a simplified genesis of quantum mechanics
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Studies in History and Philosophy of Modern Physics40 2009) 151-166
Studies in History an d Philosophy
of Modern Physics
A simplified genesis of quantum mechanics
Olivier Darrigol
CNRS
Rehseis Paris rance
RT ICLE
IN FO
Articlehistory
Received 18 December 2008
Received in revised form
5 March 2009
eyword
History of quantum theory
S T R C T
The bewildering
complexity
of
th e
history of quantum
theory
tends to
discourage
its us e as a
means
to
understand or t ea ch t he f ou nd at io ns of quantum mechanics. The p re se nt p a pe r is an attempt at
simplifying this history so as to
make
it more helpful to physicists
an d
philosophers. In particular,
Heisenberg s
notoriously difficult derivation of t h e f u nd a me n ta l e q ua ti o ns of quantum mechanics, or
later derivations of i ts statistical interpretation are rep laced with s h or t er a nd m o r e direct arguments to
th e
same purpose.
s th e implied amputations and
distortions do no t
i m pl y m a jo r anachronisms,
they
should facilitate
the grasping of th e
main
historical steps without
excluding
a reasonable assessment of
their historical or l ogical necessity.
©
2009 Elsevier Ltd. Ail
ri gh t s reserv ed.
When citing this paper, please us e th e full journal title Studies in History and Philosophy of Modem Physics
1. Introduction
Quantum mechanics is a difficult theory, th e history of which
is even more difficult. The genesis of this theory spans more
than
a quarter of a ce ntur y. It im pl ie s a baffling v ar ie ty of phys ical
problems including blackbody radiation, atomic collisions, atomic
an d
mol ec ul ar spe ct ra , o pt ic al d ispe rsio n, the p ho to -e le ct ri c
effect, th e interaction of X-rays with matter, the low-temperature
b eh av io r of sol id s an d gases, a to mi c struc tu re , an d chemical
periods. It implies a complex socio-institutional
structure with
at
least four different poles in Copenhagen, Munich, G6ttingen, an d
Berlin, w it h d en se epistolary networks, and w ith
tw o
distinct
tr ends of res earc h leading to two diff erent forms of
quantum
mechanics. It is often highly technical: for instance, it implies th e
d ec ip he ri ng of i nt ri ca te s pec tr a, an d it relies on advanced
methods of celestial mechanics. It involves enormous conceptual
difficulties bound to the fail ure of c la ssic al i nt ui ti on s of mot io n
an d i nt erac ti on . It is impregnated with th e sub tl et ie s of Niels
Bohr s philosophy an d methods.
To m os t his tor ians of physics, this complexity makes
th e
history of
quantum
mec ha ni cs a fasci na ti ng topic. They see it as
an opportunity to observe the con str uct ion of a t he or y in s lo w
motion,
an d
to compare two strikingly different scenarios
converging to a common finale. In c on trast, p hy si ci st s an d
p hi lo so ph ers can only d ep lo re the o pa ci ty
that
results from this
complexity.
Quantum
m ec ha nic s be in g a not or iou sl y s tr an ge
E maii
address: darr [email protected]
theory, they w is h t he y ha d a simpler history that c ou ld h el p
them
in t he ir t hi nk in g an d teaching. The p re se nt p ap er is an
attempt at suc h a simpl ifie d ge nesis, from Plan ck s quantum to
Dirac s transformation theory.
The simplification I have in mind implies
th e
selection of
significant events an d p ro ce sses, as well as
th e
occasional
substitution of more d irec t rea so ni ng for u nn ec essa ri ly c ompl i
cated reasoning. It does not imply any arbitrary invention, an d it
avoids common misconceptions about the origin of
quantum
discontinuity or about the m ea ni ng of th e correspondence
principle. I have selected a few important s te ps , in s uc h a manner
that
any given
step
can be
seen
as a c on se qu en ce of the
anterior
steps in a given situation, and not so much of the silenced
developments. I only
mention
false t ra il s to the e xt en t that their
termination n arro we d c on st ru ct iv e p ossi bi li ti es. The resul t is a
double-branched history leading to th e
matrix
an d wave forms of
quantum
mechanics. I have simplified each
step
in three manners:
by e li mi na ti ng
r ed un da nc y a nd
selecting
the most
telling
a rg um en ts , by a ll evi ati ng th e notation without significant
anachronism), and by replacing some convoluted reasoning with
more direct an d more transparent reasoning that could still have
been done at the time.
In the same spirit, the Cottingen physicist Friedrich Hund long ago provided
a very l uci d hist ory of quantum theory Hund, 1967, 1974), as well as a h is to ry
based outline of quantum mechanics Hund, 1967, 1974, Appendix). The present
a tt em pt is i nt er me di at e in l en gt h and purpose. T he kind of simplification is
diff erent: w hereas H und appreci at ed t he essenti als of t hi s hist ory as a f or mer
participant, I have judged from the historians methodic studies. I thank jiirgen
Rennfor making me aware of the significance ofHund s appendix.
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The latter kind of simplification is especially important,
because it eliminates stumbling blocks in understanding the last
stages of the formation of quantum mechanics. For example, in his
derivation of what would later become the commutation rule of
quantum mechanics, Heisenberg used a clever but unnatural
procedure that can be replaced by a simple appeal to Bohr’s
frequency rule. Another example is the intricate, multi-step, and
multi-component derivation of the statistical interpretation of
quantum mechanics, which I have replaced with two simpleprocedures: one borrowed from Dirac and requiring the Heisen-
berg picture only; the other being an extension of Born’s
treatment of the scattering problem in the Schrodinger picture.
What are the advantages and drawbacks of such simplifica-
tions? Undoubtedly, they run against basic principles of historical
writing according to which linear, great-men accounts should be
avoided, the diversity of contexts and approaches should be
emphasized, and the intricacy of the historical issues should
be respected. I agree with my colleagues about the importance of
respecting these principles in a full history of quantum mechanics.
Fortunately, much work has already been done in this direction,
and much is still being done by competent scholars. I do not
doubt that this effort is necessary to a proper understanding of
the relevant cognitive processes. I nonetheless hope that the
simplified genesis can serve a few legitimate purposes: clarify
conceptual connections, convey genuine features of the genesis of
quantum mechanics to physicists and philosophers, naturalize
quantum mechanics by capturing the unavoidability of some of its
features, and ease a historical approach to foundational issues.
The second and third sections of this paper are accounts
of the genesis and interpretation of the matrix and wave forms
of quantum mechanics. (Regarding the interpretation, these
accounts are confined to the rules for applying the formalism to
conceivable experiments; they do not include the more philoso-
phical and more controversial interpretation of these rules as
found in Heisenberg’s uncertainty paper and in Bohr’s comple-
mentarity.) The third section briefly indicates how a fuller history
would depart from my simplified genesis. In the fourth and last
section, I discuss the necessity of quantum mechanics in the lightof the simplified genesis. For the sake of brevity, I have limited the
bibliography of secondary literature to a few representative
works.2
2. The quantum approach
2.1. Blackbody radiation and the failure of ordinary electrodynamics
In the second-half of the nineteenth-century, electromagnetic
radiation was known to reach a well-defined state of thermal
equilibrium by interaction with matter. This state is the so-called
blackbody radiation, which can be observed within a uniformly
heated cavity with absorbing walls. As a consequence of a
thermodynamic theorem established by Gustav Kirchhoff around
1850, the spectrum of this radiation is universal: namely, the
energy per unit volume has a well-defined (continuous) distribu-
tion over frequencies. By the end of the century, this distribution
was empirically known to decrease exponentially for high
frequencies. In addition, the joint application of (macroscopic)
thermodynamics and electrodynamics implied theoretical restric-
tions on the form of this distribution (Wien’s displacement law,
and the Stefan–Boltzmann law) that were well verified by
experiments. It was therefore hoped that the theoretical study
of the electromagnetic interaction between material radiators and
radiation would yield the form of the universal spectrum.3
In the years 1905–08, this hope was destroyed by a series of
proofs that the interaction of thermalized matter with electro-
magnetic radiation yielded an absurd distribution for which the
spectral energy density diverged quadratically for high frequen-cies (the so-called Rayleigh–Jeans law). The first assumption of
these proofs was that the interaction between matter and
radiation obeyed the laws of the Maxwell–Lorentz theory of
electrodynamics. The other assumption was some sort of
ergodicity: the interaction had to be such that almost every
initial condition of the global micro-system (radiation plus
material entities) would lead to the same macroscopic behavior
in the long run. The three main proofs differed substantially in the
details. Albert Einstein’s proof of 1905 relied on the random
interaction between cavity radiation and thermalized resonators;
James Jeans’s proof of 1905 rested on Maxwellian statistical
mechanics applied to a gas interacting with radiation; Hendrik
Lorentz’s proof of 1908 was based on the application of Josiah
Willard Gibbs’s statistical mechanics to an electron gas interacting
with cavity radiation. The convergence of these three proofs
increased the plausibility of their puzzling conclusion.4
The first and perhaps most convincing of these proofs,
Einstein’s, goes as follows. Consider a set of linear electric
oscillators with a broad, quasi-continuous range of frequencies,
and suppose that these oscillators interact with the radiation
included in a cavity with mirroring walls and also with a gas that
has the well-defined temperature T . According to Kirchhoff’s
radiation theorem, the final spectrum of the radiation does not
depend on the thermalizing entities and should therefore be
identical with the universal blackbody spectrum. From measure-
ments of the specific heats of solids at moderate temperatures, the
equipartition of energy predicted by Maxwell and Boltzmann was
known to hold for the degrees of freedom of the interacting gas
and oscillators. Consequently, each oscillator should have the(time-) average energy
U ¼ kT , (1)
where k is Boltzmann’s constant. Call un dn the energy per unit
volume of the radiation whose frequency is comprised between nand n+dn. According to Max Planck, the interaction between one
of the resonators and the surrounding radiation implies the
relation
un ¼ ð8pn2=c 3ÞU (2)
between the average energy U of the resonator and the spectral
density un of the radiation at the frequency n of the resonator.
Consequently, the equilibrium spectrum of the radiation should
be given by
un ¼ ð8pn2=c 3ÞkT , (3)
ARTICLE IN PRESS
2 There are two major treatises covering the whole history of quantum theory:
Jammer (1966), Mehra & Rechenberg (1982–1987), henceforth abbreviated as CD
(conceptual development ) and HD (historical development ). For a brief history and a
bibliography, see Darrigol (2003). A history of the constructive role of classical
analogies is found in Darrigol (1992), henceforth abbreviated as CQ ( from c-
numbers to q-numbers).
3 Cf. Kuhn (1978); CQ , part A.4 Cf. Kuhn (1978). Earlier derivations of the Rayleigh–Jeans distribution by
Lord Rayleigh in 1900 and Lorentz in 1903 were only meant to apply to low-
frequency radiation. Although Jeans in 1905 and Lorentz in 1908 removed this
limitation, they speculated that an exceedingly long time might be needed for the
thermal energy transfer from matter to high-frequency radiation. Their critics
noted that in this case the observed blackbody spectrum would no longer be a true
state of equilibrium, so that its universality and its compliance with thermo-
dynamic laws would be very difficult to understand. Lorentz soon recognized the
helplessness of the situation. Jeans did the same a couple of years later.
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in contradiction with the observed blackbody spectrum and with
the obvious requirement that the total energy should be finite.5
A derivation of Planck’s relation (2) is given in an appendix.
This derivation may be criticized for assuming the definiteness,
smoothness, and isotropy of the distribution of the radiation over
various modes without proof.6 However, the validity of these
assumptions is strongly suggested by the empirical definiteness
and universality of the blackbody spectrum.
2.2. Quantum derivations of the blackbody law
In 1900, Planck obtained by obscure theoretical means a
blackbody law that fitted experiments excellently. In 1906,
Einstein derived this law by assuming that blackbody radiation
was in equilibrium with electric resonators whose energy was
restricted to be an integral multiple of the quantum hn, wherein h
is Planck’s constant and n the frequency of the resonator. The
following is a slightly modified version of his considerations.7
Consider a harmonic oscillator with the frequency n, the mass
m, and the quadratic Hamiltonian
H ðq; pÞ ¼ p2=2m þ 2p2mn2q2, (4)
and suppose that this resonator is in contact with a thermostat atthe temperature T.
The Gibbs–Boltzmann distribution law for the energy of this
oscillator yields the average energy
U ¼
R H eH =kT dq d pR eH =kT dq d p
. (5)
Owing to the quadratic character of the Hamiltonian, the surface
in the (q, p)-plane comprised between the ellipses H ( p, q) ¼ E andH ( p, q) ¼ E +dE does not depend on the value of E . Consequently,
the average energy of the resonator may be rewritten as
U ¼
R 10 E eE =kT dE
R 1
0 eE =kT dE , (6)
with the result
U ¼ kT . (7)
Now assume with Einstein that the energy of the oscillator is
restricted to the discrete values nhn. The natural discrete
counterpart of Eq. (6) is
U ¼
P1n¼0nhnenhn=kT P1
n¼0enhn=kT ¼
hn
ehn=kT 1. (8)
Together with relation (2), this yields
un ¼ 8phn3
c 31
ehn=kT 1, (9)
which is Planck’s law. As Einstein did not fail to see, a weakness of
this derivation is that the quantization of the resonator contra-dicts the classical derivation of relation (2).
This argument by Einstein was the first intimation of a sharp
concept of quantization, according to which the states of a
microphysical entity are restricted in a discrete manner depend-
ing on the quantum of action. Planck preferred to assume that the
states of the resonator with an energy differing by less than a
quantum counted only as one state in the combinatorial
computation of the resonator’s entropy. By the Solvay congress
of 1911, most experts agreed that some discontinuity had to be
introduced in the dynamics of microphysical entities in order to
save the phenomena (blackbody radiation and also the low-
temperature behavior of specific heats), although there was much
variety of opinion about the manner in which this discontinuity
should be introduced and on whether it should be reducible to
some underlying mechanism of a more familiar kind.8
Einstein’s discrete quantization was most daring, as it made itdifficult to understand how electromagnetic radiation could
interact with quantized resonators, unless radiation itself had a
discontinuous structure. As we will see in a moment, Einstein had
already speculated on radiation quanta. Whereas this speculation
long remained marginal, the discrete quantization of the states of
material entities gained ground in the early 1910s: a few theorists
including Arthur Erich Haas, William Nicholson, Niels Bjerrum,
and Niels Bohr began to use the quantum of action for the purpose
of discrete selection among the possible states of classical models
of atoms or molecules. This selection was supposed to determine
the normal state of atoms and also to explain the discrete
character of atomic or molecular spectra. The remarkable success
of Niels Bohr’s attempt in this direction largely contributed to
establish discrete quantization for matter.9
2.3. The frequency rule
Scattering experiments performed in Ernest Rutherford’s
Manchester laboratory in the early 1910s suggested that atoms
were made of a central positive nucleus with a few electrons
orbiting around it. In 1912–13, Bohr tried to model atoms as series
of concentric electronic rings the filling of which corresponded to
the chemical periods. This model being inherently instable (both
mechanically and radiatively), he used Planck’s quantum of action
to select rings endowed with a special, classically unaccountable
kind of stability. The precise way he did that in his famous trilogy
of 1913 was inspired by Balmer’s formula for the visible spectrum
of the hydrogen atom, which is the m ¼ 2 case of the more generalRydberg formula:10
nmn ¼ K 1
m2
1
n2
, (10)
where nmn is the frequency of the observed lines, m and n are two
integers, and K is the so-called Rydberg constant.
In analogy with the quantization of a harmonic oscillator, Bohr
assumed that the single electron he admitted in the hydrogen
atom could only exist in a series of ‘‘stationary states’’ determined
by a rule of the form:
E n ¼ anh¯ nn, (11)
where E is the binding energy of the electron, ¯ n its orbital
frequency, a a numerical constant, and n a positive integer. Bohr
further assumed that ordinary mechanics applied to the motion of the electron in the Coulomb field of the nucleus. The resulting
Kepler motion satisfies the relation
ðE =¯ nÞ3 ¼ p2me4=2 ¯ n, (12)
if m denotes the mass of the electron and e its charge in
electrostatic units. Together with the quantum rule (11), this
relation implies the quantized energy values
E n ¼ Kh=n2; with K ¼ p2me4=4a3h3. (13)
ARTICLE IN PRESS
5 Einstein (1905), which also has the lightquantum. Cf. Klein (1963); CD, chap.
1.3; HD , vol. 1; Buttner, Renn, & Schemmel (2001).6 Planck’s electromagnetic H-theorem provides such a proof, but only at the
price of the hypothesis of ‘‘natural radiation.’’ Cf. CQ , chap. 3.7 Einstein (1906). Cf. Klein (1965). Einstein originally discretized the micro-
canonical expression of the entropy of a resonator. He used a discretization of the
canonical distribution (as done below) in Einstein (1907).
8 Cf. Einstein (1911); Kuhn (1978); Barkan (1993).9 Cf. Heilbron (1964, 1977).10 Bohr (1913). Cf. Heilbron & Kuhn (1969), Pais (1991, chap. 8), CD , chap. 2;
HD, vol. 1, chap. 2; CQ , chap. 5.
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The Rydberg formula can then be rewritten as
hnmn ¼ E n E m. (14)
Bohr found that the choice a ¼ 1/2 yielded an accurate value of
the Rydberg constant K .
From these formal considerations, Bohr inferred that the
emission of a spectral line involved two stationary states, and
that the frequency of this line depended on the energy of these
two states according to the frequency rule (14). This was a verydaring assumption as it contradicted the classical identity
between radiation frequency and orbital frequency. Yet it is
difficult to imagine any plausible alternative in the context of
discrete quantization. One could try (as Fritz Hasenohrl did in
1911)11 to identify the spectral frequencies with the orbital
frequencies of discrete states depending on two indices. This
would be artificial, however, as there would be no natural way to
relate radiation to change of state.
In favor of the frequency rule, Bohr could have argued that it
resulted from energy conservation applied to the emission of a
lightquantum. He did not do so, however, because he agreed with
most of his contemporaries that Einstein’s lightquantum was
incompatible with the well-established wave properties of
radiation. In Bohr’s view, Maxwell’s theory of free electromagneticradiation was necessary to the very definition of the concepts
necessary to define the properties of emitted and absorbed
radiation. Lightquanta being too paradoxical, Bohr rather left the
coupling between continuous radiation and quantum jumping in
the dark.12
Despite this inherent incompleteness, Bohr’s theory quickly
achieved new successes that enhanced its credibility. It correctly
predicted that some spectral lines originally attributed to
hydrogen had to be ascribed to traces of ionized helium. It gave
a handle on the X-ray spectra of higher elements. And it correctly
explained the inelastic collisions between electrons and mercury
atoms observed by Philipp Frank and Gustav Hertz. This last
success was especially important as it indirectly confirmed
the strange frequency rule: the energy communicated by the
electrons to the mercury atoms turned out to be identical to the
frequency of the resonance line of these atoms multiplied by
Planck’s constant.13
Yet, until 1915 Bohr did not believe in the generality of his
frequency rule. For instance, he ascribed the splitting of spectral
lines in magnetic and electric field to a correction to this rule. In
1916, two masterful contributions to his theory proved the
generality of the frequency rule. Einstein showed that simple
probabilistic assumptions on the relation between the quantum
jumping of the atoms of a gas and the intensity of the surrounding
radiation led to Planck’s blackbody law if and only if the frequency
rule applied to the radiation emitted or absorbed during the
quantum jumps. Arnold Sommerfeld and his collaborators
showed that relativistic, electric, and magnetic perturbations of
the Kepler motion in the hydrogen atom could be quantized insuch a manner that the frequency rule applied generally.14
In 1918, Bohr clearly formulated the two ‘‘fundamental
assumptions’’ of his theory:
(1) that an atomic system can exist permanently only in a certain
series of states corresponding with a discontinuous series of
values of its energy, and that any change of the energy of the
system including absorption and emission of electromagnetic
radiation must take place by a transition between two such
states. These states are termed ‘‘the stationary states’’ of the
system.
(2) that the radiation absorbed or emitted during a transition
between two stationary states is ‘‘unifrequentic’’ and pos-
sesses a frequency n, given by the relation E 0–E 00 ¼ hn, where h
is the Planck constant and where E 0 and E 00 are the values of
the energy of the two states under consideration.
The first assumption is the existence of stationary states, the
second is the frequency rule. Bohr regarded them as the
unshakable pillars of his theory. They were indeed more directly
related to experiments than other assumptions of his theory.
Until at least 1925, they remained the two basic postulates of
the quantum theory, despite the vicissitudes of most other
assumptions. From the beginning, Bohr was not sure to which
extent ordinary mechanics should apply to the motion in
stationary states. He was more confident that this motion could
be represented by well-defined trajectories, whatever the appro-
priate mechanics might be. It should be emphasized, however,
that his postulates did not depend on this assumption. As we will
see in a moment, they survived the failure of the orbitalrepresentation in the years 1924–25.15
2.4. The correspondence principle
In 1913, Bohr noted that his assumptions, despite their evident
incompatibility with ordinary electrodynamics, were able to
reproduce the predictions of this theory in the limiting case for
which the quantum numbers are very high and the quantum
jumps are very small. Specifically, the frequency of the light
emitted during a transition from the n state to the nt state
approaches the frequency of the t harmonic component of the
motion in theses states when n is very large. This convergence
only holds if the constant a in the quantum rule (11) is exactly
one-half. Bohr originally used this argument to consolidate thischoice of the value of a in his theoretical expression of the
Rydberg constant. As he further noted in 1914, the asymptotic
form of the energy of the stationary states can in fact be derived
from the asymptotic agreement between classical and quantum-
theoretical spectrum. The argument goes as follows.16
The asymptotic agreement between classical and quantum-
theoretical spectrum requires
E n E ntth ¯ nn. (15)
Since the large number n can be regarded as a quasi-continuous
variable, this is equivalent to
dE ndn
h ¯ nn. (16)
Differentiation of relation (12) for the Kepler motion yields
dðE n= ¯ nnÞ ¼ dE n=2 ¯ nn, (17)
so that
dðE n= ¯ nnÞ ðh=2Þdn, (18)
which implies the asymptotic validity of the quantum rule
E n ¼ nh ¯ nn=2 (19)
and of the resulting energy spectrum. This sort of argument can be
used to guess the quantum rule for other periodic systems, as
Bohr soon did.
ARTICLE IN PRESS
11 Cf. Heilbron (1964, p. 211).12 On Einstein’s lightquantum and its difficult reception, cf. Klein (1963, 1964);
Wheaton (1983).13 Cf. CD, chap. 2; HD, vol. 1, chap. 2; Pais (1991, chap. 8); Heilbron (1974).14 Einstein (1916, 1917); Sommerfeld (1916); Schwarzschild (1916). Cf. Klein
(1979); CD, chap. 3.1; HD , vol. 1, chap. 5.1; CQ , chap. 6.
15 Bohr (1918, p. 5).16
Bohr (1913, pp. 8–9, 1914). Cf. CQ , pp. 87–89.
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As was already mentioned, in 1916 Sommerfeld and his
collaborators managed to quantize a wider class of systems, the
so-called multiperiodic systems which comprise the relativistic
Kepler problem, and the hydrogen model perturbed by a constant
electric or magnetic field (Stark and Zeeman effects). In these
cases, the blind application of the generalized quantum rules and
of the frequency rule yields far many more spectral lines than
observed experimentally. For instance, in the case of the (normal)
Zeeman effect quantum jumps are allowed in which the magneticquantum number varies by any amount, whereas the only
observed lines correspond to the variations 1, 0, +1of this
quantum number (triple splitting of the unperturbed lines).
Bohr noticed that in such cases the classical spectrum was in
better qualitative agreement with the true spectrum than the
quantum-theoretical spectrum. Indeed, the effect of a constant
uniform magnetic field on the Kepler motion is a uniform
precession at the Larmor frequency ¯ nL, following which the
Fourier spectrum of the orbital motion involves the frequency
triplet ¯ n0 ¯ nL; ¯ n0; ¯ n0 þ ¯ nL (where ¯ n0 denotes the frequency of
the unperturbed motion). In order to remedy this defect of the
quantum theory, Bohr complemented it with the idea that the
possibility and probability of a given quantum transition should
be determined by the existence and intensity of the corresponding
harmonic component of the (dipolar moment of) the motion in
the initial stationary state. The precise definition of this
‘‘correspondence’’ derived from the condition that the classical
and quantum-theoretical spectrum should asymptotically agree.17
In the simplest case of a single quantum number n, this
condition implies that the frequency of the line emitted during a
jump from n to nt should be asymptotically equal to the
frequency of the t harmonic of the orbital motion:
nn;ntt ¯ nn. (20)
It also requires the asymptotic proportionality of the probability
Annt of this quantum jump with the intensity jat(n)j2 of the
corresponding harmonic of motion:
n3n;nt An
nt / jatðnÞj2. (21)
For moderate values of the quantum number n, the former of
these two relations is no longer valid; Bohr nevertheless assumed
that the latter relation approximately held and that it was exact
whenever the classical intensity jat(n)j2 vanished. In other words,
he excluded any quantum jump for which the ‘‘corresponding’’
harmonic component of the classical electric moment vanished;
and he generally estimated the probability of a quantum jump
from the intensity of this harmonic component.
This assumption is what Bohr named ‘‘correspondence princi-
ple’’ in 1917. Contrary to a common belief, this principle was not
the mere condition that classical and quantum-theoretical spectra
should asymptotically agree. Rather, the principle required that
some of the relations satisfied by the classical harmonics of motion should be preserved (exactly or approximately) for the
‘‘corresponding’’ quantum-theoretical intensities. The implied
‘‘correspondence’’ associated the quantum jump from n to ntwith the t harmonic of the classical motion.
Thanks to this principle, Bohr could derive the much needed
selection rules, according to which certain quantum numbers,
such as the magnetic quantum number m or the azimuthal
quantum number k, can only vary by certain amounts (Dm ¼ 0,71; Dk ¼ 71). With Hendrik Kramers’s help, he also derived the
approximate values of the intensities of the lines of the hydrogen
atom. Broadly speaking, the correspondence principle is a relation
between the periodicity properties of a classical model associated
with the atomic system on the one hand, and the true spectrum of
this system on the other hand. Bohr sometimes used the principle
deductively, as a way to deduce properties of the spectrum from
computable properties of the classical motion. Some other times,
he used it inductively as a way to infer properties of the orbital
motion from the empirically known spectra. He for instance did so
in the perturbation theory and in the theory of the helium atom
which he developed with Kramers. Altogether, the principle wasvery useful, and Bohr took it as a hint to a future, complete
quantum theory that would be a rational generalization of
classical electrodynamics.18
2.5. Quantum mechanics
Bohr originally believed that the motion of the electrons in
stationary states should be represented by well-defined orbits,
whose periodicity properties were the proper basis for the
application of the correspondence principle. This is why in 1924
he resisted Wolfgang Pauli’s pressure to reject the orbital model.
At the beginning of 1925, the difficulties of the orbital model
became so severe that Bohr capitulated. Yet the correspondence
principle did not follow the orbits to the grave. Bohr now believedthat the classical orbital model could still have a ‘‘symbolic’’
relation to the true motion in stationary states. In the preceding
months, Kramers, Max Born, and Werner Heisenberg had shown
that certain classical relations between harmonic components of
motion in the classical model could be translated into exact
quantum-theoretical relations between intensities. The key to this
translation was Bohr’s ‘‘correspondence’’ between t harmonic and
n-nt jump. In his contribution to this development, Born saw
the dawn of a new Quantenmechanik.19
The correspondence principle thus became a tool for the
symbolic translation of classical relations into quantum-theore-
tical relations, regardless of any descriptive value of the classical
motion. In the spring of 1925, Heisenberg had the brilliant idea of
applying this symbolic translation directly to the classicalequations of motion. The following is a simplified version of his
reasoning.20
Consider a mechanical system whose configuration depends on
the single coordinate q and whose every motion is periodic
(anharmonic oscillator). The equation of motion can be written
under the form
€q ¼ a1q þ a2q2 þ (22)
Any given solution of this equation can be developed into a
Fourier series
q ¼Xþ1
t¼1
qt; with qt ¼ ate2pit ¯ nt . (23)
In terms of the harmonic components qt, the equation of motioncan be rewritten as
ð2pt ¯ nÞ2qt ¼ a1qt þ a2
Xt0 þt00 ¼t
qt0 qt00 þ (24)
The correspondence principle yields the translation rules:
t ¯ n ! nn;nt; jqtj2 ! Anntn
3n;nt (25)
ARTICLE IN PRESS
17 Bohr (1918). Cf. CD , chap. 3.2; HD , vol. 1, chaps. 2–5; CQ , chap. 6; Meyer-
Abich (1965); Petruccioli (1993).
18 Cf. CQ , chaps. 6–7; Darrigol (1997).19 Kramers (1924); Born (1924); Kramers & Heisenberg (1925). Cf. CD , chap.
5.1; HD, vol. 2, chap. 3.5; Dresden (1987), chap. 8; CQ , chap. 9. On the
contemporary crisis, cf. Hendry (1984); CQ , chap. 8. On John van Vleck’s
contemporary use of the correspondence principle, cf. Duncan & Janssen (2007).20 Heisenberg (1925). Cf. CD, chap. 5.1; HD, vol. 2, chap. 3.5; CQ , chap. 10;
Rudinger (1985).
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inspired from the asymptotic relations (20) and (21). The
translation of the preceding form of the equation of motion
requires a slight extension of this correspondence as
ate2pit ¯ nt ! qn;nt ¼ an;nte2pinn;ntt , (26)
in which an,nt is a complex ‘‘quantum amplitude’’ involving a
phase factor. The choice of n is immaterial in this translation rule
since any jump from n to nt corresponds to a t harmonic as long
as n
is larger than t. We may therefore translate the form (24) of the equation of motion as
4p2n2n;ntqn;nt ¼ a1qn;nt þ a2
Xt0þt00¼t
qn;nt0 qnt0;nt0t00 þ
(27)
The translation choice qt00 ! qnt0;nt0t00 ensures that the oscilla-
tion frequency of every term in this equation should be the same.
Indeed Bohr’s frequency rule implies
nn;nt ¼ nn;nt0 þ nnt0;nt. (28)
In terms of the matrix q whose elements are qm,n, Eq. (27) reads
€q ¼ a1q þ a2q2 þ (29)
Hence, the quantum-mechanical equation of motion is simply
obtained by substituting the matrix q for the coordinate q in the
equation of motion and replacing ordinary products with matrix
products.21
In the same spirit, to any dynamical variable g (t ) of the system
we may associate the Hermitian matrix g (t ) whose elements have
the form g mnð0Þe2pinm;nt . In particular, to the invariable energy H ,
we associate a diagonal matrix H whose diagonal elements are the
energies of the various stationary states. As a consequence of the
frequency rule
E m E n ¼ hnm;n; ð14Þ
the equation
_ g mn ¼ 2pinm;n g mn (30)
can be rewritten as22
_g ¼ ði=_Þ½H;g . (31)
We will now see that the compatibility of this equation of
evolution with Eq. (29) requires a certain quantum rule.
Introducing the momentum matrix p ¼ m _q, the Hamiltonian
matrix H has the form
H ¼ p2=2m þ V ðqÞ. (32)
We therefore have
p=m ¼ _q ¼ ði=_Þ½H;q ¼ ði=_Þ½p2=2m;q, (33)
or
p½p;q þ ½p;qp ¼ 2i_p. (34)
The identityd
dt ½p;q ¼ ½ _p;q þ ½p; _q ¼ 0 (35)
further implies that the commutator ½p;q is diagonal. Calling knthe diagonal elements of this commutator, condition (34) implies
pmnkm þ pmnkn ¼ 2i_ pmn. (36)
Granted that every quantum transition is allowed ( pmna0), this
reduces to the condition:
km þ kn ¼ 2i_ (37)
for every choice of m and n such that man. This can only be true if kn ¼ i_ for any n , namely23:
½q;pnn ¼ i_. (38)
Remembering that the commutator is diagonal, we get the
quantum rule24
½q;p ¼ i_. (39)
TakingV ðqÞ ¼ 1
2ma1q2 1
3ma2q3 , (40)
the equation of motion (29) is equivalent to Hamilton’s
equations:
_q ¼ @H
@p; _p ¼
@H
@q. (41)
These two equations are easily seen to derive from the equations
_q ¼ ði=_Þ½H;q; _p ¼ ði=_Þ½H;p (42)
combined with the quantum rule (39). Therefore, the fundamental
equations of Heisenberg’s quantum mechanics (in the Born–
Jordan form) can be written as25:
_g ¼ ði=_Þ½H;g for any g ðq;pÞ; and ½q;p ¼ i_: ð31; 39Þ
Remembering that the matrix H is a diagonal matrix whose
diagonal elements represent the energies of the successive
stationary states and that the squared modulus of the elementqmn of the matrix q is proportional to the probability of a
transition between the states m and n, the basic problem of
Heisenberg’s quantum mechanics (in the Born–Jordan form) is to
find two infinite matrices q and p such that [q,p] ¼ i_ and the
matrix H(q,p) is diagonal.
As Schrodinger remarked in 1926, this problem can be
described in a more abstract manner as the following three-step
problem26:
(i) Find two Hermitian operators q and p such that [q,p] ¼ i_ in
an infinite-dimensional vector space.(ii) Diagonalize the Hermitian operator H(q,p).
(iii) Find the matrix elements of the operator q in the basis that
diagonalizes H.
A simple way to accomplish the first step is to pick the
operators
q : cðqÞ ! qcðqÞ and p : cðqÞ ! i_@c=@q (43)
that act in the Hilbert space of the complex functions c(q) whose
squared modulus is integrable. The second step then amounts to
solving the (time-independent) Schrodinger equation
H ðq; i_@=@qÞcðqÞ ¼ E cðqÞ. (44)
Granted that the spectrum of possible values of the energy E isdiscrete, the Hermitian character of the operator H warrants that
these values are real and that the eigenfunctions are orthogonal
with respect to the Hermitian scalar product. The latter property
reads:Z c
mðqÞcnðqÞ dq ¼ 0 if man. (45)
ARTICLE IN PRESS
21 Matrices do not explicitly occur in Heisenberg’s paper.22 Cf. Born & Jordan (1925) for a different derivation of this equation, and
Hund (1974, pp. 227–229).
23 Heisenberg obtained this relation in a different way and in a different form.
See below p. 11.24 Hund (1974, pp. 227–229) shows that the choice of this quantum rule
implies the equivalence between the quantum version of Hamilton’s equation and
Eq. (31) applied to p and q . But he does not deal with the reciprocal implication.25 Cf. Born & Jordan (1925).26
Schrodinger (1926c).
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Lastly, the matrix elements of the operator q can be obtained
through the formula
qmn ¼
R c
mðqÞqcnðqÞ dqR
cmðqÞcnðqÞ dq
. (46)
So far, we have followed Heisenberg in regarding the quantum
operators as time-dependent and the vectors on which they act as
time-independent. In this picture, the equation of motion
_g ¼ ði=_Þ½H;g ð31Þ
can be integrated as
g ðt Þ ¼ eiHt =_g ð0ÞeiHt =_. (47)
Consequently, the unitary transformation eiHt =_ absorbs the time-
dependence of the operators. The resulting time-dependence of
the vectors is
jcðt Þi ¼ eiHt =_jcð0Þi, (48)
or
i_@jci
@t ¼ Hjci. (49)
In the wave representation of the vectors, this gives the (time-
dependent) Schrodinger equation
27
i_@cðq; t Þ
@t ¼ H q; i_
@
@q
cðq; t Þ. (50)
To summarize, Bohr’s two quantum postulates and the
correspondence principle suggest a symbolic translation of the
classical equation of motion (for a bound system with one degree
of freedom) in which quantum amplitudes oscillating at the Bohr
frequencies correspond to the harmonic components of the
classical motion. There is only one such translation that agrees
with the two postulates. The result of this translation is the
Heisenberg–Born–Jordan form of quantum mechanics. The cano-
nical commutation rule derives from Bohr’s frequency rule alone
(there is no need to translate the semi-classical Bohr–Sommerfeld
rule). Heisenberg’s quantum mechanics implicitly contains the
fundamental equations of Schrodinger’s wave mechanics.
1.6. Interpretation
In Heisenberg’s quantum mechanics, the time average of the
matrix g is a diagonal matrix whose nth diagonal element g nnrepresents the time average of the dynamical variable g in the nth
stationary state. This interpretation derives from the correspondence
principle, which makes the large n limit of g nn the zero-frequency
component of the Fourier development of g (t ) in the nth stationary
state (in the Bohr–Sommerfeld theory). It was part of Paul Dirac’s
genius to understand that the transformation properties of quantum
mechanics generate a complete interpretation of quantum me-
chanics from this tiny bit of interpretation. In the simplest case of
one degree of freedom, his reasoning goes as follows.28
Consider the diagonal elements
g a0a0 ¼ ha0jg ja0i (51)
of the operator g in a scheme for which the matrix a corres-
ponding to the dynamical variable a(q, p) is diagonal29:
aja0i ¼ a0ja0i. (52)
Evidently, the elements g a0a0 do not depend on the choice of the
Hamiltonian. We may therefore consider a as a fictitious
Hamiltonian. In this case, the elements g a0a0 represent the time
average of the variable g , and the canonically conjugate variable b
varies linearly in time (in the Bohr–Sommerfeld theory). There-
fore, these elements also represent the average value of g when
the variable a takes a given value a0 and the value b0 of the
variable b is uniformly spread. The latter interpretation no longerrefers to the fictitious Hamiltonian and therefore remains valid
when the true evolution is restored.
Accordingly, the element dðg g 0Þa0a0 represents the b-average
of d( g – g 0) when a ¼ a0. By definition of Dirac’s d function, the only
values of b that contribute to this average are those for which g is
close to g 0, and these contributions have equal weight. Therefore,
the result should be the relative probability that g ¼ g 0 whena ¼ a0 (and b is uniformly spread). Using the completeness
relationZ j g 0ih g 0j d g 0 ¼ 1 (53)
for the vectors j g 0S that diagonalize g , we have
ha0jdðg g 0Þja0i ¼Z
ha0j g 00idð g 00 g 0Þh g 00ja0id g 00 ¼ ha0j g 0ih g 0ja0i ¼ jha0j g 0ij2.
(54)
Therefore, the expression j/a0j g 0Sj2 represents the probability
that the variable g takes the value g 0 when the variable a takes the
value a0. Dirac regarded this result as the full interpretation of the
quantum formalism. To this day, it remains the basis for concrete
applications of quantum mechanics. It also inspired Heisenberg’s
and Bohr’s subsequent considerations on the intuitiveness and
completeness of quantum mechanics, which will not be discussed
in this paper.
3. The wave approach
3.1. Wave-particle duality
As is well known, Erwin Schrodinger obtained his wave
mechanics without any (initial) recourse to the quantum
mechanics of Heisenberg, Born, and Jordan. It all started with
Einstein’s suggestion, formulated in 1905, that in some respects
light of a given frequency n behaved as if it were made of discrete
quanta hn. The inference was based on the expression
S ðuÞ S ðV Þ ¼ kðE =hnÞ lnðu=V Þ (55)
of the entropy variation of the low-density blackbody radiation
(obeying Wien’s law) of energy E and of frequency comprised
between n and n+dn, when the volume varies from V to u. Einstein
then used Boltzmann’s relationS ¼ k ln W (56)
to derive
W ¼ ðu=V ÞE =hn (57)
for the probability W of a fluctuation in which the radiation is
confined within the fraction u/V of the volume V of the cavity.
Comparing this result with the probability (u/V )N that the N
molecules of a gas be found within a fraction u/V of the available
volume, Einstein hypothesized that the energy E of the radiation
was made of distinct quanta hn. He used this ‘‘heuristic
assumption’’ to explain the existence of a frequency threshold in
the photo-electric effect and predict the relation
hn ¼ P þ eV (58)
ARTICLE IN PRESS
27 A similar derivation is found in Dirac (1927).28 Dirac (1927). Cf. Kragh (1990), chap. 2; CD, chap. 6.2; HD, vol. 4, part 1; CQ ,
chap. 12.29 Although the notation best fits the case of continuous spectra for the
operators g and a, the case of discrete or mixed spectra is easily covered by
replacing the delta functions with Kronecker deltas and the integrals with discrete
sums. Dirac (1927) did not introduce state vectors. His entire reasoning was based
on the transformation matrices that left the fundamental equations (31, 39)
invariant.
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between the frequency n of the incoming radiation, the work P
needed to extract an electron from the surface of the metal, the
charge e of the electron, and the potential V needed to stop the
photoelectrons.30
Most of Einstein’s contemporaries rejected the lightquantum
hypothesis, and Einstein himself came to admit his inability to
conciliate it with the well-established wave properties of light.
The situation changed in the early 1920s when new experiments
on the interaction between X-rays and matter strongly supportedthe hypothesis. Some of these experiments, concerning the photo-
electric effect induced by X-rays, were done by Maurice de Broglie
in Paris. In 1923, his younger brother Louis de Broglie speculated
that the wave-particle duality of light extended to matter.31
Louis imagined that both matter and light involved corpuscles
sliding on waves, the relativistic 4-momentum p of the (free)
corpuscles being related to the 4-wavevector k of the (plane
monochromatic) wave through the covariant relation
p ¼ _k, (59)
whose time- and space-components yield
n ¼ E =h and l ¼ h=jpj (60)
for the frequency n and the wavelength l of the wave as functionsof the energy E and the 3-momentum p of the corpuscles.32
Emboldened by this relativistic way of associating waves to
particles of matter, de Broglie suggested that the quantum
conditions of Bohr and Sommerfeld resulted from the synchroni-
city of the motion of electrons orbiting around a nucleus with the
associated wave motion.
De Broglie also noted that Maupertuis’s principle of least
action and Fermat’s principle of least time were equivalent when
the former was applied to the motion of a corpuscle (in a
potential) and the latter to the motion of the associated wave.
Formally,
d
Z pmd xm ¼ 0 is equivalent to d
Z kmd xm ¼ 0; (61)
as long as the relation p ¼ _k can be generalized to particles
moving in a conservative field of force. De Broglie only expected
this correspondence to hold in the case for which the associated
wave can locally be approximated by plane waves, as happens in
the ray-optics limit of wave optics. In the general case, he
expected the matter waves to undergo diffraction. With foresight,
he wrote: ‘‘The new dynamics of the material point is to the old
dynamics (including Einstein’s) what undulatory optics is to
geometrical optics.’’33
3.2. Wave mechanics
In 1926, Erwin Schrodinger took this suggestion seriously and
translated it into the task of finding the wave equation that wouldyield the classical trajectories in the approximation of locally
plane waves. An easy way to achieve this goal is to compare the
Hamilton–Jacobi equation of classical mechanics, which yields
the classical trajectories, with the eikonal approximation of wave
optics, which yields the trajectory of rays as predicted by
geometrical optics.34
Forgetting about polarization, the optical wave equation in a
medium of variable index n(r ) reads
Dj n2
c 2@2j
@t 2 ¼ 0, (62)
where c is the velocity of light in a vacuum. For a monochromatic
wave of pulsation o, this reduces to
Dj þ
n2o2
c 2 j ¼ 0. (63)
In the approximation of optical geometry, the undulation can be
locally approximated by a plane monochromatic wave. The wave
j can then be written under the form
jðr ; t Þ ¼ aðr Þ cos½ot þ xðr Þ, (64)
x(r ) being a quickly varying phase and a(r ) a slowly varying
amplitude. And the wave equation can be replaced with the
‘‘eikonal equation’’
ðr xÞ2 ¼ n2o2=c 2. (65)
For a given index function n(r ), the rays are obtained by solving
this equation and drawing the lines orthogonal to the surfaces
x ¼ constant.35
For a non-relativistic particle of mass m moving in a potentialV (r ) and for a given value of the energy E (which is conserved), the
possible trajectories can be similarly determined by solving the
Hamilton–Jacobi equation
ðr S Þ2 ¼ 2mðE V Þ (66)
and drawing the lines orthogonal to the surfaces S ¼ constant. It is
therefore tempting to regard the dimensionless ratio S /_ as a
phase and to seek the (time-independent) wave equation of which
the Hamilton–Jacobi equation is the eikonal approximation. The
correspondence
2mðE V Þ=_22n2o2=c 2 (67)
gives this equation as
Dc þ2mðE V Þ
_2
c ¼ 0; or _2
2mD þ V
!c ¼ E c. (68)
This is the time-independent form of the Schrodinger equation.36
In the case for which V is the Coulomb potential of an electron in
the field of the nucleus, Schrodinger found that this equation
admitted a solution if and only if the energy E was positive or else
belonged to a discrete series of negative values identical with those
obtained in Bohr’s non-relativistic theory of the hydrogen atom.
In conformity with the Einstein–de Broglie relation between
energy and frequency, Schrodinger assumed that to a solution of
energy E corresponded an oscillation of frequency E /h. As a simple
ansatz for the most general oscillation he took
c ¼ Xa
ua
eiE at =_, (69)
where ua is a solution of the time-independent Schrodinger
equation with the energy E a. This oscillation is the general integral
of the equation
i_@c
@t ¼
_2
2mD þ V
!c. (70)
Schrodinger assumed the validity of this equation for non-
conservative systems in which the potential V depends on time.
He showed that Kramers’s dispersion formula resulted from the
ARTICLE IN PRESS
30 Einstein (1905). Cf. Klein (1963, 1982); Kuhn (1978), chap. 7; Pais (1982),
chap. 6; Stachel (1986); CD, chap. 1.3; HD , vol. 1.31 Broglie (1923a, 1923b, 1923c, 1924). Cf. Stuewer (1975); Wheaton (1983),
part 5; HD, vol. 1, chap. 5.4; Darrigol (1993).32 Einstein had already used these relations in the limited case of mass-less
lightquanta, e.g. in Einstein (1909a, 1909b, 1917).33 Broglie (1923b, p. 83).34
Cf. Hanle (1977); Wessels (1979); Kragh (1982); HD , vol. 5.
35 Cf. Landau & Lifshitz (1959, chap. 7, par. 43). On the history of this approach,
cf. Kragh (1982).36
A similar derivation is found in Schrodinger (1926b).
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first-order solution of this equation when V described the dipolar
interaction with an incoming monochromatic electromagnetic
wave. In this calculation, he assumed the original c to be the wave
function of the fundamental state of the hydrogen atom, and he
generated the outgoing electromagnetic field through the dipolar
moment
M ¼ e Z r cðr ; t Þc ðr ; t Þd3r . (71)
The latter expression derived from the interpretation of the
product ecc* as some electric density within the atom, about
which more will be said in a moment.37
As Schrodinger promptly realized, in a system of several
interacting particles the matter waves no longer exist in three-
dimensional space. The analogy between the action function and
the phase of the wave indeed makes the relevant space identical
with the 3n-dimensional configuration space, where n is the
number of particles (when spin is ignored). The Schrodinger
equation then reads38
i_@cðr 1; r 2; . . . r n; t Þ
@t ¼
_2
2m
Xn
k¼1
Dk þ V ðr 1; r 2; . . . r nÞ
!cðr 1; r 2; . . . r n; t Þ.
(72)
3.3. Interpretation
Being guided by the analogy between matter and light,
Schrodinger originally assumed that some real wave process
occurred within the atom. This naıve interpretation did not square
with the 3n-dimensional character of the relevant space. In the
end, Schrodinger assumed that jc2j was ‘‘some sort of weight-
function’’ in configuration space, as suggested by the conservation
of its integral (owing to the Hermitian character of the
Hamiltonian). More concretely, he assumed that the true electric
density and current (in ordinary space) derived from this weight-
function. This assumption yields the correct dispersion formula;
but it fails to explain spontaneous emission because the resulting
electric distribution is stationary in any stationary state. Inorder to derive the frequencies and intensities of the emitted
lines, Schrodinger had to import extraneous elements such as
the Bohr frequency rule or Heisenberg’s relation between the
matrix elements of the polarization and the intensity of spectral
lines.39
Max Born’s scattering theory of 1926 offered another handle on
the interpretation of the wave function. The object of this theory is
the scattering of electrons through a potential. Born developed the
scattered wave as a sum of plane monochromatic waves:
c ¼ eiEt =_Xk
ak eik r , (73)
the sum being extended to all the vectors k compatible with
the asymptotic energy E . In conformity with de Broglie’s idea thata plane wave represents the motion of a free particle, Born
decided that jak j2 should be proportional to the probability that
the particle be scattered in the direction of the vector k .40
Dirac offered a third bit of interpretation of the wave function
in his own version of the time-dependent perturbation theory.
While in his dispersion theory Schrodinger interpreted the
perturbed wave function
cðr ; t Þ ¼X
n
c nðt Þunðr Þ (74)
as giving the relative weight jc(r , t )j2 of the various positions r of
the electron, Dirac identified jc n(t )j2 with the probability that the
system be found in the n stationary state. Again, the Hermitian
character of the total Hamiltonian warrants the conservation of
the sum of these probabilities. In the case of an atom perturbed by
a plane electromagnetic wave, Dirac used this interpretation to
derive the value of Einstein’s Bmn coefficients for the probability of
induced quantum jumps.41
Born and Dirac thus inaugurated the interpretation of the wave
function as a means to derive the probability of certain classically
defined variables, namely a scattering angle or the final energy of
the atom. Schrodinger formally did the same for the position of
the electron, although by ‘‘weight-function’’ he probably meant a
spread of the substance of the electron and not a statistical
outcome of position measurements. The question now is: Is there
a way to extend the statistical interpretation to any classically
defined variable of the system?A simple (though somewhat formal) way to answer this
question is to imagine a very brief and very small interaction of
the system whose potential takes significant values only when the
variable takes a given value. For the sake of simplicity, suppose
there is only one degree of freedom. A given dynamical variable is
a function g (q, p) of the coordinate q and the momentum p. The
potential of the imagined interaction is proportional to d[ g (q, p)
g 0]. If c(q, t ) is the normalized wave function of the system at
the time t of the perturbation, first-order perturbation theory a la
Schrodinger–Dirac yields
P cð g 0Þ ¼
Z c
ðq; t Þd½ g ðq; i_@=@qÞ g 0cðq; t Þ dq (75)
for the probability that the system leaves the state c(q, t ) (whichcan be regarded as a stationary state at the time scale of the
perturbation). Owing to the definition of the perturbation, this
integral also represents the probability that the variable g takes
the value g 0.
Call u g (q) the normalized eigenfunctions of the Hermitian
operator g (q,i_q/qq) (the index g is sufficient in the non-
degenerate case), and assume that they form a basis in the space
of wave functions. Then the unique decomposition
cðq; t Þ ¼
Z c g ðt Þu g ðqÞd g (76)
leads to
P cð g 0Þ ¼ Z c
g
dð g g 0Þc g d g ¼ jc g 0
j2. (77)
In Hilbert-space language, this means that the probability
(density) that the dynamical variable g takes the value g 0 when
the system is in the state c is the square of the modulus of the
projection of this state over the eigenstate of the operator g (q,i_q/qq) that has the eigenvalue g 0. This rule is a slight
generalization of the rule obtained by Dirac through algebraic
quantum mechanics (for Dirac, the c state would be characterized
by a specific value of another dynamical variable). There are of
ARTICLE IN PRESS
37 Schrodinger (1926e). Although Schrodinger assumed the frequency E /h in
his earlier papers, equation (70) only appears in the last installment of his theory.
He earlier preferred a second-order equation that only applies to stationary states
and still contains the energy E .38 Schrodinger (1926b) for the time-independent version; Schrodinger (1926e)
for the time-dependent version.39 Schrodinger (1926e, p. 135). The frequency rule occurs in Schrodinger
(1926a); Heisenberg’s expression of intensities occurs in Schrodinger (1926d,
p. 465).40 Born (1926a, 1926b). In the first paper, Born succinctly treated the problem
of the scattering of an electron wave by an atom originally in a stationary state. In
the second paper, he also treated the simpler case of scattering by a fixed center of
( footnote continued)
force. Cf. CD, chap. 6.1; Konno (1978); HD , vol. 3, chap. 5.6; Gyeong Soon (1996);
Beller (1990).41 Dirac (1926). Dirac introduced the description of indistinguishable particles
by symmetric or antisymmetric waves in the same paper.
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course some mathematical difficulties bound with the discrete,
continuous, or mixed nature of the spectrum of the g (q,i_q/qq)
operators. In general, part of the above-written integrals over g
would have to be replaced by a discrete sum; and, as is well
known, the eigenfunctions can only be normalized in the discrete
case.
We have thus arrived at a fully interpreted quantum mechanics
through the wave approach. The result is entirely equivalent to
that obtained through the quantum approach. As was alreadymentioned, Heisenberg and Bohr soon argued that the statistical
interpretation yielded the maximal answer to any conceivable
experiment on quantum systems. Heisenberg’s arguments were
more dependent on the quantum approach, and Bohr’s on the
wave approach. But they did not question the formal and
predictive equivalence of both forms of quantum mechanics.42
4. What has not been said
In my simplified account of the emergence of quantum
discontinuity, Planck’s extensive work on blackbody radiation in
the period 1895–1900 has hardly been mentioned. The reason for
this silence is that Planck did not assert a breakdown of ordinary
electrodynamics in his theory. Nor did he suggest that his
resonators should only have a quantized energy. The following
remains true, however: he introduced the quantum of action; he
associated this quantum with some essentially new microphysics;
he formally introduced energy elements hn in a combinatorial
derivation of the resonators’ entropy and thus obtained the
correct blackbody law.43
I have said nearly nothing on early uses of the quantum
between Einstein’s lightquantum paper and Bohr’s theory of 1913.
In reality, various applications of the quantum to the problem of
specific heats and to the interaction between matter and radiation
(both light and X-rays) contributed to the acceptance of a radically
new quantum behavior of microphysical entities. Moreover, the
few tentative applications of the quantum to the problem of
atomic structure (for instance Haas’s and Nicholson’s) wereimportant sources of inspiration for Niels Bohr. My presentation
of Bohr’s theory of 1913 is highly selective: I have insisted on the
derivation of the Balmer formula, because it no doubt was the key
to the novel frequency rule. However, in the life span of the old
quantum theory Bohr’s main concern was the construction of
atoms and the explanation of the periodic table of elements. In his
eyes, spectra mainly counted as a means to explore atomic
structure.44
I have given much importance to the correspondence principle
without mentioning the difficulties of its reception by Bohr’s
contemporaries. In Munich, Sommerfeld and one of his disciples
(Adalbert Rubinowicz) managed to derive some selection rules
through conservation laws. For a while, they regarded the
correspondence principle as a ‘‘magic wand’’ that enabled Bohr,in the early 1920s, to mysteriously derive a plausible classification
of elements. Physicists who had no personal contact with Bohr
had difficulty making sense of correspondence arguments. In
1923–24, Max Born and two disciples of Sommerfeld, Heisenberg
and Pauli, extended the perturbation technique of the old
quantum theory and obtained results that contradicted Bohr’s
use of the correspondence principle in the construction of the
helium atom. Moreover, Pauli showed that another organizing
principle, the exclusion principle, gave a better classification of
elements than Bohr’s. In 1924–25, Bohr, Kramers, Born, and
Heisenberg were fairly isolated in their persisting trust in the
correspondence principle.45
I have been exceedingly concise on the contributions of
Sommerfeld’s school to quantum theory. I have only mentioned
the importance of the extension of Bohr’s theory to multiperiodic
systems, although Sommerfeld and his collaborators contributed
in many other ways to discussions of the intricacies of atomic
spectra. They designed simplified, multiperiodic models of higher
atoms and saved the phenomena by ad hoc modifications of thesemodels (non-mechanical constrains, half-integral quantum num-
bers, etc.). The difficulties they experienced in dealing with the
anomalous Zeeman effect notoriously endangered the credibility
of well-defined electronic orbits in the atom. Together with the
failures of the Bohr-Kramers theory of the helium atom and of the
BKS theory (more on this soon), these difficulties determined
Bohr’s and Heisenberg’s conviction that the correspondence
principle should be used in a symbolic manner in which the
classical orbits no longer represented the true motion in the atom.
The reason why I have largely neglected these important
developments in my simplified account is that they only played
a filtering role in the construction of quantum mechanics: they
served to eliminate strategies that were no longer viable, but they
did not provide the needed constructive tools.46
I have completely neglected a dramatic episode of the pivotal
year 1924: Bohr, Kramers, and John Slater (BKS) published a
quantum theory of radiation which seemed to remove the
contradiction between the continuous character of electromag-
netic character and the discontinuity of the quantum jumps. BKS
assumed that the emission of radiation occurred during the
sojourn of atoms in stationary states, not any more during
the quantum jumps. Although this radiation was only virtual
(it occurred without compensation in the stationary states), it was
held responsible for quantum jumping in other atoms (in a
manner that could only be statistical). Bohr placed much hope in
this theory, for he regarded it as a space-time implementation of
the ‘‘correspondence’’ relation between the motion in stationary
states and the properties of radiation. In early 1925, intratheore-
tical paradoxes and the experimental refutation of a consequenceof the theory (the lack of energy conservation in individual
Compton processes) forced Bohr to abandon this project and to
support the symbolic version of the ‘‘correspondence’’ that
Heisenberg would soon bring to maturity. Thus, the BKS episode
is more important by its failure than by any positive element it
brought to the construction of quantum mechanics.47
My presentation of the quantum mechanics of Heisenberg,
Born, and Jordan departs in several manners from the true
historical process. Heisenberg originally avoided matrix algebra,
presumably because the correspondence principle favored thean,nt notation over the anm notation for the quantum amplitudes.
Born and Jordan were responsible for writing quantum relations
in matrix form, with evident algebraic benefit. Most important,
Heisenberg did not derive the quantum rule [q, p] ¼ i_ in thesimple manner I have indicated (based on the frequency rule). He
only derived the diagonal elements of this relation, and he did so
by symbolic translation of the expression
J ¼
I p dq (78)
of the action-variable J which takes the quantized value nh in the
Bohr–Sommerfeld theory. In terms of the Fourier coefficients at of
ARTICLE IN PRESS
42 Cf. Beller 1999.43 Cf. Kuhn (1978); Needell (1980); CQ , part A; Gearhart (2002).44
Cf. Klein (1977); Heilbron & Kuhn (1969).
45 Bohr, Kramers, and Slater (1924); Cf. Kragh (1979); Hendry (1984); Heilbron
(1983); CQ , pp. 137–145, chap. 8.46 Cf. Eckert (1993); Forman (1968, 1970); Serwer (1977).47 Cf. Klein (1970b); Dresden (1987, chaps. 6, 8); CD, chap. 4.3; Hendry (1984,
chap. 5); CQ , chap. 9.
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the coordinate q, this expression reads
J ¼ 4p2mXt
tðt ¯ nÞjatj2. (79)
The translations rules
t ¯ n ! nn;nt; at ! an;nt ð25; 26Þ
are unfortunately insufficient to find the quantum-mechanical
counterpart of this relation. Heisenberg astutely replaced relation
(79) with
1 ¼ @
@ J
I pdq ¼ 4p2m
Xt
t @
@ J ðt ¯ njatj2Þ, (80)
in which p and q are regarded as functions of the action-angle
variables J y. He already knew the translation of the differential
operator t@/q J through the following considerations.48
As is known from the theory of action-angle variables, the
frequency ¯ n of the motion is related to the Hamiltonian E ( J ,y)
(which in fact depends on J only) through the relation
¯ n ¼ @E =@ J . (81)
This relation played an important role in Bohr’s theory of
(multi)periodic systems because it warranted the asymptotic
agreement between the quantum-theoretical spectrum and theclassical spectrum. Indeed it implies
nn;nt ¼ ½E ðnhÞ E ðnh thÞ=h t@E =@ J J ¼nh
¼ t ¯ nn. (82)
The latter relation suggest the translation rule
t@=@ J ! ð1=hÞDt, (83)
where Dt is the finite-difference operator of increment t such that
Dt f (n) ¼ f (n) f (nt) for any function f (n). Kramers and Heisen-
berg had successfully used this translation rule in their dispersion
theories.49
When applied to relation (80), this rule yields (together with
the rules (25))
h ¼ 4p2mXt
½janþt;
nj2nnþt;
n jan;n
tj2nn
;n
t, (84)
which is easily seen to be equivalent to ½q;pnn ¼ i_. By reasoning
later provided by Born and Jordan, [q, p] must be diagonal (see
p. 6 above), so that the quantum rule ½q;p ¼ i_ holds.
In the particular case of an anharmonic oscillator, Heisenberg
showed that his quantum equation of motion and his quantum
rule implied energy conservation and the Bohr frequency rule.
Born and Jordan offered the following general derivations. The
quantum rule ½q;p ¼ i_ and the quantum version of Hamilton’s
equations imply
_q ¼ ði=_Þ½H;q; _p ¼ ði=_Þ½H;p; ð42Þ
as well as
_g ¼ ði=_Þ½H;g ð31Þfor any quantum variable g (q, p). The special case g ¼ H yields_H ¼ 0. In addition, comparison of relation (31) with Heisenberg’s
_ g mn ¼ 2pivm;n g mn ð30Þ
yields Bohr’s frequency rule50
E m E n ¼ hnm;n: ð14Þ
This historical reasoning has two disadvantages. Firstly, Heisen-
berg’s derivation of the quantum rule involves clever but
unnatural steps. Heisenberg must have been aware of this
weakness, since he did not fail to mention that Willy Thomas
and Werner Kuhn had already obtained relation (84) in a different
way: by taking the high-frequency limit of Kramers’s dispersion
formula. Secondly, Heisenberg’s presentation makes the truth of
Bohr’s frequency rule depend on his quantum-mechanical
equations of motion and on the quantum rule, whereas in reality
his reasoning assumes much of this rule from the beginning.
Indeed the inspiration for his new quantum product came from
the combination rule
nn;nt ¼ nn;nt0 þ nnt0;nt; ð28Þ
which is a direct reflection of Bohr’s frequency rule. This being
understood, we may as well take the frequency rule as a starting
point, and derive the quantum rule from it as was done in my
simplified genesis.51
I have ignored a few important contributions to the new
quantum mechanics by Pauli, Norbert Wiener, Dirac, and others,
and I have not told the story of spin and statistics despite the
importance of these concepts in any application of quantum
mechanics to systems of electrons.52
I have simplified the historical relation between early quantum
mechanics and Schrodinger’s equation. Although Dirac did show
how to derive the equation from the matrix-operator ( q-number)
theory, this did not happen before Schrodinger had already
obtained the equation by different means and several authors
including Schrodinger, Heisenberg, and Dirac had shown that
Schrodinger’s (time-independent) equation could be used to solve
the equations of the matrix theory. Yet this chronology may be
regarded as a historical accident. If Schrodinger had not proposed
his equation in early 1926, there is little doubt that the matrix orq-number theorists would have obtained it in the same year as a
purely formal consequence of their own formalism. In fact,
Heisenberg, Born, and Jordan came very close to it in theirDreimannerarbeit of late 1925.
As for the interpretation of quantum formalism, I have again
widely exaggerated the separation between the matrix and the
wave approaches. Dirac and Jordan reached their final interpreta-tion after various bits of interpretation occurred either in the
matrix approach or in the wave approach. Moreover, wave-based
interpretation partially depended on relations obtained from
matrix mechanics: Schrodinger needed Bohr’s frequency rule
(his justification by analogy with acoustic beats does not hold
water), as well as Heisenberg’s polarization matrix (Schrodinger’s
stationary waves could not radiate). It remains true that Dirac’s
ingenious interpretation does not require any explicit recourse to
wave-based intuitions. It is not impossible to reach the same
result by mostly wave-based reasoning, although the relevant
section of my simplified genesis is purely fictitious.53
A last set of simplifications concerns the genesis of wave
mechanics. Full understanding of de Broglie’s motivations would
require more attention to early, marginal interest in Einstein’slightquantum, especially in the context of X-ray studies in which
de Broglie’s brother played a major role. The development of
quantum theories of gas degeneracy by Planck, Schrodinger, and
Einstein should also be taken into account as a further incentive to
develop the analogy between matter and light. As for the
Schrodinger equation, the derivation presented in my simplified
genesis was not the first that occurred to Schrodinger.54
ARTICLE IN PRESS
48 Heisenberg (1925).49 Cf. CQ , pp. 115–116, chap. 9.50
Born & Jordan (1925).
51 Admittedly, Heisenberg’s procedure has the advantage of not presupposing
the invariance and the diagonal form of the quantum Hamiltonian H.52 Cf. CD, chaps. 3.4, 5.2; HD, vol. 4, part 1; Kragh (1990); CQ , chap. 12; van der
Waerden (1960).53 Dirac (1927); Jordan (1927). Cf. Beller (1999).54
Cf. Hanle (1977); Wheaton (1983); Darrigol (1993); Kragh (1982).
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Schrodinger’s first derivation is found in notebook entries
written at the turn of the years 1926–27. The first assumption is
that matter waves of a given frequency should obey a wave
equation of the form
Dc þ o2
C 2ðr Þc ¼ 0, (85)
which is for instance known to apply to monochromatic sound
waves in a isotropic heterogeneous elastic medium. In order todetermine the function C (r ), we may consider large values of the
pulsation o for which there exist solutions that can be locally
approximated by plane waves. The local value of the phase
velocity of these waves is C (r ). According to de Broglie, it is also
given by the ratio E / p of the energy and the momentum of the
associated particle (since E ¼ _o and p ¼ _k). Taking into account
the relativistic relation
ðE V Þ2 ¼ p2c 2 þ m2c 4 (86)
between the energy and the momentum of a particle of mass m
immersed in the potential V (r ), we have
o2
C
2 ¼
p2
_
2 ¼
1
c 2
_
2 ½ðE V Þ2 m2c 4. (87)
Injecting this expression into the wave equation (85), we get the
time-independent form of the Klein–Gordon equation for a
particle in the potential V . Schrodinger managed to solve this
equation in the case of the hydrogen atom, and found an energy
spectrum which intolerably departed from the one given by
Sommerfeld’s successful theory of the fine structure of the
hydrogen atom. He therefore took the non-relativistic limit of
expression (87), which leads to the wave equation
Dc þ2m
_2
ðE mc 2 V Þc ¼ 0. (88)
This is the time-independent Schrodinger equation (save for the
inclusion of mc 2 in the energy E ). Schrodinger never published this
derivation, although it is more elementary than that based onreversing the eikonal approximation. Perhaps he did not want
reasoning based on de Broglie’s relations, since his attempt to
preserve the essentially relativistic character of these relations
had led to an unacceptable relativistic wave equation.55
Here and elsewhere, my only contention is that the fuller
history (which has largely been done) would not much alter the
basic constructive steps described in my simplified genesis. It
would better explain when and why these steps were taken, and it
would more faithfully represent the efforts devoted to various
directions of research. But it would not much help in under-
standing why quantum mechanics was reached in the end.
5. Historical necessity
Having identified a series of crucial innovations in the history
of quantum theory, one may wonder whether these innovations
were in some sense necessary. The question is not easy to answer
because the category of the implied necessity may vary. In a first
category, the novel elements are deduced from well-defined
physico-mathematical principles. In a second category, the novel
elements are induced from well-established empirical data
(together with well-confirmed, lower-level theories). In a third
category, they may result from the intractability or unavailability
of alternative approaches. In a fourth and last category, they may
be the resultant of psychological or social factors, implying for
instance the authority of a leader. In a philosophical dream-world,
the two first categories would be dominant. Needless to say that
in the real world the first and second kind of necessity are often
contaminated by the third and fourth. Also, the distinction
between deductive and inductive necessity can only be a loose
one, because induction usually requires established principles,
and principles often have a partly empirical origin.
An additional difficulty results from the variability of the kind
of necessity of a given innovative step when a fine time scale isused. Most frequently, the step is initiated by a single actor for
reasons that have to do with his personal itinerary, his cultural
immersion, and his psychological character. At this early stage, he
may be the only one to regard his move as inductively or
deductively necessary, while other actors may be skeptical and
regard the move as arbitrary. At a later stage, a critical debate
usually occurs at the end of which the majority of experts agree
that the step must be taken for reasons which may vary from case
to case: confirmation of the move by new empirical data,
theoretical consolidation of the original deduction, availability of
independent deductions that lead to the same result, compat-
ibility with independent, fruitful developments.56
In most of the following discussion, I will judge the necessity of
the innovative steps at the end of this second stage. My simplified
genesis does not provide enough information on the first stage.
Whether, for instance, Bohr’s familiarity with Harald Høffding’s
philosophy or Born’s awareness of the anti-causal philosophies of
the Weimar period justified their most daring moves could only
be judged from a much more detailed and diversified history. At
any rate, the closest approximations to inductive or deductive
necessity are more likely to be found in the justification stage.57
The early twentieth-century conclusion that ordinary electro-
dynamics could not yield equilibrium for thermal radiation comes
close to the ideal of deductive necessity. As was mentioned, the
lack of rigor in the implied deductions was compensated by
the multiplicity and variety of derivations of the same result.
Moreover, the status of one of these derivations, the Gibbsian
provided by Lorentz, rose with the conviction that Gibbs’s
ensembles correctly represented thermodynamic equilibriumdespite the lack of a firm foundation.
The introduction of quantum discontinuity obeyed a weaker
necessity of the inductive kind. Einstein’s and Bohr’s discrete
quantization was the simplest way to account for Planck’s
blackbody law and for the spectrum of the hydrogen atom.
But it was also most problematic for two reasons: it implied a
non-classical selection among classically defined states, and it
made it very difficult to imagine a plausible mechanism for the
interaction between atoms and radiation. For the latter reason,
Planck long preferred a division of phase space into cells of equal a
priori probability.
As is well known, in 1911 Paul Ehrenfest and Henri Poincare
proved that the canonical distribution of energy over resonators
could not yield a finite energy for cavity radiation unless therewas a finite energy threshold for the excitation of the resonators.58
This proof is largely illusory, because it depends on an un-
warranted extension of Gibbs’s canonical distribution law to
systems that no longer obey the laws of classical dynamics.59 The
ARTICLE IN PRESS
55
Schrodinger’s relevant notebook is lucidly analyzed in Kragh (1982).
56 These two stages are vaguely similar to Hans Reichenbach’s distinction
between context of discovery and context of justification.57 Social constructivists would agree with me that this second stage is
essential in stabilizing the basic constructs of science. However, their analysis of
the stabilizing process tends to underestimate rational constrains or to reduce
them to socially defined systems of beliefs.58 Cf. Klein (1970a); CD, p. 53.59 In his statistical mechanics, Gibbs assumed the validity of Hamiltonian
dynamics. Although Einstein did not in his own statistical mechanics, he still
assumed continuous evolution, invariance of the volume element in phase space,
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true reason why Einstein’s idea of a sharp quantization came to
dominate over more timid attempts was the multiple, successful
applications it received in the context of the Bohr–Sommerfeld
theory.
Similar comments can be made about Bohr’s frequency rule. It
is tempting to say that Bohr read the rule in the Balmer–Rydberg
formula. In reality, this inference has the typical underdetermina-
tion of any inductive reasoning: the hydrogen spectrum can only
be derived from the combination of the frequency rule with a fewother assumptions including the existence of stationary states and
the truth of the laws of wave optics for the emitted radiation. At
any rate, we saw that Bohr himself did not believe in the
generality of the frequency rule until he became aware of
Sommerfeld’s and Einstein’s contributions to his theory in 1916.
The assumption of stationary states and the frequency rule gained
credibility and became Bohr’s two ‘‘postulates’’ when their
simultaneous application yielded correct results in an increasing
variety of situations involving spectra, atomic structure, and
atomic collisions. This happened despite the evident incomplete-
ness of the theory (it left the radiation mechanism in the dark)
and despite its reliance on classical concepts belonging to an
incompatible electrodynamics.
The very definition of stationary states and the statement of
the frequency rule required classical concepts: energy and
frequency. Bohr struggled to show that these concepts could be
defined in the quantum realm through a limited use of classical
theory that did not contradict the quantum postulates. Most
important, his correspondence principle pointed to a deep formal
analogy between classical electrodynamics and the evolving
quantum theory. He hoped that in the long run this analogy
would project the consistency of the former theory over the latter.
The quantum postulates would remain intact in this process.
Although Bohr obtained the correspondence principle by
analogy with classical electrodynamics, he insisted on the formal
character of this analogy and emphasized the contrast between
the quantum postulates and the continuity of classical radiation
processes. In order to judge the necessity of this principle, one
must first be aware that in Bohr’s original view this principle wasa relation between the periodicity properties of the motion in
stationary states (whether or not this motion obeyed classical
mechanics) and the properties of the emitted radiation. There
were three arguments in favor of the necessity of this principle: it
warranted the asymptotic agreement between the empirical
predictions of classical and quantum theory; in the deductive
mode, it provided the selection rules and good estimates of the
intensities of some spectral lines; through Bohr’s more obscure
appeal to the inductive mode, it led to a plausible classification of
elements.
As has already been mentioned, the magic of the correspon-
dence principle did not catch well outside Copenhagen. By 1924,
the idea of well-defined orbits in the atom, which the principle
seemed to require, was much under criticism. Even Bohr came toreject this idea in early 1925. Yet in Bohr’s circle the confidence
never died that correct quantum-theoretical relations could be
extracted by analogy with classical multiperiodic systems,
whether or not the motion of such systems truly represented
the motion in stationary states. This confidence even increased in
1923–24 when Kramers, Born, and Heisenberg managed to
translate some classical relations into what they (correctly!)
believed to be exact quantum-mechanical relations. One reason
for the latter belief was the empirical relevance of these relations.
Another was the automatic agreement between the large-
quantum-number limit of these relations and the corresponding
classical relations. Still another was the fact, first emphasized by
Kramers, that these relations only involved the basic quantities
entering Bohr’s postulates and no longer referred to the suspicious
orbits. In the spring of 1925, Heisenberg’s conviction that he had
discovered quantum mechanics resulted from these three quali-
ties of the symbolic translation, together with the consistency and
completeness of the resulting computational scheme.Heisenberg’s quantum mechanics may be regarded as a
necessary consequence of Bohr’s two postulates (discrete sta-
tionary states, and frequency rule) and of a rule for translating the
equations of motion of a classical periodic system (expressed in
Fourier form) into relations between ‘‘quantum amplitudes’’
directly related to the observable quantities that enter the two
quantum postulates. This rule itself derived from the correspon-
dence principle, whose plausibility rested on the asymptotic
validity of classical electrodynamics and on successful applica-
tions (of a different kind) in the earlier quantum theory. One
might then wonder why quantum mechanics was not discovered
earlier, say in 1917, when Bohr already had the two postulates
as the pillars of his theory and the correspondence principle
as a constructive tool. One reason is that before 1924 no one
proscribed orbital parameters from quantum-theoretical rela-
tions. Another is that no one guessed, before Heisenberg, that the
‘‘correspondence’’ counterparts of the Fourier components of a
periodic classical motion would completely characterize the
quantum-mechanical motion just as these components them-
selves sufficed to define the classical motion.
On the side of wave mechanics, the story began with de
Broglie’s extension of the wave-particle duality to particles of
finite mass. Although the extension was natural from a formal,
relativistic point of view, it could easily pass for a crazy
speculation. The receptivity of Paul Langevin, Einstein, and
Schrodinger depended on a few favorable circumstances. Firstly,
the lightquantum, which provided the basis for de Broglie’s
extension, was gaining momentum (literally and metaphorically).
Secondly, de Broglie’s successfully applied his notion to a widespectrum of problems including the derivation of the Bohr–
Sommerfeld rule, an analogy between Fermat’s and Maupertuis’s
principles, and a derivation of Planck’s quantum cells (for the
statistics of gas molecules). Thirdly, Einstein retrieved the de
Broglie waves through a different route: in 1925 he designed a
quantum theory of gas degeneracy by analogy with Satyendra
Nath Bose’s corpuscular derivation of Planck’s law, and found that
the theoretical fluctuation of his quantum gas implied wave
behavior in conformity with de Broglie’s relations. Being also
involved in quantum-gas theory, Schrodinger measured the force
Einstein’s reasoning.60
It would nonetheless be excessive to speak of a deductive or
inductive necessity of de Broglie’s waves. In 1925 they still were a
bold assumption without direct experimental counterpart.61 DeBroglie was himself shy in his suggestion of electron diffraction.62
A stronger necessity can be seen in the deduction of the
Schrodinger equation. De Broglie’s idea that the classical
dynamics of a particle should be to wave mechanics what
geometrical optics is to wave optics automatically leads to the
time-independent Schrodinger equation in the non-relativistic
ARTICLE IN PRESS
( footnote continued)
and some weak ergodicity. In the 1910s, there already were reasons to doubt the
validity of any of these requirements in the case of quantum systems.
60 Einstein (1924, 1925a, 1925b). Cf. CD, pp. 248–249; H D, vol. 1, chap. 5.3;
Forman (1969); Hanle (1977).61 Walther Elsasser nonetheless tried to relate de Broglie waves to anomalies
observed in Gottingen for the scattering of low-energy electrons. Cf. Born (1926b);
CD, pp. 249–251; Russo (1981).62 The suggestion only appears in de Broglie (1923b, p. 549), not in the These
(de Broglie, 1924).
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limit. Moreover, the success of this equation in determining the
stationary states of the hydrogen atom could hardly be regarded
as a coincidence. One may still be perplexed by the coincidence
that made the Schrodinger equation appear just a few months
after Heisenberg’s quantum mechanics. There is little to justify
this timing besides the contemporary willingness to renounce
electronic orbits in atoms.
A last question of special philosophical interest is the necessity
of the now standard probabilistic interpretation of the formalismof quantum mechanics or wave mechanics. In Dirac’s quantum-
mechanical approach, the starting point is the allegation that for a
sharply defined value of the energy (corresponding to a stationary
state) the conjugated phase is uniformly spread. The ensuing
deduction of the whole interpretation only requires the transfor-
mation properties of the fundamental equations of quantum
mechanics (invariance by unitary transformations). Hence the
necessity of this interpretation should be measured by the
necessity of the starting point. Dirac justified his starting point
through the correspondence principle, arguing that in the large
quantum number limit a stationary state may be represented
by a revolving electron whose phase varies uniformly in
time. Thus, Dirac was willing to admit that energy and phase
retained a meaning in quantum mechanics. More generally, he
assumed that any dynamical variable and its canonical conjugate
retained a meaning in quantum mechanics although it was
impossible to have initial conditions in which both variables were
completely determined. There is no evident necessity for this
persisting relevance of classical concepts in the quantum context.
Nevertheless, the harmony of Dirac’s statistical interpretation
with the transformation properties of quantum mechanics
pleaded for the uniqueness of this interpretation.
In the matter-wave approach, Born’s probabilistic interpreta-
tion of scattered electron waves seems unavoidable. Indeed the
naıve interpretation of the wave as dilute matter would imply that
only a fraction of an electron is detected at a given angle. Any
attempt to save the naıve view by building wave packets of very
small size would fail because of the unavoidable spreading of the
wave packets. Similarly but less stringently, Dirac’s statisticalinterpretation of the perturbed Schrodinger-wave of an irradiated
atom seems hard to avoid, granted that long after the interaction
the atom can only be found in a stationary state. By itself Born’s
probabilistic interpretation of scattered waves leads to the full
statistical interpretation of wave mechanics through the earlier
given idealization of the measuring process. Although this
idealization is far remote from any concrete measurement device,
it seems legitimate as long as the concept of external potential is
admitted in the theory.
One is left with a feeling of the unavoidability of the standard
statistical interpretation of wave or matrix mechanics. Its ability
to correctly represent the outcome of experiments in the quantum
regime has rarely been contested. The apple of later discord rather
was the possibility of defining or measuring physical quantitiesmore than quantum mechanics allows.
To conclude, the historical genesis of quantum mechanics can
be regarded as a series of bold, imaginative, but firmly supported
steps. The first two constructive steps, the introduction of discrete
stationary states and the frequency rule, were taken with full
awareness of their problematic character and later consolidated
by multiple successes of their combined application. These
assumptions have counterparts in modern quantum mechanics,
although stationary states are no longer regarded as the only
possible states. In contrast, the auxiliary reliance on classical
concepts posed more and more problems and led to the severe
crisis of 1924–25. In the middle of this crisis, Heisenberg
strikingly confirmed Bohr’s idea that the correspondence principle
opened the path toward a ‘‘rational generalization’’ of classical
electrodynamics. The contemporary but largely independent
invention of a closely related wave mechanics strengthens the
air of inevitability of quantum mechanics. The statistical inter-
pretation of this theory largely derives from its mathematical
structure, combined with some correspondence arguments.
The simplified genesis on which these conclusions are based is
apt to shed light on the later philosophy of some of the
participants. In particular, it explains Bohr’s insistence on the
primacy of classical concepts, the holism he assumed for quantumphenomena, and his insistence on the statistical character of the
outcome of measurements. This history might also inspire
derivations of necessary features of the quantum world, although
more rigor would then be needed than is implied in looser sorts of
historical necessity.
Acknowledgment
I thank Jurgen Renn and an anonymous reviewer for useful
suggestions.
Appendix. Derivation of Planck’s relation between blackbody
spectrum and resonator energy
In order to derive the relation
un ¼ ð8pn2=c 3ÞU ð2Þ
between the spectral energy density un of cavity radiation and the
average energy U of an immersed resonator, we may assimilate
the resonator with an electron of charge e and mass m constrained
to move on the x axis and elastically attached to the origin of this
axis. The equation of motion of this electron in the external field E
(supposed to be uniform at the scale of the resonator) is
mð€ x þ o20 xÞ ð2e2=3c 3Þ
_ _ _
x ¼ eE x, (89)
if o0 denotes the pulsation of the free oscillations of the electron.
The second term represents the damping force due to theemission of radiation by the accelerated electron. For the
frequencies of interest, it is very small compared to the elastic
and inertial force, so that the resonator only interacts with the
Fourier components of the radiation that have a pulsation very
close to o0. The exciting field E x can be written as a sum of
contributions from the proper modes of the mirroring cavity in
which it is immersed, labeled by the discrete index s:
E x ¼X
s
as cosðost þ jsÞ. (90)
Accordingly, the permanent part of the solution of the equation of
motion (89) can be written as
x ¼X
s
as
j Z jðosÞ cosðost þ csÞ, (91)
where
Z ðoÞ ¼ ðm=eÞðo20 o2Þ þ ið2e=3c 3Þo3. (92)
On the one hand, the average energy U of the resonator reads
U ¼ mo20 x2 ¼ mo2
0
Xs
a2s
2j Z j2ðosÞ: (93)
On the other hand, the quadratic average of the exciting field reads
E 2 x ¼ 1
2
Xs
a2s ¼
Z 1
0 J ðoÞ do; (94)
if J (o)do denotes the contribution of the modes s such that
ooosoo+do.
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The comparison of these two expressions leads to
U ¼ mo20
Z 1
0
J ðoÞ
Z j j2ðoÞdo. (95)
Granted that the distribution J (o) is well-defined and smooth,
the narrowness of the resonance leads to the simpler expression
U ¼ mo20 J ðo0ÞZ
1
0
do
j Z j2ðoÞ. (96)
Using again the narrowness of the resonance, the latter integral
can be computed under the approximation
Z ðoÞ ¼ ð2mo0=eÞðo0 oÞ þ ið2e=3c 3Þo30. (97)
The resulting expression of the average resonator energy is
U ¼ ð3pc 3=4o20Þ J ðo0Þ. (98)
Granted that the radiation is isotropic, the x-component of the
electric field contributes one sixth of the energy density ( E 2+B2)/
8p of the electromagnetic field. Therefore, the spectral energy
distribution of the radiation is given by
undn ¼ ð1=8pÞ 6 J ðo0Þdo0 ðwith o0 ¼ 2pnÞ, (99)
or
un ¼ ð8pn2=c 3ÞU ; ð2Þ
as was to be demonstrated.63
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