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

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the quantum story

De Broglie’s other examiners were Jean Perrin, Élie Cartan and Charles Mauguin.

The longest wavelength standing wave is therefore equal to twice the

length of the string or pipe. Such a wave has no ‘nodes’—points where

the wave amplitude passes through zero—between the ends. The next

wave is characterized by a wavelength equal to the length of the string

or pipe, with one node between the ends. The third has one-and-a-half

wavelengths and two nodes, and so on.

De Broglie saw that whole-number requirement in Bohr’s theory

of atomic structure would emerge naturally from a model in which

an electron wave is conﬁ ned to a circular orbit around the nucleus.

Perhaps, he reasoned, the stable electron orbits of Bohr’s theory

represented standing electron waves, just like the standing waves in

strings and pipes responsible for musical notes. The same kinds of

arguments would apply. For a standing wave to be produced in a cir-

cular orbit, the electron wavelengths must ﬁ t exactly within the orbit

circumference.

Put simply, by the time the electron wave has performed one complete

orbit and returned to the starting point, the value of the associated wave

amplitude and the phase of the wave (its position in its peak–trough

cycle) must be the same as at the starting point. If this is not the case, the

wave will not ‘join up’ with itself: it will interfere destructively and no

standing wave will be produced.

To satisfy the requirement, the wavelength of the electron wave must

be such that an integral number of wavelengths will ﬁ t into the circum-

ference of the orbit. This is called the resonance or phase condition.

Bohr’s quantum numbers could therefore be thought of as the number

of wavelengths of the electron wave present in each orbit. ‘Thus,’ wrote

de Broglie, ‘the resonance condition can be identiﬁ ed with the stability

condition from quantum theory.’

De Broglie published his ideas in a series of three short papers in the

Comptes Rendus (proceedings) of the Paris Academy in September and

October 1923. He collected these papers together, extended his ideas and

presented them as a PhD thesis to the Faculty of Science at Paris University.

He produced three typed copies and passed one of these to the prominent

French physicist Paul Langevin, who was to act as an examiner.

la comédie française

Langevin didn’t know quite what to make of de Broglie’s bold (and,

possibly, crazy) ideas and sought guidance from Einstein, now professor

at the University of Berlin. Einstein asked for a further typed copy to be

sent to him.

Einstein was by now famous for his work on both the special and

general theories of relativity, propelled into the public consciousness by

the observations of the bending of starlight by the sun during the total

solar eclipse of 1919 that had spectacularly conﬁ rmed the predictions

of the general theory. The peculiar action-at-a-distance demanded by

Newton’s theory of gravity, denounced at the time as an ‘occult agency’

and made even more peculiar through the non-existence of the ether,

had now been replaced by the motions of gravitational bodies in a

curved space–time.

Einstein may have recognized a little of his former rebelliousness in

de Broglie’s work. He wrote back to Langevin offering encouragement:

‘He [de Broglie] has lifted a corner of the great veil’. This was enough

for Langevin. He accepted the thesis and de Broglie was duly awarded

his doctorate in November 1924. De Broglie’s thesis was published in its

entirety in the French scientiﬁ c journal Annales de Physique in 1925.

In a December 1924 letter to Dutch physicist Hendrik Lorentz, Ein-

stein remarked that:

. . . de Broglie has undertaken a very interesting attempt to interpret the

Bohr–Sommerfeld quantum rules (Paris dissertation 1924). I believe it is a

ﬁ rst feeble ray of light on this worst of our physics enigmas.

Not everyone was convinced. Most physicists were rather sceptical: ‘some

wit, in fact, dubbed de Broglie’s theory “la Comédie Française”.’

De Broglie’s ideas were illuminating but they were far from representing

a solution. Assuming that material particles like electrons could possess

associated wave-like properties had led de Broglie to identify the quantum

numbers of Bohr’s theory with the resonance condition for standing waves.

But this was nothing more than a theoretical connection. De Broglie had

not derived the quantum numbers from some kind of wave theory of the

atom. And he had as yet no explanation for quantum jumps:

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From this we see why certain orbits are stable; but, we have ignored

passage from one to another stable orbit. A theory for such a transition

can’t be studied without a modiﬁ ed version of electrodynamics, which

so far we do not have.

Einstein was already concerned about the implications of quantum

jumps for the principles of causality, the notion that every effect in the

physical universe is traceable to a direct physical cause. It was clear that

an electron in a high-energy atomic orbit will fall to a more stable, lower-

energy orbit. It is caused to do so by the mechanics of the atom. But the

theory allows only the calculation of the probability that a spontaneous

jump will take place. The exact moment of the jump and the direction of

the consequently emitted light-quantum appeared to be left entirely to

chance. These were things that could not be predicted.

Einstein was not at all comfortable with this. In 1920 he had written

to German physicist Max Born on the subject, noting that he: ‘would

be very unhappy to renounce complete causality’. From the very begin-

ning, Einstein had viewed the quantum hypothesis as provisional, to be

replaced eventually by a new, more complete theory that would properly

explain quantum phenomena.

After pioneering quantum theory through one of its most testing

early periods, Einstein was beginning to have grave doubts about what

the theory implied.

One thing was clear. The quantum jumps between stable atomic orbits

simply could not be accommodated in accepted theories of the dynamics

of electrically charged particles. As de Broglie had acknowledged in his

thesis, a modiﬁ ed electrodynamics was required. Familiar models were

no longer of any use when applied to the mysterious interior of the atom.

A radical new vision was needed.

Despite the success of the Bohr–Sommerfeld rules, the ‘old’ quantum theory (as it came

to be known) was creaking under the strain of a growing body of inexplicable experi-

mental results. Whilst the spectrum of the simplest atom—that of hydrogen—could be

‘explained’ using these rather ad hoc rules with some degree of conﬁ dence, the spectrum of

the next simplest atom—helium—could not. The spectra of certain other types of atoms,

such as sodium and the atoms of rare-earth elements, showed ‘anomalous’ splitting when

placed in a magnetic ﬁ eld.

The old quantum theory, created by shoe-horning quantum

rules into a classical mechanical structure, simply could not account for this effect.

The problems with theory spilled over into confrontation. Bohr’s general reluctance to

embrace Einstein’s light-quantum led him to formulate an alternative approach which

did not invoke point-like particles to explain phenomena such as the photoelectric effect.

He found, however, that there was a high price to be paid for this reluctance. In early 1924,

together with Dutch physicist Hendrik Kramers and American John C. Slater, Bohr had

proposed a theory in which the principles of conservation of momentum and energy

within individual atomic events had been abandoned, replaced instead by the appearance

of conservation as a result of statistical averaging. Einstein was furious.

The Bohr–Kramers–Slater (BKS) proposal was subsequently shot down by further

experiments in early/mid-1925. Bohr, whose personality and power of persuasion had

A Strangely Beautiful Interior

Helgoland, June 1925

In fact, this ‘anomalous’ Zeeman effect (named for Dutch physicist Pieter Zeeman) is actu-

ally the normal behaviour. It is the general behaviour of simpler atoms, regarded by the physi-

cists of the time as normal, that is actually anomalous.

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convinced his junior colleagues of the merits of the proposal against their better judge-

ment, agreed to give the effort an ‘honourable funeral’. There was now little alternative but

to accept the reality of light-quanta.

The BKS proposal was symptomatic of the physicists’ growing sense of desperation.

It was becoming increasingly clear that a new quantum theory was needed, constructed

somehow from the ‘bottom up’, from a new set of principles that did not depend on con-

tradictory classical pictures of charged, material particles in planetary-like orbits around

a central nucleus.

There was revolution in the air.

It had become apparent in 1917 that the cramped working conditions at

the University in Copenhagen could not accommodate Bohr’s vision for

Danish physics. He had set off on the long and rather tortuous journey that

would lead eventually to the construction of a new Institute for Theoretical

Physics. The Institute was completed in 1920 and inaugurated in March

1921. Bohr had involved himself directly in the construction planning. He

secured further funding in the early 1920s from the Carlsberg Foundation

and the International Education Board (founded by John D. Rockefeller in

1923) to purchase scientiﬁ c equipment, provide student grants and funds

for visiting scholars, and to extend the Institute further to accommodate

this burgeoning activity.

There followed a grand procession of aspiring young physicists, beat-

ing a path to the neo-classical style doors of Bohr’s Institute, to study

with the master of atomic theory. One young German physicist recently

arrived during the Easter vacation in 1924, experiencing problems with

Danish customs and unable to speak the language, realized that all he

had to do was mention Bohr’s name:

. . . as soon as it became clear that I was about to work in Professor Bohr’s

Institute, all difﬁ culties were swept out of the way and all doors were

opened to me. And so from the very outset I felt safe under the protection

of one of the greatest personalities in this small but friendly country.

Werner Heisenberg had ﬁ rst met Bohr in Göttingen in June 1922, during

a scientiﬁ c festival held in Bohr’s honour. Just 20 years old, he had been

studying for his doctorate in physics under Sommerfeld at the University

of Munich, and had rather impudently stood at the end of the third of

a strangely beautiful interior

Bohr’s lectures to make a critical remark. Bohr had replied hesitantly,

and after the lecture had invited Heisenberg to join him on a walk over

the nearby Hain Mountain. Bohr explained that he fully understood the

nature of Heisenberg’s doubts, and invited the young physicist to join

him in Copenhagen for a term’s study leave.

In July 1923 Heisenberg completed his doctorate in Munich, youth-

ful arrogance and disdain for aspects of experimental physics caus-

ing him some problems during his oral examination. Unable to derive

the equations for the resolving power of a microscope to the satisfac-

tion of the aged Wien, Heisenberg achieved a poor pass result. It was

a salutary experience for the young doctor, dismissed by Wien as an

upstart.

From Munich Heisenberg had travelled to Göttingen to work with

Max Born with the intention of qualifying as a university lecturer. ‘He

looked like a simple peasant boy,’ Born later recalled, ‘with short, fair

hair, clear bright eyes and a charming expression.’ In Göttingen he had

his ﬁ rst experiences wrestling with the problems that beset the quantum

theory of the atom. When in January 1924 Bohr repeated his invitation

for Heisenberg to come to Copenhagen, he accepted and made arrange-

ments to travel to Denmark during the Easter break.

Heisenberg was received warmly at Bohr’s Institute, but was over-

whelmed and somewhat intimidated by what he found there. The

Institute was a bustling centre for atomic physics and he encountered: ‘a

large number of brilliant young men from every part of the world, all of

them greatly superior to me, not only in linguistic prowess and worldli-

ness, but also in their knowledge of physics.’

He saw little of Bohr in those ﬁ rst few days of his stay in Copenhagen.

But Bohr eventually found some time and asked Heisenberg to join him

on another lengthy walk, through the island of Zealand to the Kronborg

Castle, long associated with Denmark’s most famous, but possibly myth-

ical, prince Hamlet.

In coming to Copenhagen Heisenberg had, for a short time at least,

escaped the vicissitudes of a Germany in social and political turmoil. In

February 1924 Adolf Hitler had been sentenced to ﬁ ve years’ imprison-

ment for his part in the failed beer-hall putsch of the previous Novem-

ber, but his trial had been a major propaganda victory for the National

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Socialist party.

Bohr was interested in what the young German had to

say about events in his country.

‘But now and then our papers also tell us about more ominous, anti-

Semitic, trends in Germany, obviously fostered by demagogues. Have

you come across any of that yourself ?’ he asked Heisenberg. Bohr was

part Jewish.

Indeed he had. Heisenberg told Bohr of his experiences during a lec-

ture on general relativity delivered by Einstein in Leipzig in the summer

of 1922. He had thought that science would be above political strife of

the kind that had led to civil war in Munich, and was greatly saddened to

witness student demonstrations against Einstein’s ‘Jewish physics’, spon-

sored by Philipp Lenard.

Heisenberg’s ﬁ rst visit to Copenhagen was short, but it was followed by

many longer visits. He returned to Bohr’s Institute in September 1924,

supported by grants from the Carlsberg Foundation and the Interna-

tional Education Board. During this second visit he worked closely with

Hendrik Kramers, a talented Dutch theoretician who by this time had

become Bohr’s right-hand man.

It was this work with Kramers that helped to convince Heisenberg

that progress in atomic theory could only be made by abandoning any

attempt to ‘understand’ the interior workings of the atom. He came to

realize that the planetary model, with its images of material particles cir-

culating around a central nucleus in a series of stable orbits, was visually

rich but analytically empty. As a classical mechanical model, its useful-

ness had been prolonged by the addition of rather arbitrary quantum

rules, but it was surely doomed to failure.

Back in Göttingen, Born had argued for a new ‘quantum mechanics’

to replace the classical theory as applied to the internal structure of the

atom. Heisenberg had elected to focus his attention on what could be

seen, rather than what could only be guessed at. The secrets of the atom

where revealed in atomic spectra, in the precise patterns of frequencies

and intensities (or ‘brightness’) of individual spectral lines. Heisenberg

now realized that a new quantum theory of the atom should deal only

Hitler served only eight months of his sentence. Whilst in prison he wrote Mein Kampf with

Rudolph Hess.

a strangely beautiful interior

in these observable quantities, not unobservable mechanical ‘orbits’ con-

forming to arbitrary quantum rules.

On Hain Mountain, Bohr had spoken of the challenges they faced.

‘These models,’ he had said, ‘have been deduced, or if you prefer guessed,

from experiments, not from theoretical calculations. I hope that they

describe the structure of atoms as well, but only as well, as is possible in

the descriptive language of classical physics. We must be clear that, when

it comes to atoms, language can be used only as in poetry. The poet, too,

is not nearly so concerned with describing facts as with creating images

and establishing mental connections.’

Heisenberg decided it was now time for a new language.

He returned once more to Göttingen in April 1925. Towards the end of

May he succumbed to a severe bout of hay fever, and asked Born for 14

days’ leave of absence to recuperate. On 7 June he arrived on the small

island of Helgoland, just off the north coast of Germany, hoping that the

clear North Sea air would facilitate a speedy recovery. His face was so

badly swollen that the landlady of the boarding house where he stayed

was convinced he had been in a ﬁ ght.

Free from distractions, he now made swift progress. He had been

working on an approach to atomic theory in which the parameters of

an unobservable interior mechanics were replaced by terms correspond-

ing to atomic events that could be observed—the jumps or transitions

between orbits which were manifested as spectral lines. He had con-

structed a rather abstract model consisting of an inﬁ nite series (called a

Fourier series) of harmonic oscillators, each characterized by an ampli-

tude and a frequency.

He had identiﬁ ed each oscillator—each term in

the Fourier series—with a quantum jump from some stable orbit charac-

terized by a quantum number n to an orbit characterized by a quantum

number m. The result was an inﬁ nite table of symbols or terms in the

Fourier series, organized into columns and rows, each term representing

a quantum transition from some initial to some ﬁ nal state.

He now assumed that he could calculate the intensities of the resulting

spectral lines as the squares of the amplitudes of the terms that appeared

More familiar examples of harmonic oscillators include a swinging pendulum and a ball

moving up and down in regular motion attached to a spring.

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in the table. In the case of a jump from state n to n − 2, for example, he

found it necessary to multiply the amplitude of the term corresponding

to the jump n to n − 1 with the amplitude of the term corresponding to

the jump from n − 1 to n − 2. More generally, he calculated the intensity

of a spectral line resulting from any quantum jump as the sum of the

products of the amplitudes for all possible intermediate jumps.

Whilst this multiplication rule seemed straightforward and satisfac-

tory, Heisenberg was aware of a potential paradox. If this same rule

is used to calculate the product of two different physical quantities (x

and y, say) then it suggests that there may arise situations in which the

product x multiplied by y is not equal to the product y multiplied by

x. Heisenberg was quite unfamiliar with this kind of result and greatly

unsettled by it.

He wasn’t quite sure where all this was leading when he arrived on

Helgoland. He had become aware that the scheme was not guaranteed

to be free of contradictions and, in particular, may not respect the need

for the conservation of energy. The lesson from the failure of the BKS

proposal was that conservation of energy was an imperative for any pur-

ported new quantum mechanics of the atom. Heisenberg now set about

the task of calculating the energies to check that everything was in order.

He worked on it long into the night:

When the ﬁ rst terms seemed to accord with the energy principle, I became

rather excited, and I began to make countless arithmetical errors. As a

result, it was almost three o’clock in the morning before the ﬁ nal result

of my computations lay before me. The energy principle had held for all

the terms, and I could no longer doubt the mathematical consistency and

coherence of the kind of quantum mechanics to which my calculations

pointed. At ﬁ rst, I was deeply alarmed. I had the feeling that, through

the surface of atomic phenomena, I was looking at a strangely beautiful

interior, and felt almost giddy at the thought that I now had to probe this

wealth of mathematical structures nature had so generously spread out

before me.

He was too excited to sleep. He quietly left his lodgings and walked in the

darkness, climbing a rock jutting out to sea on the southern tip of the

island. He watched as the sun rose.

a strangely beautiful interior

Heisenberg hastily wrote up his calculations and passed a copy of the

manuscript to Born. Despite his experience on Helgoland, he still wasn’t

certain if this rather intuitive approach made any real sense. Born recalled

Heisenberg telling him that: ‘he had written a crazy paper and did not

dare send it in for publication; I should read it and, if I liked it, send it to

Zeitschrift für Physik.’ Born was enthusiastic about the paper but puzzled

by the multiplication rule that Heisenberg had used. It seemed familiar.

On 10 July 1925 he ﬁ nally remembered. Born had been taught this multi-

plication rule as a student. It was the rule for multiplying matrices.

A matrix is a square or rectangular array of numbers organized in

columns and rows. Like ordinary numbers, matrices can be added, sub-

tracted, multiplied, and divided. Speciﬁ c rules are necessary to guide

matrix multiplication, showing how each element in each matrix must

be combined to give the corresponding elements in the ﬁ nal, product

matrix. Unlike ordinary numbers, it is possible to ﬁ nd matrices (for

example, x and y) that do not commute; that is, the product of x multi-

plied by y is not equal to y multiplied by x.

Born now worked with his student Pascual Jordan to recast Heisen-

berg’s calculations into the language of matrix multiplication. They

discovered that the matrix for the energy of the system is diagonal—all

the elements in the matrix are zero but for those along the diagonal of

the array, which are time-independent and represent the stable quantum

states (the ‘orbits’) of the system.

They also discovered that Heisenberg had been right to be disconcerted.

They found that the matrix equivalents of classical physical quantities

such as position and momentum do not commute. Denoting the posi-

tion matrix as q and the momentum matrix as p, they discovered that the

difference between the product pq and the product qp (called the com-

mutation relation) is equal to −ih/2p, where i is the square-root of minus

1 and h is Planck’s constant.

Classical mechanics (in which the values of

position and momentum are represented by ordinary numbers which

always commute) was once again seen as a limiting approximation of

quantum mechanics, in which Planck’s constant h is assumed to be zero.

Strictly speaking, pq−qp = −ih1/2p, where 1 is the unit or identity matrix.