and then substract it from
the first. The element ๐ป12โฒ will then disappear from the first row, but the first
element is converted into
๐ป11โฒ โ ฮ๐ธ1 โ
๐ป12โฒ ๐ป12
โฒ
๐ธ20 โ ๐ธ1
0
Next multiply the third row by ๐ป13โฒ /(๐ธ3
0 โ ๐ธ10) and substract it from the
first. The result will be disappearance of ๐ป13โฒ and addition โ๐ป13
โฒ ๐ป31โฒ /(๐ธ3
0 โ ๐ธ10) to
the first element. This process is continued until all non-diagonal elements of the
first row have disappeared. We now have
๐ป11โฒ โ ฮ๐ธ1 โโ
๐ป1๐โฒ ๐ป๐1
โฒ
๐ธ๐0โ๐ธ1
0
โ
๐=2
(๐ธ20 โ ๐ธ1
0)(๐ธ30 โ ๐ธ1
0)โฏ = 0
If ๐ธ10is non-degenerate, as we are supposing, none of the parentheses except
the first can be zero. We therefore conclude
ฮ๐ธ1 = ๐ป11โฒ โ โ
๐ป1๐โฒ ๐ป๐1
โฒ
๐ธ๐0โ๐ธ1
0โ๐=2 (11 โ 106)
and this is the Rayleigh-Schrรถdinger perturbation formula. The quantity ๐ป11โฒ is ofen
called the first-order perturbation, the sum of the right is called the second-order
perturbation. By retaining more elements in (104) third and higher orders may be
computed, but these are rarely used. When the approximation (106) is not sufficient
it is generally preferable to return to the variation scheme, or to find a more
successful way of evaluating the determinant (104). Formula (106) may, of course
, be used to calculate the perturbation in any energy level which in non-degenerate;
to show this fact it may be written in the form
๐ธ๐ = ๐ปโฒ๐๐
Where we have also used the Hermitian property of ๐ปโฒ๐. The prime on the
summation symbol indicates that the term in which = ๐ฆ should be omitted.
Next, let us find the coefficients ๐ in (103). They are obtained from (96) which
now reads
โ ๐ (๐ธ๐พ๐ฟ๐พ0 + ๐ปโฒ๐ - ๐ธ๐ฟ๐ ) = 0, k = 1,2,...
In accordance with the approximations which led to eq. (106) we put E =๐ธ10 and
neglect every ๐ปโฒ๐ unless one of the subscripts is 1. We then find
๐1๐ปโฒ21 + ๐2 (๐ธ20 - ๐ธ1
0 ) = 0 if k = 2
๐1๐ปโฒ31 + ๐3 (๐ธ30 - ๐ธ1
0 ) = 0 if k = 3 , etc. (11-107)
Hence
๐ = ๐ปโฒ1
๐ธ10โ ๐ธ
0 ๐1 , 1
Or in general, if we are interested not in ๐ธ1 but i ๐ธ๐
๐ = ๐ปโฒ๐
๐ธ๐0โ ๐ธ
0 ๐๐, k (11-108)
The coefficient ๐๐ must be chosen so that is normalized. Since all other ๐ are
small, its value is very nearly unity and may be taken as such.
Formulas (107) and (108) have been derived by assuming that the level, k,
whose perturbation is being calculated, wa non-degenerate. For degenerate levels
both formulas obviously fail, for they contain terms with vanishing denominators
(several ๐ธ0 being equal to๐ธ๐
0). To deal with the case of degeneray we have to return
to the fundamental determinant (104). If the functions ๐ข1, ๐ข2, . . ., ๐ข๐, all belong to
the same energy ๐ธ10 (we then say that the level ๐ธ1
0 has an ๐ โfold degenaracy),
these functionsare equally concerned in the perturbation, and if we formerly
retained all matrix elements of the form๐ปโฒ1, we must now retain๐ปโฒ2, ๐ปโฒ3, . . . ,
๐ปโฒ๐, also. But for most purposes sufficient accuracy results if we neglect all
elements connecting a state of the degenerate group with all states not belonging to
that group. Eq. (104) reduces in this case to
|
|
๐ปโฒ11 โ ๐ธ ๐ปโฒ12 ๐ป
โฒ13,.. . . ๐ป
โฒ1๐
๐ปโฒ21 ๐ปโฒ22 โ ๐ธ ๐ปโฒ23 ,. . . . ๐ปโฒ2๐๐ปโฒ31 ๐ป
โฒ32 ๐ป
โฒ33โ๐ธ . . . ๐ป
โฒ3๐
. . . . . . . . . . . . . . . . . . . . . ๐ปโฒ1๐ ๐ปโฒ2๐ ๐ป
โฒ3๐. . . ๐ป
โฒ๐๐ โ ๐ธ
|
| = 0 (11-109)
The ๐ roots of this equations (of which some may coincide) are the energies into
which ๐ธ10 will โsplitโ as the result of the perturbation. They canot, of course, be
represented by a general formula.
These energies are said to represent the first-order perturbation. If greater
accuracy is desired the work may be continued in this way. By substituting the fist-
order energies into eq. (96) and neglecting all states not belonging to the degenerate
group, ๐ sets of coefficients ๐1,๐2, . . . ๐๐, are found, each set belonging to a single
first-order energy. This yields ๐ functions
i =
uan
i1
If now we construct matrix elements with the ๐ โfunctions , dHvvH jiij ' , these
will be diagonal; for solving (109) is the well-known procedure for diagonalizing
the matrix Hโ. (See Chapter 10.) Hence, when the ๐ฑ โ functions are chosen to
represent the ๐ degenerate states, the second order perturbation can be computed
by formula (107), from which the terms with vanishing denominator sre now absent
because every ๐ปโฒ๐ corresponding to them is zero.
11.23. Example : Non-Degenerate Case. The stark Effect.
Let ๐ป0 represent the Hamiltonian operator for anyone โelectron system, and let
๐1,๐2, be its eigenfunctions. When a uniform electric field along X is applied, the
term Hโ = eFx is added to ๐ป0, e being the electronic charge and F the field strength.
The normal state of the system is non-degenerate, hence formula (107) may be used.
Denoting the normal state by the subscript zero, we find
2
0
0
0
022
000 '
EE
xFeeFx (11-110)
Here ๐ฅ0๐ = ๐0๐ฅ๐๐d. The first term on the right is usually zero because|๐0|
2 is an
even function of ๐ ; thus the โ first-order Stark effectโ is absent.
In classical physics, theincrement in energy of an atom due to a static
electric field is expressed in terms of the polarizability ๐ถ in the form
2
2
1aF
On comparing this with (110) we find for the polarizability of the normal
state of our system
โ= 2๐2โ|แตช0๐|
2
๐ธ๐0โ๐ธ๐
0
๐
For an oscillator, this takes a particularly simple form, since all แตช0๐ vanish
with the exception of แตช01=โ1/2๐ฝ (cf. chapter 3, eqs. 92 and 93). Also, ๐ธ๐0 =
(๐ +1
2) โ๐ฃ. Thus
โ= 2๐2๐ฅ012
โ๐ฃ
Comparison with the problem of sec. 21 shows that second-order perturbation
theory gives in this instance the same result as the variational method with the trial
function โ0 ๐0 +โ1 ๐1 in general, however, the use of a simple variation function
yields a much poorer result for the polarizability than the method of sec. 22
11.24. Example: Degenerate Case. The Normal Zeeman Effect.
The energy states of the hydrogen atom were found to be
๐ ๐,๐(๐)๐๐(๐, ๐)
To a given ๐, there belong 2๐ + 1 spherical harmonics of the form
๐๐ = โ ๐๐
๐
๐=โ1
๐๐๐(๐๐๐ ๐)๐๐๐๐ , (๐๐
โ๐ = ๐๐๐
And each such combination with its own set of Coefficient,๐๐ Forms a proper
eignefunction when multiplied by๐ ๐,๐. The energy does not depend on m; the state
under consideration has therefore a 2๐ + 1 fold degeneracy.
Let us choose the 2๐ + 1 functions in the simplest possible way, namely by
letting each ๐๐ contain only one term, as follows:
๐ ๐,๐. ๐โ๐๐๐๐๐โ๐๐๐, ๐ ๐,๐. ๐โ๐+1๐๐
๐โ1๐โ๐(๐โ1)๐ , . . . . , ๐ ๐,๐. ๐1๐๐๐๐๐๐๐
And label them๐1, ๐2. . . , ๐2๐+1, in that order.
The Zeeman the splitting of the energy levels of an atom in a magnetic field.
When a uniform magnetic field along the Z-axis and of strength F is applied to the
hydrogen atom, its unperturbed Hamiltonian takes on the extra term.23
๐ปโฒ = โ ๐ฤง๐
2๐๐๐น ๐
๐๐โก โ๐๐ด
๐
๐๐
Each matrix element ๐ปโฒ = โซโ ๐โ๐ปโฒdr contains the factor โซ๐ ๐,
2 ๐๐2๐๐
Witch, by virtue of the normalization of the radial functions, is unity. If we
form, e.g. ๐ป12โฒ we obtain an integral over ๐ timeโซ ๐๐๐๐
๐
๐๐ ๐โ๐(๐โ1)๐๐๐
2๐
0, and this
vanishes in the same way all other non-diagonal matrix elements are seen to be zero.
The diagonal element ๐ป11โฒ is
โ๐๐ด. (โ๐๐).โซ๐1โ๐1๐๐ = โ๐ด๐
And the others are similarly constructed.
When these elements are substituted into (109) we have
|โ๐๐ด โ โ๐ธ 0 0
0 โ(๐ โ 1)๐ด โ โ๐ธ 00 0 โ(๐ฟ โ 2)๐ด โ โ๐ธ
000| = 0
โฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆโฆ0
0 0 0โฆโฆโฆโฆโฆ ๐๐ด โ โ๐ธ
The determinant is already diagonal, our choice of functions was a fortunate
one. The perturbed energies are clearly
23 See Vleeck, J.H., โThe Theory of Electric and Magnetic Susceptibilities,โ oxford,
1932. We write here M for the electron mass to avoid conflict with the summation
index m (magnetic quantum number). Note that Hโ is the quantum representation
of ๐
2๐๐ F.L, where L is the angular momentum vector of sec. 11.3
โ๐ธ = ๐๐ด = ๐ฤง๐๐น
2๐๐โฒ๐ = โ๐,โ๐ + 1, . . . 0, 1. . . ๐
Classically, an electron in a magnetic field F performs a uniform precession of
angular frequency ๐๐ฟ = ๐๐น/2๐๐ known as the Larmor frequency. Thus we see
that โ๐ธ = ๐ฤง๐๐ฟ
Problem. Calculate the Stark effect of the rigid rotator (cf. sec. 11.12), for
the state ๐ = 3, adopting the same choice for the spherical harmonics as above.
Here ๐ปโฒ = ๐๐๐น ๐๐๐ ๐, provided the electric field F is along Z. the determinant will
not be diagonal. To calculate the matrix elements, use formulas (3-48 and 53).
Include in your calculation successively more states: ๐ = 2,3,4; ๐ = 1,2,3,4,5.
TIME-DEPENDET STATES. SCHRำฆDINGERโS TIME EQUATION
11.25. General Considerations.
In all preceding considerations we have assumed that the states of the
systems in question were stationary ones, that the time coordinate could be
disregarded in describing them. In generalizing the theory so to make it applicable
to states which change in time it is well to look back and see why a time-free
description was possible thus far.
It is important to note that the time, t, in classical mechanics is canonically
conjugate to the energy, E, in the same sense that x is conjugate to ๐๐ฅ. Let us then
for the moment consider the operator ๐๐ฅ = ๐ฤง(๐
๐๐ฅ). Its eigenstates were seen to be
(cf. eq. 90 ๐๐ = ๐๐(๐
ฤง)๐๐ฅ
. What do they tell us about the distribution of the system
in x? The answer is, it is uniform.
Whatever is true at the point x1, is also true at the point x2. This is the
meaning of the uncertainty principle applied to the case at hand: if the momentum
is known with certainty, the state function is entirely noncommittal with regard to
x. If in the calculation of the mean value of an operator Q,
๏ฟฝ๏ฟฝ = โซ๐๐โ๐๐๐๐๐
Q did not depend on x, we could have afforded to neglect the factor ๐(๐
โ)๐๐ฅ
of
๐๐ altogether. It had to be included, however, because most operators of interest do
depend on x
But this trivial situation existed with regard to the time coordinate in all the
Schrodinger problems considered heretofore. The states were those in which the
energy was known with certainty, and for this reason the state functions were
completely indiscriminate in respect to t. What was true at t1 was also true at t2.
Moreover, the other operators used were independent of t. This condition will
always be present as long as we are dealing with closed systems, for the energy will
then be constant in time.
When the system is an open one, the present method must clearly fail. But
the last remarks contain the hint that we should, perhaps, associate with E the
operator --ih (๐/๐๐ก). This would lead to the eigenvalue equation.
โ๐โ ๐
๐๐ก๐ = ๐ธ๐
Which is certainly too simple because the energy depends on other things
beside the time. The example above gives us no definite lead at this point because
pa does possess the single dependence on x. There is, however, only one reasonable
way to include these other variables, namely, to put them into E, which thereby
ceases to be an eigenvalue: E must be replaced by the Hamiltonian operator H. We
then arrive at Schrodingerโs time equation
โ๐โ ๐
๐๐ก๐ = ๐ป๐ (11-111)
H is to be constructed as before by replacement of every Cartesian
coordinate pi by โ๐โ (๐ /๐ ๐๐) and the dependence on t is to be introduced
explicitly.
It is immaterial, of course, whether we choose eq. (111) or its complex conjugate
equation. The latter choice has certain advantages and will here be made.
Furthermore, we shall use the symbol a (more or less generally) for time-dependent
state functions and thus record Schrodingerโ time equation in the form
๐โ๐๐ข (๐1โฆ๐๐๐ก)
๐๐ก= ๐ป (โ๐โ
๐
๐๐1, โฆโ ๐โ
๐
๐๐๐; ๐1โฆ๐๐; ๐ก) (๐1โฆ๐๐; ๐ก) (11-112)
This equation, being of the first order in t, permits prediction of the state u at any
future (or past) time when u is known as a function of the coordinates at present.
Although it is closely related to the preceding deve10pments, eq. (112) is a new
postulate not derivable from those already given. The present theory must be valid
also in the special case when H does not contain t. When that is true eq. (112) is
separable. On writing ๐ข = ๐(๐1 . . . ๐๐). ๐(๐ก)it becomes equivalent to the
equation.
๐ปฯ
๐=iโ
๐๐๐๐ก
๐
Each side of which must represent a constant. But in View of the form of ~
the left-hand side, that constant must be one of the eigenvalues of the operator H,
say EA, so that
๐ป๐๐๐๐ ๐๐
๐๐ก=โ๐๐ธ๐โ
๐
The general solution of eq. (112) for the special case in which H .is
independent of the time is
๐ = โ ๐๐ ๐๐๐๐(0๐๐ธ๐/โ)๐ก (11-113)
We have formerly said that any state function, such as u, could be expanded
in the orthonormal system of functions 114.. This expansion was written as
๐ =โ๐
๐
๐๐๐
We now see that this is indeed true even when the analysis is made on the
basis of eq. (012), but the coefficients a) are always functions of the time: ๐๐ =
๐๐ โ (๐๐ธ๐โ)๐ก. The mean value of E, computed for the state
๏ฟฝ๏ฟฝ =โ|๐๐| 2๐ธ๐ =โ|๐๐| 2๐ธ๐๐๐
It is independent of t. But the probability of finding the system at the point
q1. . . qnof configuration space, uโ๐ข = โ ๐๐โ
๐ ๐๐๐๐โ๐๐๐
(๐
ฤง)(๐ธ๐โ๐ธ๐)๐ก
is a
superposition of oscillating functions of the time. The only way for this time
dependence to be obliterated would be to have ๐๐ = ๐ฟ๐ in which case.
๐ข โ ๐ข = ๐๐โ๐๐
Thus, whenever a state is formed by superposition of energy eigenstates, the
mean energy of the system remains constant, but the configuration of the system
changes in time. The reader should note, of course, that the solution of the
Schrodinger equation (12) when multiplied by ๐โ(๐๐ธ๐/โ)๐ก is also a solution of (112),
but that the solution of (112) does not in general satisfy (12).
Problem. Let the time-dependent Hamiltonian be H = H o + V (t), where H0
acts only on space coordinates and has eigenfunctions 1h, eigenvalues Ex. Show
that
๐ข =โ๐
๐
๐๐๐๐โ(๐โ)(๐ธ๐๐ก+โซ๐๐๐ก
11.26. The Free Particle; Wave Packets.
The Eigenfunction of the energy of a free mass point (cf. sec. 11.9) moving
in one dimension without restriction are ๐๐ = ๐โ๐๐๐ฅ its energies๐๐ =โ2
2๐๐2, and
there is no quantization. The general solution of eq. (112) for the free particle is
therefore,
๐ข = โซ ๐(๐)โ
โโ๐๐[๐๐ฅโ(
โ2
2๐๐2๐ก]๐๐ (11-114)
A function constructed after the manner of (113) but with an integral instead
of a sum. An integral very similar to this has been already encountered in the
mathematical formulation of waves (cf. eq. 7-38) and of diffusion phenomena (eq.
7-53). It is interesting to inquire what form it will have at some time t if at t = 0 it
is given by ๐ข = ๐ข0(๐ฅ). the coefficient ๐(๐) may be determined by Fourier analysis.
We have
๐ข0 = โซ ๐(๐)๐๐๐๐ฅ๐๐โ
โโ
Whence by eq. 8-13
๐(๐) =1
2๐โซ ๐ข๐(๐)๐
โ๐๐๐โ
โโ
๐๐
Eq. 114 therefore reads
๐ข(๐ฅ, ๐ก) =1
2๐โฌ ๐ข๐(๐)๐
โ๐[๐(๐ฅโ๐)โ(โ/2๐)๐2๐ก]โ
โโ
๐๐๐๐
In this instance, the integration over to cannot be performed (as it could in
the diffusion problem, sec. 7.14). To proceed further it is necessary to introduce the
function ๐ข๐ explicitly.
Assume that ๐ข0 = ๐โ๐ฅ2/2๐2. Then, with the use of the formula.
โซ ๐โ๐๐ฅ2+๐๐๐ฅ๐๐ฅ = โ
๐
๐๐โ๐
2/4๐โ
โโ (11-115)
We find
๐(๐) =๐
โ2๐๐โ๐
2๐2/2
Hence
๐ข =๐
โ2๐โซ ๐
โ๐2[๐2
2+๐โ(
โ2๐
)๐ก]+๐๐๐ฅ๐๐
โ
โโ
= (1 + ๐โ
๐๐2๐ก)โ1/2
expโ [๐ฅ2
๐2+ฤง2
๐2๐2๐ก2] (11-116)
Again with the aid of (115).
Eq. (114) represents a superposition of waves of wave length 27r/k and
frequency ๐ฃ = (โ/47๐๐) ๐2.the form of no here chosen describes a concentration
of waves about the origin, a phenomenon called a โwave packet.โ Such a wave
packet does not retain its spatial distribution; eq. (116) is characteristic of the
manner in which it diffuses.
From the point of View of quantum mechanics, ๐ข02 is the probability density
of the particle at t = 0. It represents a Gauss error function of โwidthโ a. At time t,
= (1 + ๐โ
๐๐2๐ก)โ1/2
expโ [๐ฅ2
๐2+ฤง2
๐2๐2๐ก2]
The probability density is still a Gauss function, but of smaller maximum and
of width [๐2 + (โ2/ ๐2๐2) ๐ก2]12โ
Problem a. Compute how long it would take an electron, localized within ๐ =
10โ10cm, to diffuse through twice that distance
b. How long would it take an object weighing one gram, localized 1 cm, to
diffuse through twice that distance?
c. Show that if๐ข0 = ๐๐๐๐๐ก, where K is a constant, the wave will be of the
form ๐ข = ๐๐๐๐กโ(โ/2๐)๐2๐ก],
If our particle is free to move in three dimensions, then as shown in sec. 11.9,
๐๐ = ๐๐๐๐๐ก๐๐๐ ๐ธ๐ =โ2
2๐๐2
Hence (114) has the form
dkekcu tkmrki 2 (11-117)
Again, if ๐ข = ๐ข0(๐ฅ, ๐ฆ, ๐ง)
๐ข0 = โซ ๐(๐)๐๐๐๐ฅ๐๐โ
โโ
Whence by 3-dimensional Fourier analysis
๐(๐) =1
8๐3โซ๐ข๐(๐, ๐, ๐ฟ)๐
โ๐๐๐ ๐๐
The vector ๐ having components ๐, ๐, ๐ฟ
Assume now, in analogy with the one-dimensional ease, that
๐ข0 = ๐โ๐2/2๐2
At the ๐ก = 0 have wave packet is a spherical concentration of waves centered about
the origin the probability packet has a similar shape and a width a. on inserting ๐ข0
into the relation for c we have
๐(๐) =1
8๐3โซ ๐
โ(๐2
2๐2)โ๐๐๐
๐๐โ
โโ
โซ๐โ๐2/2๐2๐๐โซโซ๐โ๐
2/2๐2๐๐
= (๐
โ2๐) 3๐โ(๐
2/2)๐2
This gives
๐ข = (2๐) โ3/2๐3โซ๐โ[(
๐2
2)+๐(
ฤง2
2๐)๐ก]๐1
2+๐๐๐๐ฅ๐๐1
Times two similar integral with k1 replaced by k3 and k3, x by y and z. hence
๐ข = (1 +๐ฤง
๐๐2๐ก)โ3/2
expโ [๐2
2(๐2 + ๐ฤง
๐๐ก)]
The interpretation of this result is not different from that of (116).
Before leaving the subject of โparticle wavesโ we should remark that every
component wave of the packet (117), being of the form ๐๐(๐.๐โ2๐๐๐ก travels in a
positive direction along k. Had we chosen the sign as in eq. (111) and not as in
(112), the waves would have been of the form ๐๐(๐.๐โ2๐๐๐ก which implies that they
travel along โk. Since ๐ฤง represents the momentum of the particle, the latter choice
is a unusuitable one. We also also note that the wave length ๐ =2๐
๐=
2๐ฤง
๐๐ค=
โ
๐๐ค
conforms to the De Broglie formula. The phase velocity of the waves is ๐ฃ๐ =ฤง๐
2๐=
๐๐ค
2๐= ๐ฃ/2 but their group velocity24 defined as 2๐ (
๐๐ฃ
๐๐) = ๐ฃ is equal to the classical
speed of the particle.25
11.27. Equation of Continuity, Current. If the state function changes in time in
accordance with the Schrำงdinger equation.
๐ป๐ข = ๐ฤง๐ข (11-118)
Will it remain normalized? If it does not, there occurs a destruction or creation of
probability, while initially there was certainty, a situation which would clearly be
physically untenable. Permanence of normalization, however, follows immediately
from (118).
For
๐
๐๐กโซ๐ข โ ๐ข๐๐ = โซ[๐ข โ ๐ข + ๐ข โ ๐ข] ๐๐ =
๐
ฤง โซ[๐ข๐ป โ ๐ข โ +๐ข โ ๐ป๐ข] ๐๐
Because of (118) and the last expression is zero on account of the Hermitian
character of H.
24 For a discussion of group velocity, see Sommerfeld, โWellenmechanischer
Erganzungsband โFriedr. Vieweg & Shon, Braunschweig, 1929, p.46. 25 This form of I is correct so long as the potential energy V is of the scalar form
here used. When H contains a vector potential, A, the term (e/c) A must be added
to the exspression for the current here given.
Having shown that ๐ข โ ๐ข is conserved we can define a probability current
by subjecting u*u, which we will ๐ for the moment, to the equation of continuity
๐
๐๐ก+ โ. I = 0 (11-119)
Whatever I turns out to be must be regarded as the current corresponding to the
โflowโ of the quantity u*u. we shall limit our consideration to the case of a single
particle so that
๐ป =ฤง2
2๐โ2 + ๐ (๐ฅ, ๐ฆ, ๐ง)
Althought generalization to many-deminsioned configuration space is easy.
Again because of (118)
๐๐
๐๐ก= ๐ข โ ๐ข + ๐ข โ ๐ข =
๐
ฤง(๐ข๐ป โ ๐ข โ โ๐ข โ ๐ป๐ข)
=๐ฤง
2๐(๐ข โ โ2๐ข โ ๐ขโ2๐ข โ) = โ [
๐ฤง
2๐(๐ข โ โ ๐ข โ ๐ข โ๐ข โ)]
To satisfy (119) we must put
๐ผ = โ๐ฤง
2๐(๐ข โ โ ๐ข โ ๐ขโ ๐ข โ) (11-120)
It is interesting to observr that a state u which has no complex devendence onn a
space variable has no current associated with it. Thus, in theFree particle problem,
๐๐๐ ๐๐ฅ and sin ๐๐ฅ represent stationary states, but ๐๐๐๐ฅ and ๐โ๐๐๐ฅ have currents.
Problem. Compute for the various regions of the barrier problems
considered in sec. 11.10
11.28. Application of Schrำงdingerโs Time Equation. Simple Radiation
Theory.
The cases in which eq. (118) can be solved exactly are not numerous and not very
interesting. When the time equation methods, the most useful of which will now be
illustrated.
Let an atom, whoose normal Hamiltonian function, free from all
perturbations, is ๐ป0, be suddenly subjected to a light wave which adds a perturbing
energy
๐(๐ฅ, ๐ก) = โ๐๐น0๐ฅ sin๐๐ก (11-121)
To H. Physically, this means the light wave is monochromatic and has
frequency ๐ฃ = ๐/2๐; its electric is along X and of amplitude ๐น0 if V did not contain
x and sin๐๐ก in product form, eq. (118) with ๐ป = ๐ป0 + ๐ would be separable; the
fusion of x and t into V spoils separability.
In solving (118) we use the following initial condition: At๐ก = 0, when the
atom was exposed to the perturbation V, the atom was certainly in an eigenstate of
the operator๐ป0, say in the state ๐1 corresponding to the energy ๐ธ1 which we shall
take to be the lowest energy of the system. Or, if we wish to include the trivial time
dependence of the state, we take
๐ข = ๐1๐โ(๐๐ธ1/ฤง)๐ก (11-122)
The solution of
(๐ป0 + ๐)๐ฃ = ๐ฤง๐ฃ (11-123)
Which we desire, is certainly available in the form
๐ฃ = โ ๐๐๐๐๐โ(๐๐ธ1/ฤง)๐ก๐ (11-124)
26 Imples that the wave length of the light is large compared with the size of the
atom. Correctly ๐ = โ๐๐น0๐ฅ sin (๐๐ก โ2๐๐ง
๐), and we are omitting the term๐ง/๐. The
legitimacy of this will be clear form the following analysis.
Provided we let the coefficients ๐ be functions of the time. This follows
immediately from the completeness of the ๐๐ with respect to function of the space
coordinates. When (124) is substituted into (123). There results
โ๐๐
๐
(๐ป0 ๐๐ + ๐ ๐๐)๐โ(๐๐ธ1/ฤง)๐ก =โ(๐๐๐ธ๐ ๐๐ + ๐ฤง๐๐ ๐๐)๐โ(๐๐ธ1/ฤง)๐ก๐
Wherein each term ๐ป0 ๐๐ on the left cancels ๐ธ๐ ๐๐ on the right. Let us now
multiply the remaining terms of the equation by ๐๐โ and integrate over configuration
space, remembering the orthogonality of the ๐๐. Then after simple rearrangement,
๐๐ = โ๐
ฤงโ ๐๐๐๐๐๐ ๐๐[
๐ธ๐โ๐ธ๐ฤง
]๐ก, ๐ = 1,2,3, .. . (11-125)
Where, as usual,
๐๐๐ = โซ๐๐โ ๐ ๐๐๐๐
If the unperturbed atom has an infinite number of states, (125) represents an
infinite set of linear differential equations, which in general canโt be solved. But we
nw recall that ๐ก = 0 ๐ฃ = ๐ข which means that all ๐๐ except ๐1 were zero at that time.
There after ๐1 decayed from 1 to some smaller value, while all other cโs grew from
0 to various finite values. We now limit our inquiry to times so small that ๐1 is still
sensibly unity, and the other cโs are small compared with it, although ๐1 may be
quite comparable with the time derivatives of other cโs. This permits the
26 Eq. (121) is a valid approximation for the purpose at hand. It neglects the energy
due to the magnetic vector of the light wave whose contribution is small compared
to (121) in the ratio ๐ฃ/๐, where ๐ฃ is the velocity of the charge composing the atom
and ๐ the velocity of light. For hydrogen ๐ฃ/๐, is 1/137. Furthermore, eq. (121)
approximation of replacing every ๐๐ on the right-hand side of (125) by its value at
๐ก = 0 while retaining every ๐๐. The equation then beomes
๐๐ = โ๐
ฤง๐๐1๐
(๐ฤง)(๐ธ๐โ๐ธ1)๐ก
To simplify writing we introduce the abbreviation
(๐ธ๐ โ ๐ธ1)
ฤงโก ๐๐
And observe that ever ๐๐ > 0, since, as we are assuming, ๐ธ1 is the lowest
energy state. In view of (121),
๐๐1 = โ๐๐น0๐ฅ๐1 sin๐๐ =1
2๐๐๐น0๐ฅ๐1(๐
๐๐๐ก โ ๐๐๐๐ก)
So that
๐๐ =๐๐น02ฤง
๐ฅ๐1[๐๐(๐๐+๐)๐ก โ ๐๐(๐๐โ๐)๐ก]
On integration,
๐๐ =๐๐๐น02ฤง
๐ฅ๐1 [๐๐(๐๐โ๐)๐ก โ 1
๐๐ โ ๐โ๐๐(๐๐+๐)๐ก โ 1
๐๐ + ๐] , ๐ โ 1
Where we have at once adjusted the constant of integration so that ๐๐ = 0
when ๐ก = 0. For physical reasons, only the first term in the square parenthesis need
be retained because it alone can attain appreciable magnitude. (Both ๐ and ๐๐ > 0
). In fact ๐๐ is large only when ๐ โ ๐๐ and this fact is accentuated when ๐๐ is
squared:
|๐๐|2 =
๐2๐น02
2ฤง2|๐ฅ๐1|
21 โ cos(๐๐ โ ๐)๐ก
(๐๐ โ ๐)2=๐2๐น0
2
4ฤง2|๐ฅ๐1|
2 ๐ ๐๐2[
(๐๐ โ ๐)2 ๐ก]
((๐๐ โ๐)
2 ๐ก)2
We now interpret this result. The coefficient ๐๐ is, in view of (124), the ๐ โ
๐กโ probability amplitude in the expansion of the state function ๐ฃ at time ๐ก in terms
of energy eigenstates of the normal atom. Hence because of sec. 5 |๐๐|2 is the
probability at time ๐ก the ๐ โ ๐กโ energy level of the atom be excited; it is the
โtransition probabilityโ from state 1 to state ๐ when the atom has been exposed to
monochromatic light of frequency ๐/2๐ fot ๐ก seconds.
Many interesting conclusions of a physical nature can be drawn from eq.
(126), of which only two will here be mentioned. First, the transition probability is
proportional to the square of the matrix element connecting the states in question.
Whenever ๐ฅ๐1 is the criterion of a โforbiddenโ transition in the second place, the
transition probability is small unless๐ โ ๐๐, which is the Bohr frequency
condition.
Problem. The reader may be surprised to find that |๐๐|2 is not a linier
function of๐ก, as might be except on physical grounds. Show that, when the incident
light forms a continuous spectrum of uniform intensity, |๐๐|2 is the proportional
to ๐ก. (For this purpose, (126) must be ntegrated over ๐ from 0 to โ; but the
integration may without appreciable error be taken from โโ ๐ก๐ + โ)
ELECTRON SPIN. PAULI THEORY
11.29. Fundamentals of the Theory.
The theory so far developed describes the general behavior of atomic and molecular
systems surprisingly well, but it makes some false predictions, particularly with
regard to the finer details of the energy states of atoms, the Zeeman Effect and the
magnetic properties of electrons. It was soon apparent that the state of a single
electron could not be represented as a function of three space coordinates alone, but
that another parameter was required whose interpretation was for some time in
doubt. Most decisive in clarifiying the situation was the spectroscopic observation
of the doubling of the energy levels of a single electron: in all alkali atoms, for
instance, two levels are found where the Schrำงdinger equation permits only one.
The energy difference between these levels was such as would be produced by a
small magnet of magnetic field present in the atom on account of the these two
energy states was known to be different it was equal to that caused by the electronโs
orbital motion, plus ฤง/2 in the other state.
Uhlenbeck and Goudsmit suggested that the electron behaves like a
spinning top having a โspinโ angular momentum of magnitude ฤง/2 which
however, can only add or subtract its whole amount, in quantum fashion, to any
angular momentum the electron already prossesses as a result of its orbital motion.
Correspondingly, the electron generates by its spin a magnetic moment of
magnitude ฤง๐/2๐๐ (๐ is the electron mass, ๐ the velocity of light), and this also
communicates itself in ๐ก0 ๐ก0 , either parallel or in opposition, to any magnetic
moment already present.
To describe the electron spin as an angular momentum of the usual kind and
to associate with it an operator like L (eq.44) proved a fruitless undertaking, chiefly
because L would have more than two eigentatates. The most successful procedure
of including the spin in the quantum mechanical formalism, aside from Diracโs
relativistic treatment of the electron, is that of Pauli which will now be describe.
What follows will refer only to the spin states of a single electron some applications
to several electrons may be found in sec. 34 and 35.
Since the three space coordinates are insufficient to specity the complete
state of an electron, we introduce a fourth, the โspin coordinatesโ and denote it
by ๐ ๐ง. It corresponds, in classical language to the cosine of the angle between the
axis of the spin angular momentum and the Z-axis of coordinates. This visual
interpretation while in no way dictated by the mathematical formalism, will be
found a useful mental aid. Thus the state function of an electron has the form.
๐(๐ฅ, ๐ฆ, ๐ง, ๐ ๐ง)
Since in all that follows, the hypothetical spin coordinates ๐ ๐ง and ๐ ๐ฆ. Are
never needed, we shall hence forth delete the subscript ๐ง on ๐ but retain the above
interpretation. Hence ๐ = ๐(๐ฅ, ๐ฆ, ๐ง, ๐ ๐ง). Finally, it is well for the moment to
abstract attention entirely from the space dependent part of the wave function, i.e.,
to consider ๐ฅ, ๐ฆ, ๐ง as fixed, cocentraining our inquiry solely upon the electron spin.
Then ๐ = ๐(๐ )
If s, like x, y and z, were permitted to assume a continuous range of values,
difficulties would result. Pauli therefore postulates-in a manner admittedly ad hoc
and designed to force success of the theory-that the range of 8 consists of only two
points: s = ยฑ11 (classical meaning: spin vector is parallel or in opposition to Z). A
function of s is therefore defined only at these two points. The most general spin
function is, accordingly.
โ (๐ ) = ๐๐ฟ๐ +1 + ๐๐ฟ๐ โ1 (11-127)
Where the ๐ฟโฒs are Kronecker symbols. Our postulates involved certain
integrals over configuration space. But an integral over configuration space
consisting of two points vanishes. It becomes necessary to redefine the integral as
a summation over the two points:
โซ๐น(๐ )๐๐ โก ๐น(โ1) + ๐(1)
If โ (๐ ) is to be normalized
โซ(|๐| 2๐ฟ๐+1๐ +(๐โ๐ + ๐โ๐)๐ฟ๐ + ๐ฟ๐ โ)๐๐ = |๐| 2 + ๐| 2 = 1| (11-128)
In a very trivial sense, eq. (127) represents an expansion of a function (Ms) in a
complete orthonormal set of functions๐ฟ๐ +1 ๐๐๐ ๐ฟ๐ โ1. To what operator do these
two functions belong as eigenstates? The answer is suggested by intuition and will
be justified by its complete success; it is the operator๐ ๐ง, which is associated with
the observable: spin angular momentum along Z. We must now give thought to the
mathematical structure of this operator.
Empirical evidence cited in the introductory paragraphs demands that its
two eigenvalues be ยฑh/2. Hence it must satisfy the two equations
๐ 2๐ฟ๐ +1 =โ
2๐ฟ๐ + 1
๐ 2๐ฟ๐ โ1 =โ
2๐ฟ๐ โ 1 (11-129)
It is possible to show that no differential operator of the type encountered previously
can satisfy these equations without giving rise to an infinite number of other
eigenstates. But why search for the operator? The simplest point of View, and that
here taken, is to regard eqs. (129) as a definition ofthe operator Sz
To simplify the notation, and to be in accord with custom, we now introยป duce the
symbol a(s) for๐ฟ๐ +1, and๐ฝ๐ โ1. Furthermore, we define a new operator.
๐๐ง =2
โ๐ ๐ง
Which has eigenvalues ยฑ1, for the simple expedient to save writing. Then, in view
of (129),
๐ฟ๐ง๐(๐ ) = ๐(๐ ), ๐ฟ ๐ง๐ฝ(๐ ) = ๐ฝ(๐ ),
It is indeed possible and often useful to find an explicit operator in form of a matrix
which will satisfy these equations. This matrix is easily formed by means of the
principles outlined in sec. 17. Our eigenstates are ๐1 = ๐,๐2 = ๐ฝ, and we construct
(๐ฟ๐ง)๐๐ = โซ๐๐โ๐๐ง๐๐๐๐ with the integral replaced by a summation. We thus obtain
the two-square matrix
๐๐ง = (1 0
0 โ 1)
To let it operate on what was formerly the function (15(3) the latter has to be
regarded as a vector whose components are its expansion coefiicients:
If the function Q) is given by ๐(๐ ) = ๐๐ + ๐๐ฝ a and b being numbers, then the
vector ๐(s) is
๐ = (๐
๐)
Thus, in the matrix representation,
๐๐ง๐ = (1 00โ1)(๐๐)
And the reader will easily verify by the rules of Chapter 10 that the two 0
eigenvectors of ๐๐งare ๐ = (๐0)๐๐๐ ๐ =(0
๐)where the values of both a27And b must
be unity because of (128). The eigenvalues are, respectively, +1 and -1. But the
functions qt corresponding to the vectors (10) and (0
1) are clearly๐ ๐๐๐ ๐ฝ, which
takes us back to the scheme (130).
It is seen that there is a complete isomorphism between the two descriptions
of the operator SZ and its eigenstates ๐: One in terms of matrices and
eigenvectorsfโwhere the rule of operations is (132); the other in terms of linear
substitution operators and eigenfunctions, Where the rule of operations is (130).
The question now arises as to the structure of the operators Sx and Sy,
associated with the other two components of the spin.28In endeavoring to construct
them it is important to recall one significant fact concerning the ordinary angular
momentum L: its components do not commute With one another. In fact (see eq. 7)
LzLy -LyLx = ihLz, LyLz -LzLy = ihLz
LzLx LxLz = ihLy
Let us assume that the components of the spin S, this being an angular momentum
operator, must be subject to the same commutation rules. In terms of a" rather than
S, we postulate
๐๐ง ๐๐ฆ โ ๐๐ฆ ๐๐ง = 2๐๐๐ง; ๐๐ง ๐๐ฆ โ 2๐๐๐ฅ; ๐๐ง ๐๐ฅ โ ๐๐ฅ ๐๐ง = 2๐๐๐ฆ
27 An operator p is in general uniquely determined when the result of is action upon
each member of an orthonormal set of function is known. The method of defining
an operator is ordinarily not useful because an infinite number of relation like (129)
would be required.
These relations imply that an eigenstate of ๐ ๐ง or e.g., a (s) or ๐ฝ(๐ ) cannot be a
simultaneous eigenstate ๐ ๐ฅ or๐ ๐ฆ, (sec. 7).
The construction of๐๐ฅ, and๐๐ฆ, ๐๐ง being given, is more easily performed in the
matrix scheme. If we set ourselves the problem of determining two matrices am and
cry, which, when combined with ๐๐ง of eq. (131), obey (133), we easily find that the
answer is not unique. But certainly the solution
๐๐ง = (1 0
0 โ 1) ๐๐ฆ = (
0 โ ๐
๐ โ 0)
Is a possible one. The ambiguity here encountered permits just enough freedom to
make possible a rotation of coordinate axes (see Chap. 15).
Let us, then, accept (134) as our solution in matrix form. Clearly ๐๐ง
Has eigenvalues 5:1, eigenvectors โ1
2(11) andโ
1
2( 1โ๐) ; ๐๐ง has eigenvalues ยฑ1
eigenvectors โ1
2(1๐) andโ
1
2( 1โ๐)๐๐งthe observable values28 Of all three components
Sx, Sy, and Sz are there foreยฑโ/2. When these results are translated into the function
language they read as follows. The equation ๐(๐ ) = ๐๐(๐ )has two possible
(normalized) solutions:
๐ = 1, ๐(๐ ) = โ12 [๐(๐ ) + ๐ฝ(๐ )]
๐ = โ1, ๐(๐ ) = โ12 [๐(๐ ) โ ๐ฝ(๐ )]
}
The equation ๐๐ฆ๐(๐ ) (s) = ๐๐(๐ ) has two possible solutions:
28 While we need only one spin coordinate, Sz, all the components of the operator
must be be introduced because they appear in te Hamiltonian and other operators.
๐ = 1, ๐(๐ ) = โ12 [๐(๐ ) + ๐ฝ(๐ )]
๐ = โ1, ๐(๐ ) = โ12 [๐(๐ ) โ ๐ฝ(๐ )]
}
โI" The equation๐๐ง๐(๐ ) (s) = ๐๐(๐ ) has two possible solutions:
๐ = 1, ๐(๐ ) = ๐(๐ )
๐ = 1, ๐(๐ ) = ๐ฝ(๐ )}
If now we write the eqs (135a) in the simpler form
๐๐ฅ๐ + ๐๐ฅ๐ = ๐ + ๐ฝ, ๐๐ฅ๐ โ ๐๐ฅ๐ = ๐ โ ๐ฝ
And solve these by adding and subtracting, we find
๐๐ฅ๐ = ๐ฝ, ๐๐ฅ๐ฝ = ๐
The same procedure applied to (135b) and (135C) yields similar relations.
Summarizing these results: The operators๐๐ฅ , ๐๐ฆ, ๐๐ง may be represented either by
the set of linear substitutions
๐๐ฅ๐ = ๐ฝ, ๐๐ฅ๐ = ๐๐ฝ, ๐๐ฅ๐ = ๐
๐๐ฅ๐ฝ = ๐, ๐๐ฅ๐ฝ = โ๐๐, ๐๐ฅ๐ฝ = โ๐ฝ
Or by the matrices
๐๐ฅ = (1 00 1) ๐๐ฆ = (0 โ ๐
๐โ0), ๐๐ฅ = (
1 00โ1)
For practical use, the set of substitutions is to be preferred. Note that the
operators and satisfy the convenient relations
๐+๐ = 0 ๐โ๐ = ๐ฝ
๐+๐ฝ = ๐ ๐โ๐ฝ = 0
They are sometimes called โdisplacement operators.โ We return to the
consideration of the general state function of an electron, which includes ๐ฅ, ๐ฆ, ๐ง and
s as argument. Such a function may certainly be expanded in eigenfunctions of ๐๐ง
i.e
๐(๐ฅ, ๐ฆ, ๐ง, ๐ ) = ๐ + (๐ฅ, ๐ฆ, ๐ง)๐(๐ ) + ๐ โ (๐ฅ, ๐ฆ, ๐ง)๐ฝ(๐ )
Normalization now requires
โซ๐โ๐๐๐ โกโ๐โ๐๐๐ฅ๐๐ฆ๐๐ง
๐
= โซ(๐+โ ๐+ + ๐โ
โ๐โ) ๐๐ฅ๐๐ฆ๐๐ง = 1
The operators ๐๐ฅ๐๐ฆ๐๐ง do not act on ๐+and ๐โwhich are only. Function of ๐ฅ, ๐ฆ, ๐ง
in other words, they commute with space coordinates. Thus, for instance
๐๐ฆ๐(๐ฅ, ๐ฆ, ๐ง, ๐ ) = ๐๐ฆ๐+๐ + ๐๐ฆ๐โ๐ฝ = ๐+๐๐ฆ๐ + ๐โ๐๐ฆ๐ฝ = ๐๐+๐ฝ โ ๐๐โ๐
In the matrix scheme, ๐(๐ฅ, ๐ฆ, ๐ง, ๐ ) is represented by the vector
๐ = (๐+(๐ฅ, ๐ฆ, ๐ง, ๐ )
๐โ(๐ฅ, ๐ฆ, ๐ง, ๐ ))
In the sense of this analysis it may be said that the introduction of the spin in the
Pauli manner causes all Schrำงdinger function to become two-component functions.
Problem. Carry out the algebra involved in finding the two
Hermitian matrix (134).
11.30. Applications.
Atom in a Magnetic Field. Our interest here is not in a complete solution of this
problem, which may be found worked out in most books on quantum mechanics,
but in its silent mathematics features. We wish to find the energies of an electron
atom (e.g., hydrogen or, with good approximation, the alkalis) when it is placed in
a uniform magnetic field. The Hamiltonian consist of two parts, one acting on the
electrons space coordinates and one acting on the spin coordinate. The former will
be called ๐ป0 the letter is the โspin energyโ. If the magnetics field แผฏ is taken along
the Z-axis, the classical energy of a particle of magnetic moment ๐ would be๐.แผฏ =
๐๐ง.แผฏ๐ง. But empiricalt, the magnetic moment associated with the spin is (ฤง๐/2๐๐)๐
. We shall here write ๐ for the constantฤง๐/2๐๐. In quantum mechanical
transcription, then, the โspin energyโ is ๐แผฏ๐ง๐๐ง where ๐๐ง is interpreted as the
operator (130) or (131)
(๐ป0 + ๐แผฏ๐ง๐๐ง)ัฐ = ๐ฌัฐ 11-138)
Let
ัฐ(x, ๐ฆ, ๐ง, ๐ ) = ๐ + (x, ๐ฆ, ๐ง)๐ผ(๐ ) + ๐ โ (x, ๐ฆ, ๐ง)๐ฝ(๐ )
And substitute obtaining
๐ผ(s)[๐ป0 + ๐แผฏ๐ง โ ๐ธ]๐+ + ๐ฝ(๐ )[(s)[๐ป0 + ๐แผฏ๐ง โ ๐ธ]๐โ = 0
Provided relations (136) are used. Since โ ๐๐๐ ๐ฝ are linearly independent,
orthogonal functions of s, their coefficients in the las equation must separately
vanish. Hence we have
๐ป0๐+ + ๐แผฏ๐ง = (๐ธ โ ๐แผฏ๐ง)๐+ ๐ป0๐โ + ๐แผฏ๐ง = (๐ธ + ๐แผฏ๐ง)๐โ
Now let ๐ธ0 be an eigenvalue of ๐ป0 ฯ0 the corresponding eigenfunction. The first
of eqs., (139) (which is nothing more than an eigenvalue equation for the
operator๐ป0) then says ๐ฌ โ ๐แผฏ๐ง = ๐ธ0 or๐ธ = ๐ธ0 + ๐แผฏ๐ง๐+ = ฯ0. On substituting
this value of E into the second equation it reads ๐ป0๐โ=(๐ธ0 + 2๐แผฏ๐ง)๐โ and this
can only be satisfied by putting ฯโ = 0 because (๐ธ0 + 2๐แผฏ๐ง) is not an eigenvalue
of๐ป0. Thus we obtain as one solution of (138)
๐ธ = ๐ธ0 + ๐แผฏ๐ง ัฐ = ๐0(x, ๐ฆ, ๐ง) โ (๐ ) (11-140a)
But we can also start with the second of eqs. (139) and assume ๐โto be ๐0 ๐ธ +
๐แผฏ๐งto be ๐ธ0 . Then ๐+ = 0 and we have
๐ธ = ๐ธ0 โ ๐แผฏ๐ง ัฐ = ๐0(x, ๐ฆ, ๐ง)๐ฝ(๐ ) (11-140b)
How does the inclusion of the spin modify the eigenvalues and eigenfunction of the
Schrำงdinger equation when there is no magnetics field? The answer is obtained by
letting แผฏ๐ง vanish in (140b). Both values of E coalesce to ๐ธ0 which now represent
the ordinary Schrำงdinger energy in the absence of a field, but the functions ัฐ
remain distinct. The spin thus introduces a degeneracy into the Schrำงdinger
representation of states. Formulas (140) account- in a primitive way-for the
doubling of the alkali energy levels, the field แผฏ๐ง being caused in that case by the
electronโs orbital motion, and not by external agencies.
Problem. Solve eq. (138) by the method of separation of variables, i.e., by
putting ัฐ = ๐(x, ๐ฆ, ๐ง)๐(๐ ) and show that (140) is the solution obtained by that
method also.
b. A Spin Problem. Having shown how spin and coordinate functions cooperate in
the description of the states of an electron, let us omit further reference to space
coordinates and inquire. What are the energies which an electron, placed in a
uniform magnetic field of arbitrary direction, may assume regardless of its
translational motion. The only energy of interest is that due to the sin. Let แผฏ be
the magnetics field strength. The Schrำงdinger equation reads.
๐แผฏ. ๐๐ = ๐(แผฏ๐ฅ๐๐ฅ +แผฏ๐ฆ๐๐ฆ +แผฏ๐ง๐๐ง)ฯ(s) (11-141)
If แผฏ is taken along Z, the equation reduces to
๐แผฏ๐๐ง . ฯ(s) = ๐ธฯ(s) (11-142)
29
29 This can be seen explicitly if the equation is multiplied by either ๐ผ(๐ ) or ๐ฝ(๐ )
and the โintegratedโ over.
The operator on the left is but a constant multiple of oz and must therefore
have the same eigenfunctions as๐๐ง, i.e ๐ ๐๐๐ ๐ฝ. The corresponding eigenvalues are
at once seen to be E = ยฑ๐โ. We shall show that eq. (141) has the same eigenvalues,
but different eigenfunctions. Make the substitution ๐ = ๐๐ผ โ ๐๐ฝ(๐ ) in eq. (141).
On using, subsequently, relations (136) the result will be โ๐๐๐๐ก: 4 + ๐๐ โ
๐๐๐ค ๐๐) + 3โฌ๐ง < ๐๐ ๐๐} โ ๐ธ๐๐ + ๐ค) = 0
๐{โ๐ง(๐๐ฝ + ๐๐) โ ๐โ๐ฆ(๐๐ฝ โ ๐๐) +โ๐ง(๐๐ + ๐๐ฝ)} โ ๐ธ(๐๐ + ๐๐ฝ) = 0
As before, the coefficients of a and 6 may be put equal to zero separately, so that
๐(โ๐ฅ๐ โ ๐โ๐ฆ๐ โโ๐ง๐ = ๐ธ๐
๐(โ๐ฅ๐ โ ๐โ๐ฆ๐ โโ๐ง๐ = ๐ธ๐}
If the equations are to have solutions a, b, which are different from zero, the
determinant of the coefficients of a, b must vanish, whence E = ยฑ๐โ
On substituting E = +๐โ into the first of eqs. (143) and then taking the square of
its absolute value, we have
(โ๐ฅ2 +โ๐ฆ
2)|๐| 2 = (โ +โ๐ง) 2|๐| 2
Let us call the angle between โ and .the Z-axis,๐, so that 2(โ๐ฅ2 +โ
2 =
โ 2 ๐ ๐๐ 2 ๐ and โ๐ง = โ
๐๐๐ 2 ๐ . Furthermore, in View of (128), |๐| 2 = 1-
|๐| 2 . When these substitutions are made and the last equation is solved, the squares
of the absolute values of a, b are found to be ๐๐๐ 2 ๐/2 and๐ ๐๐ 2 ๐/2, respectively.
Let us then puta = cos๐/2, b = ๐๐๐ฟsin ๐ฟ/2, treating ๐ฟ as a phase constant. With
the further substitutions โ๐ง = ๐ป ๐ ๐๐ ๐ cos๐, โ๐ฆ = ๐ป sin ๐ ๐ ๐๐ ๐ where ๐ is the
azimuth of the field, we find from (143) that ๐ = โ๐
In a similar way, when ๐ธ = โ๐๐ป, ๐ฟ = ๐ โ ๐, ๐ =๐ ๐๐๐
2๐ = โ๐โ๐๐
๐๐๐ ๐
2
We conclude that eq. (141) has the eigenvalues ๐ธ1 = ๐๐ป, ๐ธ2 = โ๐โ and the
corresponding eigenfunctions
๐1 = cos๐
2. ๐(๐ ) + sin
๐
2๐โ๐๐ ๐ฝ(๐ )
๐2 = sin๐
2. ๐(๐ ) + cos
๐
2๐โ๐๐ ๐ฝ(๐ )
} (11-143)
Notice that, when the field โ is the reversed in direction (i.e ๐ โ ๐ โ ๐, ๐ โ ๐ +
๐), and ๐1 and ๐2 exchange their roles.
Problem. Solve eq. (141) by diagonalizing the matrix โ๐ง๐๐ง +โ๐ง๐๐ง +
โ๐ง๐๐ = (โ๐ง
โ๐ง + ๐โ๐ฆ
โ๐ง โ ๐โ๐ฆ
โ๐โ๐ง) and show that it leads to the same result.
THE MANY-BODY PROBLEM AND THE EXCLUSION PRINCIPLE
11.31. Separation of the Coordinates of the Center of Mass.
In classical mechanies, a system containing many particles and subject only to
internal forces behaves in such a way that its center of mass moves uniformly on a
straight line. As a corollary of this theorem every classical two-body problem may
be reduced to a one-body problem.30A similar fact may be proved in quantum
theory.
The Schrำงdinger equation for a system of n particles of masses ๐1. . . ๐2
reads:
(โโฤง2
2๐๐
๐๐ โ1
2 + ๐)๐ = ๐ธ๐ (11-145)
Where โ12=
๐2
๐๐ฅ๐2 +
๐2
๐๐ฆ๐2 +
๐2
๐๐ง๐2 the potensial energy, V, is to be regarded as a function
of the relative coordinates ๐ฅ๐ โ ๐ฅ๐, ๐ฆ๐ โ ๐ฆ๐ , ๐ง๐ โ ๐ง๐ we first transform to a new set of
coordinates, defined as follows:
๐ฅ =1
๐โ ๐๐๐ฅ๐๐1 ๐ = โ ๐๐
๐1 (11-146)
๐ฅ21 = ๐ฅ2 โ ๐, ๐ฅ3
= ๐ฅ3 = ๐ฅ3
โ ๐,โฆ . ๐ฅ๐1 = ๐ฅ๐
= โ๐
White similar relations for the y and z components. Note that ๐ฅ3 is missing; the
coordinates of one particle have been eliminated by the introduction of the center
of mass coordinates X, Y, Z. In computing the sum of the lapcacian operator
occurring in (154) in terms of the new coordinatos we observe:
๐๐
๐๐๐=๐๐
๐๐ฆ๐=๐๐
๐๐ฅ๐=๐๐
๐;๐๐ฅ๐
๐
๐๐ฅ๐=๐๐ฆ๐
๐
๐๐ฆ๐=๐๐ง๐
๐
๐๐ง๐= ๐ฟ๐๐ โ
๐๐
๐
Using these relations, simple differentiation yield
๐2๐
๐๐ฅ12 =
๐12
๐ 2(๐2๐
๐๐ฅ12 โ 2โ
๐2๐
๐๐ฅ๐๐๐ค +โ
๐2๐
๐๐ฅ12๐๐
๐ค
๐
๐=2
๐
๐=2
)
๐2๐
๐๐ฅ12 =
๐12
๐ 2(๐2๐
๐๐ฅ12 โ 2โ
๐2๐
๐๐ฅ๐๐๐ค +โ
๐2๐
๐๐ฅ12๐๐
๐ค
๐
๐=2
๐
2
)
+2๐๐
๐(๐2๐
๐๐ฅ๐๐๐ค-โ
๐2๐
๐๐ฅ12๐๐
๐ค) +๐๐=2
๐2๐
๐๐ฅ12+
๐2๐
๐๐ฅ12
30
And similar expression for the derivatives with respect to y and z. When these are
combined we obtain, in place of (145), the equation
EVzzyyxxMmM
n
ji jijiji
n
ii
2,
2222
2
22
22
''''''222'
(11-147)
Here 2 is the Laplacian with respect to the center of mass coordinates, 2'
i , with
respect to the primed coordinates. While V is not directly a function of the primed
coordinates, it may be expressed in term of them because ijij xxxx '' . A
difficulty might seem to appear in connection with 1xxi because ix' is absent
from the primed set. But it is easily seen that ii
n
xmxm '2
11 , whence
30 So long relativity effect are neglated.
jj
n
ji
ii xmm
xxx ''1 . Therefore V, when expressed in term of new coordinates,
will not contain X, Y, or Z.
As result, eq. (147) is separable; therefore may be written as
).''(),,( 2 nzxZYX
Correspondingly, 'EEE c , where Ec is energy associated with ),,( ZYX ,
determined by
cEM
22
2
This is Schrแฝdinger equation of a free particle of mass M, it produces, as
we know, no quantization. The remainder of (147) describes the internal motion
of the particles :
'22 2,
''2
2
'2
EVMm
n
ji
ji
n
i
i
(11-148)
It differs from the normal from Schrแฝdingerโs equation by presence of the terms in
''
ji and by the fact that V has a different functional form in the primed
coordinates than in the unprimed ones.
The coordinates (146) measure the position of the i-th particles relative to
the center mass. It also possible to use a less symmetrical but physically more useful
set coordinates, which is closely related to (146). If we put
,,,,
,1
113
'
312
'
2'
11
xxxxxxxxx
mMxmM
X
nn
n
i
n
ii
(11-149)
Thus measuring all coordinates relative to that one which has been eliminated ( 1x
), we obtain in the same manner the equation
EVMmM
n
ji
ji
n
ii
2,
''2
2
22
22
222'
(11-150)
This form is particularly useful when it is desired to calculate the energy of a many-
electron atom, for particle 1 may then be taken to be the nucleus and summations
in (150) are extended electrons. The equations remaining after separation of the
motion of the center of mass is now
'11
2 ,
''
12
22
' EVmm
n
ji
ji
n
i
Where m is the mass of an electron, 1m , that of nucleus. If may be written in terms
of the reduced mass.
1
1
mm
mm
As follows :
'22
''
1
22'
2
EVm ji
ji
i
i
(11-151)
The terms in the double summation play an important role in the isotope effect of
heavy atoms.31
They are present whenever the number of electrons is greater than one. For the case
of hydrogen, eq.(151) has the same form as Schrแฝdingerโs equation for a stationary
nucleus, except for the replacement of the electron mass by . Hence the true
energies of the hydrogen atom are not exactly given by eq.(64), but by that equation
with written from m. Note that the function V is different in (148) and (151), and
that the trms of the double summation have opposite signs. Nevertheless the
31 See Huges, A. L., and Eckart, C.,Phys. Rev. 36,694(1930)
equivalence of these two equations for the two-body problem may be seen as
follows. Write for the potential energy in (151)
),',','( zyxVV
Where
12' xxx , etc.
The V-function of (148) must then be expressed in terms ZzYyXx 222 ,, .
Now .2
1
2112 Xx
m
mmxx
Therefore we must use in (148)
,',','1
21
1
21
1
21
z
m
mmy
m
mmx
m
mmVV
And the equation reads
',','',','',','11
2
'2'
212
2
zyxEzyxzyxVmmm
Where
.1
21
m
mm If here we put ''',''',''' xzzyyxx it becomes :
','','',''''11
2
2
212
22
EzyxVmmm
Which is identical with eq. (151)
11.32. Independent System
Physical system are independent, or isolated from another, if Hamiltonian
operator of one contains no terms referring to another system. There is then no
interaction between them. Consider n independent systems, and let the coordinates
of the r-th system (including the spin coordinate) be symbolized by the single letter
qr. If its Hamiltonian operator is Hr its Schrแฝdingerโs equation will be :
)()( )()()(
r
r
i
r
ir
r
ir qEqH (11-152)
)(r
iE being the i-th eigenvalue of the r-th system.
The state function describing the entire assemblage of n systems will satisfy
the equation
nnn qqqEqqqHHH 212121 ,, (11-153)
To find its solutions we put n
n
n qqqqqq )(
2
)2(
1
)1(
21,
tentatively. Substitution in (153) and use of the fact that 1H acts only on 1q , etc.,
leads at once to the equation
EHHH
n
n
n )(
)(
)2(
)2(
2
)1(
)1(
1
Which show that each term )(
)(
r
r
rH
is separately a constant, say )(rE and that
the sum of all these constant is E. But if )(
)(
)(r
r
r
r EH
then
)(r must be one of
the energies )(r
iE , Therefore
)()2()1(
)(
2
)2(
1
)1(
21 ,
n
sji
n
n
n
EEEE
qqqqqq
(11-154)
This result in indeed what intuition would lead us to expect. For clearly the total
energy of a number of isolated system is the sum of the individual energies.
Furthermore, if 1w is the probability that system 1 be found at 21, wq that system 2
be found at 2q , then the probability that both of these statements be true
simultaneously is product 21ww . Hence the individual functions, whose squares
are these probabilities, must like-wise combine as factors.
This latter circumstance is dictated also by the time dependence of the
Schrแฝdingerโs states eq. (113). For only the product of the individual function
.,,,)2()1( )/()2()/()1( etcee tEitEi will have the
r
r tEi
e
)()/(
required in
Eti
n eqqq )/(
,,2,1 )( .
11.33. The Exclusion Principle.
When two independent systems occupy the energy states )1(
iE and )2(
jE
respectively, the combine system has an energy
)2()1(
ji EEE
And a state function
)2()( )2(
1
)1( qq ji (11-155)
We shall suppose for the moment that the individual states )1(
i and )2(
j are
non-degenerate. Then, unless there happen to be two energies )1(
lE and )2(
kE whose
sum is precisely the same as)2()1(
ji EE , the combined state (155) will also be non-
degenerated. This will generally be the case when the two systems are different in
a physical sense.
But if they are similar, e.g., both electrons or both hydrogen atoms, another
situation arises. We may then drop all superscripts in the description of the states,
and write (155)
)()(, 21
)2()1( qqEEE jiji (11-156)
This state is degenerate, although i and j are not; for if we interchange
the result a different function but not a different energy. This degeneracy,
which is peculiar to the description of any aggregated of similar systems, is known
as exchange degeneracy. Classically it implies that the energy of the total system is
unaltered when two individual constituents exchange place and spins.
In the more general case where iE has ig and jE has jg linearly
independent function associated with it, the number of ' s corresponding to E will
be, not ji gg but ji gg2 .
Returning to the case of non-degeneracy of i and j we not that the two
functions
)()(),()( 2121 qqqq ijIIjiI
Which are linearly independent, are equally good representatives of the state
in which ji EEE . Moreover, any linear combination of the two satisfies the
Schrแฝdingerโs equation for this value of E, and has just claim to be considered. Of
course, only two such combinations can be linearly independent. Let us then
consider the function
III ba
Where we shall assume 122 ba to assure normalization. On โexchangingโ the
two systems, III and III , hence the function above transforms itself
into
III ab
The numerical value of which for any given configuration 21,qq will in general be
different from III ba . Physically, this implies that the configuration which
result when the two systems exchange places has an altogether different probability
than the original, a consequence that is clearly objectionable.
However, among all linear combinations there are two which avoid this
dilemma. They are the symmetri32
combination
IIIs qq 2
1, 21
And the โ antisymmetric โ one
IIIA qq 2
1, 21
They are independent and indeed orthogonal; the first remains unaltered on
exchange of systems, the second change its sign. Both, therefore, yield probabilities
2 which are insensitive to exchange.
Consider now, not two, but n independent similar system, in states
,,, sji . The assemblage has energy sji EEEE , and is describe by the
state function
nsjin qqqqqq 2121, (11-157)
Where every permutation of the qโs among the ' s on the right will produce a new
function belonging to the same E, provided the subscripts, sji , are all different
(which we shall assume for the moment). Hence, if P np qqq 21, the function
which result from (157) when this permutation is made, then
p
ppn aqqq 21, (11-158)
32 A function is said to be symmetric with respect to a given operation if the operation leaves it
unchanged; it is said (in quantum mechanics) to be antisymmetric if the operation changes its sign
without altering it in any other way.
Where the pa are arbitrary constants, one for each permutation (arbitrary except for
the normalization condition), represents an acceptable state function for the energy
E. since there were originally n! linearly independent function, there will also be n!
linearly independent combinations of type (158)
Fortunately, most of these are uninteresting, for they cause
2
21, nqqq
To change when an exchange is made among any of the 'q s. There are certainly
two combinations, however, which preserve probabilities on exchange. One is
symmetrical, the order the antisymmetrical combination. The symmetrical ones is
formed by making all the coefficient pa in (158) equals :
p
pns nqqq 21
21 !, (11-160)
The antisymmetric one giving opposite signs to even and odd permutations (cf.
Chapter 15) :
nssss
njjjj
niiii
A
qqqq
qqqq
qqqq
n
321
321
321
21
! (11-160โ)
Which the reader will easily recognize as equivalent to the expansion (160).
It is to these functions, s and A , that we must confine our attention. Lest
the simplicity of our formalism obscure significant details, we recall that rq stands
for all coordinate of the r -th system. Thus, if the systems were electrons, )( rj q
would be an abbreviation for a combination of space and spin functions:
rrrrjrrrrj szyxszyx ,,,,
In the notation of sec. 29, and an interchange of rq and pq means that rx is to be
exchanged against rp zx , against rp zy , against pz and rs against ps .
There is no a priori way of deciding which of the two function, (159) or
(160), is preferable. But here the exclusion principle, early recognized by Pauli,
creates simplicity in a most effective way. It states that if the individual systems
belong to a certain class (see below), only antisymmetric functions may be used in
describing the assemblage. This principle is of the nature of postulate; it has not yet
been deduced from more fundamental axioms, although one might hope, from a
mathematical point of view, that this will prove possible.33
Why nature insist upon antisymmetric states for some and symmetric states for
other among its creatures is at present a puzzle.
The elementary systems which Pauliโs principle is known to apply are;
electrons, positrons, protons, neutrons, neutrinos, and mu-mesons; photons, on the
other hand, and several kind of meson, are described by symmetrical state functions.
Perhaps the most important consequence of the exclusion principle is this.
Suppose our assemblage consist of electrons, two of which are described by the
same function i (i.e., the functions are identical with respect to positional and spin
factors). The determinant (160โ) will then have two equal rows, and hence will
vanish. We may therefore say: two systems obeying the Pauli principle cannot be
in the same state. This fact governs the structure of atoms and molecules; each
electrons added to the shell of an atom must have its own set of quantum numbers.
33 A very searching and interesting examination of the principle in the light of other fundamental
issue has been given by Pauli, Phy. Rev. 58, 716 (1940)
The exclusion principle makes it impossible to distinguish two states which
differ only by an interchange of two constituent systems, a fact which has already
been noted.
Photons, which are described by the symmetrical function (159), may exist
in identical states, because that function does not vanish when two sets of indices
like i and j, contained in p become equal.
11.34. Excited States of the Helium Atom.
To show how the Pauli principle is applied we treat some of the excited
states of the helium atom. The letter is to be regarded as a simple assemblage of 2
electrons moving in the Coulomb field of the nucleus (and under their mutual
repulsion), hence the considerations of the foregoing section apply. However, in the
first part of our treatment we shall ignore both the electron spin and the exclusion
principle.
The Schrแฝdingerโs equation has already been given (eq. 90); it is
E
r
eHH
12
2
21 (11-161)
Where
i
iir
e
mH
22
2 2
2
If the term 12
2
re were absent the two electrons would be independent, and would
be a product of the form Eqq ji ,21 being ji EE . Moreover i and j
would be hydrogen eigenfunctions with atomic number Z = 2, for 1H and 2H are
Hamiltonian operators for a single electron in a Coulomb field. To retain the
notation of sec. 19 we shall now write u for the individual electron functions, so
that, in the absence of the interaction term,
222111 zyxuzyxu ji (11-162)
Function of this type will be used as variation function with the complete
Hamiltonian (161). Let us first give thought to the proper choice of the individual
functionu . The state corresponding to the lowest energy of a single electron is (ef.
eq. 67a)
0
221
3
0
3
10
2 ar
ea
u
(11-163)
We are writing here, in place of the single subscript i , the values of the two quantum
number 1n and 0l . The first excited state either
),(02020 YRu
Or
),(12121 YRu
The spherical harmonic is a constant, but 1Y is any linear combination of the three
function cos,cos 0
1
1
1 PeP iand ieP cos1
1 . It will be convenient to choose
the following normalized combinations
r
zPY
r
y
ePePiY
r
xePePY
x
ii
x
ii
x
4
3cos
4
3cos
16
3
4
3
sinsin4
3coscos
16
3
4
3coscos
16
3
0
1
1
1
1
1
1
1
1
1
And the define34
34 21R is given in eq. (65); its explicit form will not be needed here.
zz
yy
xx
YRu
YRu
YRu
YRu
212
212
212
02020
(11-164)
As the four independent, orthonormal functions describing the first excited state of
the one-electron system. The product (162) can be formed by combining 10u with
any one of the four functions (164); furthermore, the arguments can be interchanged
in each of the functions thus constructed. We are therefore concerned with the
following eight functions, each of which is a solution of eq. (161) with the term
12
2 re deleted, and belongs to the energy
HEa
eE 5
4
11
2
0
2
0
(11-165)
2121
2121
2121
2121
10282107
10262105
10242103
1020220101
uuuu
uuuu
uuuu
uuuu
zz
yy
xx
(11-166)
In writing them we have indicated the arguments 111 ,, zyx and 222 ,, zyx simply
by (1) and (2). A combination of these functions
8
1
a
Will be used as a variation function in the sense of sec. 20. The best energies of the
system are given by (97), and this reduces at once to the form (104) because the
are orthonormal and belong to the operator 21
0 HHH . The perturbing term is
12
2
'r
eH
The next step in solution of our problem is the calculation of the matrix
elements 222111
* ' dzdydxdzdydxH ji using functions (166), the details of which
may be left for the reader.35
Symmetry arguments may be used to show that
8877665544332211 '','','','' HHHHHHHH
And that only functions in the same line of (166) give non-vanishing elements.
Furthermore the volume element adopted in the evaluation of I (sec. 19) is
convenient in proving:
785634772211 '',';''' HHHHHH
Since the are real''
jiij HH . We are left, therefore, only with the following
matrix elements:
'2121'
'21'
2121'
21'
102
12
22
2
2
1034
12
22
2
2
1033
1020
12
22
20
2
1012
12
22
20
2
1011
Kduur
euuH
Jdr
euuH
Kduur
euuH
Jdr
euuH
xx
x
In a sense previously defined, (see sec. 11.21) J and Jโ are Coulomb integrals, K
and Kโ exchange integrals.
35 See Heisenberg, W., Z. Phy. 39, 499 (1926)
The determinant eq. (97) becomes
0
'00
'00
00'
00'
'00
'00
00
00
eJK
KeJ
eJK
KeJ
eJK
KeJ
eJK
KeJ
(11-167)
provided we write e for .0EE All elements not written are zeros. The determinant
has two single roots: ', 21 KJeKJe and two triple roots:
'',' 43 KJeKJe The perturbation 12
2 re may
Therefore be said to change the one unperturbed level 0E into four perturbed levels:
,40302010 ,,, eEeEeEeE as indicated qualitatively in the diagram (Fig. 6).
To find the functions corresponding to the eight roots e we must return to
equations (96) :
etceJaKa
KaeJa
eJaKa
KaeJa
0''
0''
0
0
43
43
21
21
On substituting 1e for e we find .0, 4312 saaaaa on substituting
2ee we find ,0, 4312 saaaaa and so forth. We thus obtain the set
of energies and normalized variation functions given in the first two columns of
Table 1.
TABLE 1
E
KJE 0 212
1 a Triplet
KJE 0 212
1 s Singlet
''0 KJE
87
65
43
2
1
2
1
2
1
a
a
a
Triplet
Triplet
Triplet
''0 KJE
87
65
43
2
1
2
1
2
1
s
s
s
Singlet
Singlet
Singlet
It now becomes necessary to include the spin into our analysis. To do this accurately
would require a modification of the Hamiltonian operator (161), for the magnetic
moments of the spinning electrons produce an interaction with the magnetic field
do their orbital motions and this interaction has not been included in (161). We shall
omit this spin-orbit interaction and refer the reader to the literature for the more
accurate treatment.36
In other words, we shall suppose that the Hamiltonian does not act on the spin
coordinates. The state function is then separable and appears as the product of an
orbital (any of the functions in the table) and a spin function, and the letter may be
taken as an eigenfunction of z for each electron. Let us consider these spin function
more closely. For the two electrons, we have four functions:
11111111 ,,, ssandssssss
These, however, do not have convenient exchange properties, for when 1s and 1s are
interchanged, the first and last remain unaltered, the second transforms into the third
two other, equivalent functions, which are symmetrical with respect to an exchange
of spin coordinates. They are, when normalized, 21212
1ssss and
21212
1ssss . We have in this way obtained four spin functions
)16811(2
1
;,2
1,
2121
3 212 21211 21
ssssA
ssssssss
The first three of which are symmetrical, only the last being antisymmetrical.
Furthermore, this set of function is orthogonal (and complete).
To include the spin we need only multiply each one of the functions in Table
1 by one of the spin functions 1to A , a procedure which yields 32 different
functions of position and spin coordinates. But here the exclusion principle effects
36 Condon, E. U., and Shortley, G. H., โThe Theory of Atomic Spectra,โ Macmillan Co. New
York, 1953
a great simplification. It says that only functions which are antisymmetrical when
all coordinates, i.e., position and spin coordinates, of the two electrons are
interchanged, are to be permitted. Hence a function of Table 1 which is symmetrical
can only be combined with A , and a function which is antisymmetrical only with
1 2, and .
3
Now the functions marked a in the table are antisymmetric; they can be
multiplied by any one of the three functions. Each of them corresponds,
therefore, to three states. For this reason the energy states ''0 KJE and
KJE 0 are said to be triplet states. If spin-orbit interaction had been included
in our calculation each of these levels would have appeared as three closely adjacent
levels, while the other energies, marked singlet, would have remained single.
It is true that the functions in Table 1 are only approximate solutions of eq.
(161). Nevertheless what we have said about their symmetry with respect to
exchange of electrons may be shown to hold rigorously. The structure of the helium
energy spectrum, and in particular the singlet-triplet character of the states, are
therefore correctly given by the simple theory of this section; the numerical values
of the energy levels will be in error.
The normal state of helium atom, whose energy was computed
approximately in sec. 19 of this chapter, is given in the present notation by
21 1010 uu , if we neglect the spin. It is clearly symmetrical and can only be
multiplied by A when the spins are introduced. Hence it is a singlet state. When the
helium atom is in a singlet state, its probability is very small, as may be shown by
an extension of the methods used in sec. 11.28. Hence triplet and singlet levels so
not โ combine, โ and helium may be said to have two distinct spectra, the triplet
spectrum, to which spetrocopist apply the term โ orthohelium โ spectrum, and the
singlet spectrum called โ parhelium โ spectrum.
Problem a. Instead of using the 8 functions (166) as linear variation
function s, star with the 32 functions obtained from (166) by multiplying each of
them by 1 2 3.,, A Show that, if these 32 functions are suitably arranged, the
determinant equation is a four-fold repetition of the one obtained above, and that it
yields the same result in regard to both energies and functions.
Problem b. the following spin operators for two electrons may be defined:
212121
2
2
2
2
2
2
2
1
2
1
2
1
2
21
2
21
2 zzyxyxxzyxzyx
zzz
where 1x is the operators z acting on spin coordinate ,1s etc. Show that
1 2 3.,, and A are all eigenstates with respect to both of these operators, in
particular that
0,8,8,8
0,2,0,2
2
33
2
22
2
11
2
33211
A
Azzzz
Are these result consistent with the classical interpretation according to which
1is the state in which both spin are parallel and along Z,
2is the state in which both spin are parallel and perpendicular to Z,
3is the state in which both spin are parallel and along to Z,
A is the state in which both spin are opposed and yield no resultant angular
momentum?
11.35. The Hydrogen Molecule,
One of the stumbling blocks of pre-quantum chemistry was the phenomenon of
homo-polar binding; it is impossible to explain on the basis of classical dynamics
the union of two hydrogen atoms to form a molecule. They only attraction which
two neutral structures like H-atoms could possibly exhibit was due to quadru-pole
forces, and these were known to be too weak to account for molecular binding. It
was shown by Heitler and London that the homo polar bond it caused by a typical
quantum-mechanical effect: the โexchangeโ of the two electrons. Its meaning will
be clear from the following discussion. The method of calculation37 to be employed
is a simple one which lays little claim to quantitative accuracy38 ,
but exposes the significant facts in a beautiful way. It is similar to the treatment of
the
2H ion, from which it differs by the presence of two electrons instead of one.
The coordinate system to be used will be clear from fig. 7; particles 1 and 2 are
electrons,
Fig. 11-7
A and B are the protons whose positions are regarded as fixed. In connection with
Fig. 7, we also wish to outline the use of a coordinate system and a volume element
which are very convenient in the numerical work involved in this problem.
The coordinate system for the two electrons will contain the six variables
,,,,,, 2112211 rBBA
2111
2211212
0
0
BRAARB
BBBrBB
The volume element ,21 ddd where
1111
2
11 sin dddAAd
37 Heither, W., and London, F., Z. Phy. 44, 455 (1927) 38 The most elaborate and accurate calculation, also employing the variational method was made
by James, H. M., and Coolidge, A. S., J. Chem. Phy. 1,825 (1933)
Now
11
22
1
2
1 cos2 RARAB
Whence
11111 sin22 dRAdBB
On eliminating 11sin d from by means of this last relation, we find
111111
1 ddBBdAA
Rd
The element 2d is obtained by writing down an expression similar to 1d but using
1B as base line:
2221212
1
2
1 ddBBdrr
Bd
Hence the product
211112122211
1 dddBBdrrdBBdAA
Rd (11-169)
Several similar volume elements can be constructed by the same method.
After this excursion, let us consider the Schrแฝdingerโs equation of the 2H
problem. It is
E
RrBABAe
mH
111111
2 121221
22
2
2
1
2
(11-170)
We endeavor to solve it by the method of linear variation functions, choosing as
constituents of the trial function simple but reasonable approximation to the correct
1d
. If H did not contain the last four items in the parenthesis multiplying 2e it would
simply be the sum of two hydrogen-atom Hamiltonian, and
21 BA uu
Where
0
2
0
1
21
3
02
13
0 2,1
B
B
A
A eaueau
Are hydrogen functions centered about A and B respectively. On the other hand,
in the terms RrBA 1111 1221 were missing from the parenthesis, H would
also be the sum of two hydrogen-atom Hamiltonians, but 21 AB uu . Both the
these โs are equally good approximations, and both must be included in the trial
function. Note that they differ with respect to an exchange of the electrons (or,
amounts in this problem to the same thing, the protons). Hence we adopt
2121 21 ABBA uucuuc (11-170)
As variation function in minimizing dH . As explained in sec. 20, the process
leads to the secular equations
0cc
0cc
2222221211
1212211111
EE
EE
(11-171)
And E given by
0
22222121
12121111
EE
EE
(11-172)
Here
21
2
12112
2221
22
11
112121
121
duudduuuu
dduu
BAABBA
BA
The letter integral is familiar from sec. 21, it is the quantity there called AB .
Hence
0
22
2
2112 ,3
1
R
e p
Next, we turn to
2111 1121 dduHuuu AABA
The 2 terms in H need not calculated; their effect upon 1Au and 2Bu is at
once obtainable from the differential equations which these function satisfy:
222
,112
2
22
2
2
1
22
1
2
BHB
AHA
uB
eEu
m
uA
eEu
m
In this way we find
R
eJJEH
2
11 '22
Where
1
1
22
21
1
1
222 121 d
B
ueddBuueJ A
BA (11-173)
And
1
1
22
21
12
222 121
' dB
uedd
r
uueJ ABA (11-174)
J is given in sec. 21, eq. (100), and Jโ has the value
64
3
8
1111' 22
2 e
R
eJ
Problems. Prove this result, using the system of coordinates and the volume
element (169).
Furthermore,
1122
As the reader will easily verify. In a similar way,
12
2
21
12122112 '22 R
eKEEH
Where
1
1
1
2 11 dBuueK BA (11-175)
And
21
12
1
12 2211 dd
r
uuBuueK BABA
(11-176)
The value of K is given in eq. (100), and
4'2'2ln6
3
13
4
23
8
25
5' 322
0
2
EiEi
ee
K
Where 5772,0 (Euler-Mascheroni constant),
22
2
123
1',
e
And xEi is an abbreviation for the exponential integral
xu
duu
exEi ,)(
Which is tabulated and discussed, for instance, in โTable of Sine, Cosine, and
Exponential Integrals,โ Federal Work Agency, New York, 1940.
Problem. Evaluate Kโ See in this connection, Sugiura, Y., Z. f. Phy. 45, 484
(1927).
The two roots of (172) are
1
'2'22
1
1
'2'22
1
21
2
12112
21
2
12111
KKJJ
R
eEE
KKJJ
R
eEE
H
H
(11-177)
Substitution into (171) shows that to 1E there corresponds them function
212112 21
1 ABBA uuuu
(11-178)
And to 2E the function
212112 21
2 ABBA uuuu
(11-179)
The energies and are plotted against R, the internuclear distance, in Pauling
and Wilson.39
It will be seen that has a minimum in the neighborhood of the experimental
internuclear distance of the 2H molecules; at this minimum is negative and
equal in other of magnitude to the experimentally known minimum which causes
the stability of the molecule. On other hand, is positive for all R, decreasing in
39 Loc. Cit., p. 344
1E 2E
1E
1E
2E
monotone fashion with increasing R.it, therefore corresponds to repulsion between
the atoms. Comparison of and shows the difference in their behavior as
function or R to be predominantly due to the presence of the K and Kโ integrals.
These would have been missing if electron had not been taken account of by
introducing the two functions constituting the of eq. (170). In that case also, there
would have been only one energy and not two. Now while (170) may be crude
approximation, the fact that two equivalent functions, differing only with respect to
electron exchange, will compose the correct solution of (170) is beyond doubt,
hence the qualitative aspect here obtained cannot be questioned. The integrals K
and Kโ are called exchange integrals.
Let us now included the spin and apply the Pauli principle. The spin function
are those already encountered in the helium problem, eq. (168). If the resultant
function is to be antisymmetrical, 1 , which is symmetrical in the position
coordinates of electrons 1 and 2must be multiplied by an antisymmetrical function
of the spins, of which there is only one, namely A . However, 2 may be multiplied
by one of the three functions 1 2 3or . It represent a triplet state while is
singlet.
To the energy , therefore, there correspond three times as many quantum
mechanical states as to 1E . From this fact may be drawn the conclusion that when
H-atoms approach they will, ceteris paribus, be three times as likely to repel as to
attract each other.
REFERENCES
To begin with source material, there are : Schrำงdingerโs charming volume
โWave mechanicsโ (Blackie and Son, London, 1932) which is a collection of this
epoch-making papers of 1926 and 1927; Heisenbergโs more popular โThe physical
Principles of the quantum theoryโ (Chicago University Press, Chicago, 1930); De
Broglie and Brillouinโs and Jordanโs โElementare Quantenmechanikโ (J. Springer,
Berlin, 1930) The foundations of the subject, both mathematical and philosophical,
1E 2E
1
2E
are treated most thoroughly but also most abstractly by Dirac in his โPrinciple of
Quantum Mechanicsโ (Clarendon Press, Oxford, Third Edition, 1947) and by J. v.
Neumann in โMathematische Grundlagen der Quantenmechanicโ (J. Springer,
Berlin, 1932)
General treatises are :
Condon, E. U., and Morse, P. M., โQuantum Mechanics,โ McGrew-Hill Book Co.,
Inc., New York, 1929.
Ruark, A. E., and Urey, H. C., โAtoms, Molecules, and Quanta,โ McGrew-Hill Co.,
Inc., New York, 1930
De Broglie, L., โTheorie de la Quantification,โ Hermann et Cie, Paris, 1932.
Frenkel, J., โWave Mechanics,โ Vols. I and II, Clarendon Press, Oxford, 1932,
1934.
โHandbuch der Physisk,โ Vol. XXIV, Parts I and II (numerous author), Julius
Springer, Berlin, 1933.
Pauling. L., and Wilson, E. B., โIntroduction of Quantum Mechanics,โ McGrew-
Hill Co., Inc., New York, 1935
Jordan. P., โAnschauliche Quantenmechanik,โ J. Springer, Berlin, 1936.
Kemble, E. C., โThe fundamental Principles of Quantum Mechanics,โ McGrew-
Hill Co., Inc., New York, 1937.
Dushman, S., โElements of Quantum Mechanics,โ John Wiley and Sons, Inc., New
York, 1938.
Sommerfeld, A., โAtombau und Spektrallinien,โ Vol. II, Vieweg und sons,
Braunschweig, 1939.
Mott, N. F., and Sneddon, I. N., โWave mechanics and its Applicationsโ Oxford
Press, 1948.
Schiff, L. I., โQuantum Mechanics,โ McGrew-Hill Co., Inc., New York, 1949
Bohm, D., โQuantum Theory,โ Prentice-Hall, Inc,. New York, 1951.
Slater, J. C., โQuantum Theory of Matter,โ McGrew-Hill Co., Inc., New York, 1951
Houston, W. V., โPrinciples of Quantum Mechanics,โ McGrew-Hill Co., Inc., New
York, 1951
Lande, A., โQuantum Mechanicsโ Pitman Publishing Corp., New York, 1951
A list books in which quantum mechanics is applied to special problems
follows.
Van Vleck, J. K., โThe Theory of Electric and Magnetic Susceptibilities,โ
Clarendon Press, Oxford, 1932.
Condon, E. U., and Shortley, G. H., โThe Theory of Atomic Spectra,โ The
Macmillan Co., New York, 1935.