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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.



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.

O. Darrigol / Studies in History and Philosophy of Modern Physics 40 (2009) 151–166 152



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.




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.




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).




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).




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.




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).




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.




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.




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).




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,




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).




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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63 A somewhat similar proof is found in Pauli (1929, pp. 1522–1525, chap. 27).

I use Gaussian units for which the Coulomb force between two point charges q and

q0 is qq 0/r 2

if their mutual distance is r .




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a simplified genesis of quantum mechanics

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