Baggott J. The Quantum Story: A History in 40 Moments

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Both Pauli and Dirac moved quickly to use the new matrix mechanics to derive key

features of the hydrogen emission spectrum, most notably the Balmer formula. Pauli sub-

mitted a paper describing his results to the journal Zeitschrift für Physik on 17 January

1926, beating Dirac, who submitted his paper to the British journal Proceedings of the

Royal Society, by just ﬁ ve days.

Pauli also used the new mechanics to account for the Stark effect, named for Ger-

man physicist Johannes Stark, in which spectral lines are split by a static electric ﬁ eld.

However, certain relationships still had to be assumed, in the guise of an ansatz, an

‘educated guess’ or hypothesis to be veriﬁ ed through agreement with experimental results.

Furthermore, the theory was not yet fully relativistic—that is, it did not conform to the

demands of Einstein’s special theory of relativity. And it did not yet accommodate Pauli’s

own exclusion principle, electron spin, and the anomalous Zeeman effect.

Whilst older physicists may have struggled with the theory’s mathematical complexity and

lack of ‘visualizability’, matrix mechanics looked set to become a romping ground for a new

generation of quantum physicists. In his paper Pauli wrote: ‘Heisenberg’s form of quantum

theory completely avoids a mechanical-kinematic visualization of the motions of electrons

in the stationary states [i.e. the stable orbits] of an atom.’ The message was clear. To make

progress we ﬁ rst must rid ourselves of the conceptual baggage of our classical physical heritage

and focus our attention only on what can be observed and measured in the laboratory.

The direction for future development of the theory seemed set. Yet the young turks of

the quantum revolution were about to be upstaged by a 38-year-old Austrian physicist

A Late Erotic Outburst

Swiss Alps, Christmas 1925

a late erotic outburst

named Erwin Schrödinger. By the time he was ﬁ nished, the landscape of the new quantum

theory had been almost completely transformed.

Schrödinger had secured his doctorate in physics at the University of

Vienna in May 1910. After a little more than a year of obligatory military

service, he returned to the University and qualiﬁ ed as a lecturer (a Privat-

dozent) in 1914. Four years of military service in the First World War then

intervened. He returned to Vienna in 1917, becoming an associate pro-

fessor in 1920. He then moved ﬁ rst to Jena, then Stuttgart, then Breslau

before securing the professorship in theoretical physics at the University

of Zurich in 1921.

He arrived in Zurich with his wife, Annemarie (known to him affec-

tionately as Anny), in October 1921. Diagnosed with suspected pul-

monary tuberculosis just a few months later, he was ordered to take a

complete rest cure. He and Anny retreated to a villa in the Alpine resort

of Arosa, near the fashionable ski resort of Davos, where they stayed

for nine months as Anny nursed him back to health. The altitude and

the relative isolation provided valuable opportunities for Schrödinger to

think, and he wrote two scientiﬁ c papers whilst on his road to recov-

ery. He returned to Zurich and resumed his heavy teaching schedule in

November 1922, delivering his delayed inaugural lecture on 9 December.

His teaching commitments left little time for research and, though

cured of his respiratory illness, he remained weak and tended to tire eas-

ily. As he settled into his new life in Zurich he may have pondered on his

place in the ﬁ rmament of physics. He had drawn praise for his versatility

as a scientist and the breadth of his knowledge and accomplishments,

but he had yet to make a noteworthy contribution to any branch of phys-

ical science with which he was familiar. As he grew older, he had little

choice but to watch as a younger generation of physicists overtook him.

It seemed that he would be sidelined, worthy of little more than a foot-

note in the history of physics. When, in 1924, he was invited to attend the

fourth in a series of conferences on physics established by the wealthy

Belgian industrialist Ernest Solvay, he was not asked to present a paper.

His marriage to Anny was also in trouble. Theirs had been a relatively

‘open’ marriage, with both Erwin and Anny indulging in extra- marital

affairs. Their inﬁ delity caused friction but had thus far generated

the quantum story

insufﬁ cient heat for either of them to sue for divorce. Schrödinger

was keen to father a child which, given their sexual incompatibilities,

seemed unlikely to happen. Yet Anny was in favour of continuing the

marriage, with all its openness. Anny was a Catholic, divorce was

expensive, and the arrangement may have appealed to Schrödinger’s

anti-bourgeois sensibilities.

Despite their incompatibilities, Anny

remained enraptured by Erwin’s looks and intellect: ‘You know it

would be easier to live with a canary than a racehorse, but I prefer the

racehorse.’ The heat was building, however.

Schrödinger would sometimes seek refuge in the bohemian nightlife

of the city, in the company of colleagues from the neighbouring ETH;

Dutch physicist Pieter Debye and mathematician Hermann Weyl. They

formed a complex social set. Anny fell in love with Weyl, and they began

an affair. Meanwhile Weyl’s wife Helene Joseph, who had studied phi-

losophy under phenomenologist Edmund Husserl, had fallen for Swiss

ETH physicist Paul Scherrer.

The turmoil in Schrödinger’s private life inevitably affected his work.

He published no papers in 1923, but managed to regain his appetite for

research the following year. In October 1925 his attention was drawn to a

footnote in one of Einstein’s recent papers. It mentioned ‘a very notable

contribution’ by de Broglie.

Intrigued, Schrödinger acquired a copy of de Broglie’s thesis. What

he read would have deep resonance. Whilst recovering from illness in

Arosa in 1922 he had written a paper, ‘On a Remarkable Property of the

Quantized Orbits of a Single Electron’, which he had submitted to the

Zeitschrift für Physik. Building on ideas he had derived from his close friend

Weyl, he had observed a relationship between the stable electron orbits

and a so-called ‘gauge factor’, a characteristic of the orbits themselves.

He had found that he could understand the relationship between the dif-

ferent electron orbits—and the origin of the principal quantum number

n—by interpreting the gauge factor as a phase factor.

Schrödinger kept a notebook detailing his sexual encounters. This was not merely a log of

sexual conquests; it was rather intended as a personal exploration of female sensuality.

We will return to the idea of gauge theories in Chapter 20.

a late erotic outburst

Although at the time he had recognized this result as important, he

had chosen not to pursue it further. As he now read de Broglie’s thesis he

understood that the relationship he had observed was simply de Broglie’s

phase condition for a standing electron wave conﬁ ned to a circular orbit

around the nucleus.

Physicists at the University of Zurich and the ETH had established joint

bi-weekly colloquia at which they would discuss current topics of mutual

interest. Debye asked Schrödinger if he would be prepared to present a

colloquium on de Broglie’s thesis, which had recently been published in

the French journal Annales de Physique. Schrödinger agreed, and wrote to

Einstein on 3 November, drawing parallels with his earlier 1922 paper on

electron orbits but noting that: ‘Naturally de Broglie’s consideration in the

framework of his comprehensive theory is altogether of far greater value

than my single statement, which I did not know what to make of at ﬁ rst.’

The seminar was held on 23 November. In the audience was a young

Swiss student called Felix Bloch, who had started at the ETH the previ-

ous year with the intention to study engineering. But he found physics

held the greater fascination, and he learned more from one of Debye’s

introductory classes than all his other courses combined. Schrödinger’s

colloquium on de Broglie’s thesis was to prove particularly memorable,

as he recalled more than ﬁ fty years later:

So in one of the next colloquia, Schrödinger gave a beautifully clear

account of how de Broglie associated a wave with a particle and how he

could obtain the quantization rules of Niels Bohr and Sommerfeld by

demanding that an integer number of waves should be ﬁ tted along a sta-

tionary orbit. When he had ﬁ nished, Debye casually remarked that this

way of talking was rather childish. As a student of Sommerfeld he had

learned that, to deal properly with waves, one had to have a wave equa-

tion. It sounded quite trivial and did not seem to make a great impression,

but Schrödinger evidently thought a bit more about the idea afterwards.

A few days before Christmas 1925, Schrödinger left Zurich for a short

vacation in the Swiss Alps, returning once more to the villa in Arosa

where he had stayed in 1922 and at which he and Anny had vacationed

during Christmas 1923 and 1924. But his relationship with his wife was

the quantum story

at an all-time low. He therefore chose to invite an old girlfriend from

Vienna to join him, leaving Anny in Zurich. He also took with him his

notes on de Broglie’s thesis.

We do not know who the girlfriend was or what inﬂ uence she might

have had on him, but when he returned on 8 January 1926, he had dis-

covered wave mechanics.

It is impossible to provide a physically rigorous derivation of the quantum

mechanical Schrödinger wave equation starting from classical physics. It is

possible, however, to follow Schrödinger’s reasoning from notebooks he kept

at the time. His starting point was the well-known equation of classical wave

motion, which interrelates the space and time dependences of any wave

form described by a wavefunction, typically given the symbol y (psi). He

chose to work initially with the time-independent version of the wave equa-

tion, suitable for describing standing (rather than travelling) waves.

Into this classical wave equation he then substituted the relationships

between the particle mass (more speciﬁ cally, particle momentum) and

wave frequency that had been deduced by de Broglie. He realized that

in order to obtain meaningful solutions of the wave equation it would

be necessary to make some assumptions about the form of the electron

wavefunction—the mathematical description of the electron as a wave.

He needed to restrict the range of functions that could be accepted.

Speciﬁ cally, the functions had to be single-valued (only one value for a

given set of spatial coordinates), ﬁ nite (no inﬁ nities), and continuous (no

sudden ‘breaks’ or discontinuities).

Like de Broglie, he adopted a fully relativistic approach, seeking to

derive a wave equation for the electron in a hydrogen atom that would

necessarily conform to the requirements of Einstein’s special relativity.

He cast the wavefunction of the electron in a coordinate system more

suited to the problem of motion in a spherical electrostatic ﬁ eld. This is

a system of so-called spherical polar coordinates, which consist of the

distance, r, between the electron and the nucleus, an angle q (theta—a

co-latitude), and a second angle j (phi—a longitude). He found that

he could factor the wavefunction into two separate functions, a ‘radial’

function depending only on r, and an angular, or ‘spherical harmonic’

a late erotic outburst

function depending only on the angles q and j. The result was a series of

highly complex differential equations.

It was here that he reached the limit of his mathematical competence.

He could solve the differential equation for the spherical harmonics but

struggled to solve the equation for the radial function. Nevertheless, he

had seen enough. On 27 December he wrote a letter to Wien:

At the moment I am struggling with a new atomic theory. If only I knew

more mathematics! I am very optimistic about this thing and expect that if

I can only . . . solve it, it will be very beautiful. . . . I hope I can report soon in a

little more detailed and understandable way about the matter. At present, I

must learn a little more mathematics . . .

He returned to Zurich on 8 January and immediately sought help from

Weyl. Within about a week he had solved the differential equation

for the radial function. Perturbed by the fact that he now had to deal

with half-integral quantum numbers, he reverted to the very similar

non-relativistic wave equation which yielded only integer values. The

solution to all the mysteries now lay before him.

By constraining the properties of the three-dimensional electron

wavefunction ﬁ tted around the central nucleus, a very speciﬁ c pattern of

possible solutions of the wave equation emerged naturally and directly.

Schrödinger found that solutions for the radial function depend on two

integer numbers, equivalent to the principal quantum number n and the

quantum number k. Solutions for the angular function depend on k and

m. The energies of the various solutions depend on the square of the

principal quantum number, thereby reproducing the Balmer formula.

In Schrödinger’s derivation, the only ansatz concerned the constraints

he had placed on the form of the electron wavefunction. All the rest—the

quantum numbers, their interrelationships, the hydrogen energy levels,

the Balmer formula—was the result of irresistible mathematical logic. It

was indeed very beautiful.

Schrödinger submitted a paper describing his results to Annalen der

Physik. It was received on 27 January 1926, barely three weeks after he

had made his initial discovery and just ten days after Pauli’s paper had

been received by the Zeitschrift. In his paper Schrödinger showed that

the quantum story

the quantum numbers, introduced in ad hoc fashion by Bohr and Som-

merfeld, emerged ‘in the same natural way as the integers specifying the

number of nodes in a vibrating string’.

The differential equations arising in Schrödinger’s wave mechanics have

a special property. A differential operator

operates on a function to yield

the same function multiplied by some resultant quantity. In the case of

the differential operator for the total energy of a dynamical system (called

the Hamiltonian operator

), the quantity returned after the operation is

the total energy of the system. The functions satisfying such equations

are given the special name eigenfunctions, and the quantities produced as

a result of the operation are called eigenvalues. Consequently, Schrödinger

chose to title his paper ‘Quantization as an Eigenvalue Problem’.

For Schrödinger, the triumph of wave mechanics meant an oppor-

tunity to put signiﬁ cant elements of classical mechanics back into the

quantum picture. He now sought to replace discontinuous quantum

jumps between unvisualizable electron orbits with a more classical, vis-

ually appealing picture of smooth, continuous transitions between the

stationary wavefunctions of the system.

In the six months from January to June 1926, Schrödinger produced

a series of six highly creative papers on the new wave mechanics, born

from what Weyl called a ‘late, erotic outburst in his life’. Among them

was a paper in which Schrödinger demonstrated the mathematical equiv-

alence between matrix and wave mechanics. In classical mechanics, the

momentum ( p) of a particle is given by its mass multiplied by velocity.

In wave mechanics this expression for momentum is replaced by a dif-

ferential operator. It is now accepted practice to distinguish between the

classical p and the wave mechanical operator equivalent by drawing a

caret,^, above the letter to indicate that it is now an operator, p̂

Needless to say, the effect on a wavefunction of a series of operations

depends on the order in which the operations are applied. In Schrödinger’s

wave mechanics, the effect of multiplying the wavefunction by the value

An operator is simply an instruction to do something to a mathematical function, such as

multiply it, differentiate it, etc.

Named for the nineteenth century Irish mathematician William Rowan Hamilton.

a late erotic outburst

of the position, q, and then applying the momentum operator, pˆ, is dif-

ferent from applying the momentum operator and then multiplying the

result by the position. In other words pˆ qy is not equal to q pˆy. In wave

mechanics the momentum operator is structured such that the position–

momentum commutation relation, pˆ q − q pˆ, is equal to −ih/2p.

‘I was absolutely unaware of any generic relationship with Heisen-

berg,’ Schrödinger wrote in a footnote to his paper, ‘I naturally knew

about his theory, but because of the to me very difﬁ cult-appearing meth-

ods of transcendental algebra and because of the lack of Anschaulichkeit

[visualizability], I felt deterred by it, if not to say repelled.’ The feelings

were mutual. As Heisenberg wrote to Pauli shortly after the publication

of Schrödinger’s equivalence paper: ‘The more I think about the physical

portion of the Schrödinger theory, the more repulsive I ﬁ nd it . . . What

Schrödinger writes about the visualizability of his theory “is [paraphras-

ing Bohr] probably not quite right”, in other words, it’s crap.’

But, despite Heisenberg’s misgivings, Schrödinger’s tour de force began to

win many adherents. Attention now turned to the electron wavefunctions

themselves. What were they and how were they meant to be interpreted?

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PART II

Quantum

Interpretation