concepts & problems in quantum mechanics-ii by manmohan dash
TRANSCRIPT
CONCEPTS & PROBLEMS IN QUANTUM MECHANICS-II
By Manmohan Dash
On Left; the electron of the Hydrogen atom observed for the first time. According to the wave function formalism the wave function is an amplitude. The square of this amplitude is “probability” per unit of some quantity. Probability; that the electron would be found in a given range of that quantity, eg location probability.
ψ
4 Top
Wave Function ψ (x, t);
A function of (x) does not
give expectation value of
variables like p, in a direct
way.
Where; p: momentum.
There is a need to define
“operators” for such
variables.
Slide 59, Part-I
For expectation value of
variables like (p) or their
function f(p);
That is; <p> and <f(p)>,
When wave function ψ = f(x, t).
Define operators for p.
Slide 60, Part-I
Equations like Newton’s
Laws, of classical world,
“exist in quanta world”.
These equations relate
expectation values of
variables such as x, p
and V.
Slide 61, Part-I
Heisenberg’s Uncertainty Relations;
expectation values of certain
variables not arbitrary with each
other;
e.g. < x > and <p> .
Their uncertainty ∆ and SD or σ
bear “inverse relationship”:
eg
2
~
x
x
p
p h
Slide 62, Part-I
Representative problems
We saw that;
<x> is “average location” of the Quantum, called as expectation value of the “location of the quantum”.
<x> is determined from wave function or amplitude ψ(x, t).
x|t)(x,|x x 2 d
How to know the expectation value of momentum of the quantum? Or, that of variable p. In “classical world”, a particle has a momentum p defined by its mass m and velocity v; p = m.v, that is, p is product of mass and velocity.
So we can define velocity < v > ;
xdt
d v,x||x x 2
d
What is the velocity of the quantum in a probabilistic interpretation of wave function?
In probability distributions; either discrete or continuous, in part-I, we defined a central tendency, called; mean or average or expectation value of a variable such as x; < x >.
What is location x, in probabilistic interpretation of wave function?
By using the “Schrodinger Equation (S)” and its complex conjugate (S*) to evaluate;
)()()( **
tt
**
t
2
t
*
SS
2
t
The probabilistic velocity leads to the probabilistic momentum;
x||x x 2 ddt
dm
dt
dmvmp
Lets take a step in that direction;
x)(x2
ix xm x
xx
*
x
2
t
*
dddt
dm
Given
Step 1,
Step 2, Integration-By-Parts;
Wave Function properties ;
After two integration by parts;
x|t)(x,|x x 2 d
x )( x2m
ix x x
xx
*
x
2
t
*
dddt
d
|xx
xx
)(x
)(x
) (x
b
a
b
a
b
a
fggdd
dfd
d
dgf
gfd
dg
d
dfgf
d
d
x ,0
x x x
*
d
m
i
dt
dv
Given < x >, after “integration by parts” two times; we have < p >;
Compare <x> and <p> ;
Thus <p> = , momentum; <p>, operator;
x x
x|t)(x,|x x
x
*
2
didt
dmvmp
d
x xi
x x x
*
*
dp
d
xi
xi
p
A quantum found at C, upon measurement !
Where was it located right before measurement?
3 philosophies; Realist, Orthodox, Agnostic
Orthodox view, or Copenhagen Interpretation; it was no where, location indeterminate prior to measurement, act of measurement brought it. Wave Function Collapsed at C.
Most widely accepted, among Physicists, most respectable view.
Experimentally confirmed, supported by Bell’s arguments.
The above figure is from DJ Griffiths
WAVE FUNCTION COLLAPSE
WAVE FUNCTION EVOLUTION
From http://www.mysearch.org.uk/
From DJ Griffiths
Agnostic view; we wouldn’t know, “how many angels on needle point”?
Rejected by Bell’s arguments. Lack of experimental support.
Realist or “hidden variable” view; It was somewhere: Present info not sufficient.
Deterministic. Not rejected by Bell’s arguments, but lack of experimental support.
The mass of a quantum is spread out. Wave Function of Copenhagen Interpretation allows such distribution.
Does an
expecation value
give an average
value of a set of
measurement on
same particle?
Ensemble Vs One Particle Average is not in the sense of mathematics but
physics. A value obtained on the first instance of measurement is indeterminate, prior to measurement ! Wave function collapses after a measurement, any further repetition of measurement, immediately, would give same value as obtained before.
If its the same particle, it must go back to the state ψ as it was prior to measurement, for any average to be calculated, by further measurement. Or an identically prepared ensemble of particles has to be taken, all particles in the state ψ.
Problem 1.6
Why can’t you do
integration by parts
directly on middle
expression in
equation 1.29,
pull the time
derivative over onto
x, note that partial
of x wrt t is zero,
and conclude that
time derivative of
expectaion value is
zero.
Why can’t you do integration by parts directly on middle expression in equation 1.29, pull the time derivative over onto x, note that partial of x wrt t is zero, and conclude that time derivative of expectaion value is zero.
So we see that the integration does not reduce to zero acording to the prescription in the problem. 1.6 !
p is now an operator, so any general variable is simply a replacement; by the operator of p Eg Q (x, p) has an expectation value given by; Classically; In probabilistic representation the operator of Kinetic Energy T is;
In classical mechanics all variables can be set as a function of location x and momenta p.
In probabilistic representation also, these variables can be represented through the x and p variables.
xi
p
x )
xi x,( p) x,( * dQQ
vm r L and 2
2
1
22
m
pvmT
2
22
2
2*
2
x2m-
x x
2
- T
T
d
m
Problem 1.7
Now let us discuss the Ehrenfest Theorem
as we have pointed out earlier. Let us prove the theorem. Problem 1.7
Problem 1.7, from slide; 21
Problem 1.7, from slide; 21
Suppose you add a constant V0 to the potential energy (constant: independent of x, t). In classical mechanics this does not chanage anything. But what happens in Quantum Mechanics?
Show that the wave funcion picks up a time dependent phase factor given below, what effect does this have on expectation value of dynamic variables?
) t/iV- ( x 0 pe
Problem 1.8
Lets see how to do Problem 1.8
Remaining part of Problem 1.8
Heisenberg Uncertainty relation
There is a fundamental way in which a classical wave shows us that any precision we have for location of a point on the wave comes from the fact that waves can be localized packets.
In that case the wavelength and consequently the momentum of the wave become very badly spread.
When we have a precise wavelength and momentum (monochromatic wave) its quite clear we wouldn’t know which location-point of the wave would give a precise location of the wave.
All the locus of the wave would suffice and we would lose any sense of precise location.
Heisenberg Uncertainty relation
Heisenberg Uncertainty relation
This purely classical wave property
thus transpires to the probabilistic systems that we have been discussing so far.
We also note that momentum and wavelength of a Quantum are related as follows, this is called as de-Broglie Relation; p = h/λ = 2πћ/λ
Thus in our probabilistic interpretations the expectation values and consequently the spread or error or uncertainties given by standard deviations of the distributions of certain variables, bear an inverse relation with each other.
This is called Heisenberg Uncertainty Relation which we will discuss in more rigor later. For now;
2x
p
Lets discuss last problem of this presentation which exemplifies some of the ideas we have discussed so far including Heisenberg’s inequality of last slide.
Part (a)
Part (b)
Part (c)
Part (c)
Part (d)
So far we have discussed 9 problems in the last two presentations and discussed in depth the ideas of Quantum; such as Schrodinger Equation, Philosophy of Quantum reality and Statistical interpretation, Probability Distribution, Basic Operators, Uncertainty Principle.
In the next lecture, we will discuss further problems that will put us in a sound situation as regards a basic footing in an introductory non relativistic quantum mechanics.
If you enjoyed the last two presentation styled lectures, leave me any note, feedback, errors to [email protected] , feel free to share this among students and teachers ,you know who could benefit from this. Also you can visit my website ; mdashf.org !!