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

Подождите немного. Документ загружается.

the quantum story

Heisenberg was at once both pleased and relieved: ‘Only much later did

I learn from Born that it was simply a matter here of multiplying matrices,

a branch of mathematics that had hitherto remained unknown to me.’ He

acquired some textbooks on the subject and quickly caught up. He was

soon collaborating with Born and Jordan and together they published a

further paper on the new quantum mechanics in November 1925.

A proof copy of Heisenberg’s paper found its way to respected English

physicist Ralph Fowler in Cambridge. Unimpressed, Fowler passed the

copy to his young student, 23-year-old Paul Dirac.

It was only when Dirac studied the proofs properly, about a week or

so after he had ﬁ rst been given them, that he realized precisely what

Heisenberg had done. He now worked intensely on the problem, identi-

fying analogies between Heisenberg’s multiplication rule and mathemat-

ics developed by French theoretician Siméon Poisson in 1809. When he

had ﬁ nished he had independently arrived at the position–momentum

commutation relation, proved the principle of energy conservation, and

had derived the relationship between the energies of the stable orbits and

frequency of emitted radiation, the relation originally deduced by Bohr.

Dirac sent a handwritten version of his paper describing these results

to Heisenberg, and was subsequently disappointed to learn that his work

had been anticipated by Born and Jordan. Heisenberg was nevertheless

full of praise: ‘. . . on the one hand, your results, especially concerning the

general deﬁ nition of the differential quotient and the connection of the

quantum conditions with the Poisson brackets, go considerably further

than [Born and Jordan’s] work; on the other hand, your paper is also writ-

ten really better and more concisely than our formulations given here.’

The new quantum revolution was now ﬁ rmly underway. The beginnings

of a true quantum mechanics had been formulated, but with new, unprec-

edented levels of mathematical abstraction. Heisenberg’s prescription, as

elaborated by Born and Jordan, came to be known as matrix mechanics

(a name that Heisenberg hated, because it sounded too mathematical).

But the new theory was not to everybody’s taste. If it was to displace

the ‘old’ quantum theory, the new quantum mechanics had to show that

it could do everything the old theory could do. Not least it had to predict

the emission spectrum of the hydrogen atom. There was no shortage of

challenges to be faced.

There was no great rush to embrace the new quantum mechanics, with its abstract,

inﬁ nite-dimensional tables of numbers and obscure multiplication rules. Bohr was

enthusiastic. Einstein was not.

As work began to deploy the new methods in the calculation of the emission spec-

trum of hydrogen, the problem of the ‘anomalous’ Zeeman effect continued to tease the

intellects of the new quantum physicists. The ‘normal’ Zeeman effect had been nicely

accommodated in the old quantum theory through the introduction of a third (magnetic)

quantum number, m. But the anomalous effect remained stubbornly intransigent. When

young Austrian physicist Wolfgang Pauli was stopped in the street in Copenhagen and

asked why he was unhappy, he replied: ‘How can one look happy when he is thinking

about the anomalous Zeeman effect?’

In an attempt to classify the nature and magnitude of the splitting of the spectral lines

of multi-electron atoms in 1920 Sommerfeld had introduced a fourth quantum number,

which he called the ‘inner quantum number’, j, and a new selection rule. Where the ﬁ rst three

quantum numbers had evolved from reference to classical conceptual models of the inner

workings of the atom, this fourth quantum number was entirely ad hoc. Sommerfeld sim-

ply assumed that the motions of multi-electron atoms are complex and characterized some-

how by a ‘hidden rotation’. In 1921 German spectroscopist Alfred Landé at the University

of Tübingen suggested that the quantum numbers m and j could take half-integral values.

Two years later he proposed that for atoms with a many-electron ‘core’ and a single outlying,

‘valence’ electron, the hidden rotation should be associated with the core electrons.

The Self-rotating Electron

Leiden, November 1925

the quantum story

Landé’s approach was relatively successful, but it was still all quite confusing and

barely comprehensible. To add insult to injury, in 1922 German physicists Otto Stern and

Walther Gerlach had studied the effects of a magnetic ﬁ eld on a beam of silver atoms, pro-

duced by effusion from a heated oven. Stern had been looking for evidence for the kind of

space quantization predicted by the Bohr–Sommerfeld theory and when they found that

the beam was indeed split into two components by the magnetic ﬁ eld, they thought they

had found it. Others, including Einstein, were convinced they had not, and were greatly

puzzled by these results.

The explanation, when it came, would be the last hurrah of the old quantum theory.

Pauli had already established a reputation as a prodigy of Einstein’s rela-

tivity when he arrived in Munich in the autumn of 1918, at the tender

age of eighteen, to join Sommerfeld’s research team. He wrote a review

article on general relativity which was published a few years later and

of which Einstein wrote: ‘Whoever studies this mature and grandly con-

ceived work might not believe that its author is a twenty-one year-old

man.’ The renowned physicist and philosopher Ernst Mach was Pauli’s

godfather.

At Munich, Sommerfeld introduced Pauli to the intricacies of atomic

theory and the number-games and selection rules of the old quantum

theory. Pauli received his doctorate in 1921, moving to Göttingen to

work with Born for a short period before moving to the University of

Hamburg. Like Heisenberg, Pauli ﬁ rst met Bohr during the festival in

Göttingen in 1922, and was deeply impressed by Bohr’s personality, wis-

dom, and insight. On Bohr’s invitation, he spent a year in Copenhagen

before returning to Hamburg in late 1923.

It was in Copenhagen that Pauli had begun to wrestle with the prob-

lem of the anomalous Zeeman effect. It caused him considerable difﬁ -

culties and regular periods of anguish and despair. Whilst he was able

to improve somewhat on the proposals of Sommerfeld and Landé, he

disliked the ad hoc nature of the theorizing that was involved. He was

looking for something more fundamental.

Back in Hamburg towards the end of 1924, Pauli’s attention was drawn

to the work of Cambridge physicist Edmund Stoner, described in the

preface to the fourth edition of Sommerfeld’s book Atombau und Spek-

trallinien. In Stoner’s original paper, published in October 1924 in the

the self-rotating electron

British journal Philosophical Magazine, he had set out a scheme describing

the relationship between the quantum numbers and the idea of electron

‘shells’, surrounding the nucleus and imagined to nest one inside the

other like a Russian matryoshka doll. The energy of each shell is deter-

mined by the principal quantum number, n. The number of possible

states or ‘orbits’ within each shell is then determined by the values that

the quantum numbers k and m can take for a given value of n.

The rules dictated that k must be an integral number greater than zero

and less than or equal to n,

and m could take a total of 2k − 1 values over

the range −(k − 1), −(k − 2), … 0, …, (k − 2), (k − 1). So, for n = 1, the only

values of k and m that are possible are k = 1 and m = 0. This implies a single

state, or orbit. For n = 2, the rules imply four different orbits, for n = 3 the

number of orbits is nine. This implies that the number of possible orbits

increases as n

But the pattern reﬂ ected in the periodic table of the elements tells a

slightly different story. German physicist Walther Kossel had earlier

argued that the striking stability and inertness of the noble gases (such

as helium, neon, argon, krypton) could be understood in terms of Bohr’s

atomic theory if these atoms were assumed to have ﬁ lled, or ‘closed’,

shells. The periodic table could then be understood as a progression of

occupancy of the electron shells, forming a pattern in which ﬁ rst two

(hydrogen, helium), then eight (lithium through to neon), then another

eight (sodium to argon), then eighteen electrons (potassium to krypton)

are added in sequence until each successive shell is ﬁ lled, or closed.

Stoner had gone one step further in his prescription. Instead of assign-

ing a single electron to each orbit he had chosen to assign two: ‘In the

classiﬁ cation adopted, the remarkable feature emerges that the number

of electrons in each completed level is equal to double the sum of the

inner quantum numbers as assigned…’. In n

orbits, Stoner suggested,

there should be 2n

electrons. For n = 1 there is only one orbit, implying

occupancy up to a total of two electrons, for n = 2 there are four orbits,

The restriction that k must be greater than zero was impossible to prove in the old quantum

theory. In modern atomic physics, k has been replaced by the orbital angular momentum (or

azimuthal) quantum number l, which takes values l = 0, 1, 2, …, n. Consequently, the magnetic

quantum number m takes the values −l, …, 0, …, l.

the quantum story

implying up to eight electrons, for n = 3 there are nine orbits and up to

eighteen electrons.

Pauli put two and two together. He believed that Landé’s model in

which a ‘hidden rotation’ was ascribed to the core of electrons in a multi-

electron atom was wrong. And yet the model appeared to work quite well.

Stoner was suggesting a doubling of the electron count ascribed to each

orbit, and hence each shell. The answer, Pauli reasoned, was to ascribe the

fourth quantum number not to the core of electrons, but to each individual

electron. He was led to the inspired conclusion that the electron must have

a curious, non-classical ‘two-valuedness’ characterized by a half-integral

quantum number. There was more. The shell structure of atoms and the

periodic table of the elements implied that each orbit could accommo-

date two and only two electrons. Pauli wrote:

There can never be two or more equivalent electrons in the atom, for

which, in strong ﬁ elds, the values of all quantum numbers . . . coincide. If

one electron is present in the atom, for which these quantum numbers (in

the external ﬁ eld) have deﬁ nite values, then this state is ‘occupied’.

This is Pauli’s exclusion principle. It says that no electron in an atom can have

the same set of four quantum numbers. An electron in a state character-

ized by n = 1, k = 1, m = 0, and j = +½ can enter the same orbit (with the

same values of n, k, and m) only by taking the quantum number j = −½. No

other values of j are possible so each orbit can contain only two electrons.

Pauli was unable to provide any formal explanation for this rule and

had no alternative but to argue that it seemed to ‘offer itself automati-

cally in a natural way’. When in December 1924 he sent a copy of the

To make explicit the connection with the 2, 8, 8, 18, . . . pattern of the periodic table it is

necessary to jump ahead to our current understanding of the atomic orbits and their relative

energies. The orbit with n = 1, k = 1 (l = 0) is spherical and named by convention as the 1s orbit.

This can accommodate up to two electrons (accounting for hydrogen and helium). For n = 2 the

possible orbits are 2s and three different 2p orbits, accommodating up to a total of eight elec-

trons (lithium to neon). For n = 3 the orbits are 3s, 3p (up to eight electrons—sodium to argon)

and ﬁ ve different 3d orbits (ten electrons). However, the 4s orbit actually lies somewhat lower

in energy than 3d and is ﬁ lled ﬁ rst. Therefore, the combination 4s, 3d, and 4p (accommodating

up to eighteen electrons in total) together account for the next row of the periodic table, from

potassium to krypton.

the self-rotating electron

manuscript of a paper describing his results to Bohr and Heisenberg in

Copenhagen, he received an exuberant, though barbed, response from

Heisenberg:

I am the one who rejoices most about it, not only because you push the swin-

dle to an unimagined, giddy height ( by introducing individual electrons

with 4 degrees of freedom) and thereby have broken all hitherto existing

records of which you have insulted me, but quite generally, I triumph that

you too (et tu, Brute!) have returned with lowered head to the land of the

formalism pedants . . .

Although its origin in the as yet undeﬁ ned quantum mechanics of the

atom remained vague, Pauli’s exclusion principle accounted for an

important feature of multi-electron atoms—the fact that they exist at all.

There had been nothing in the earlier atomic theory to provide a reason

why all the electrons in a multi-electron atom should not simply collapse

down into the lowest-energy orbit.

For neutral atoms, increasing the number of electrons implies an

increase in the total positive charge of the nucleus. The greater the nuclear

charge, the smaller the radius of the innermost orbit, as the electrons are

pulled more tightly towards the nucleus. Now we would expect that the

repulsion between increasingly closely packed electrons would tend to

resist collapse of the atom into ever-smaller orbits and hence ever-smaller

volumes, but it is relatively straightforward to show that electron–electron

repulsion cannot prevent heavier atoms from shrinking dramatically in

size as the charge of the central nucleus is increased. The repulsion between

neighbouring electrons simply isn’t strong enough to overcome the force

of attraction. Atomic volumes, easily calculated from atomic weights and

densities of the elements, follow a complex variation with atomic charge

but they do not systematically shrink with increasing charge.

By preventing the electrons from collapsing or condensing into the low-

est-energy orbit, the exclusion principle allows complex multi-electron

atoms to exist in the pattern described by the periodic table. It enables

the existence of a marvellous variety of elements, the multitude of pos-

sible chemical combinations, and hence all material substance, living and

non-living. This was a fantastic achievement.

the quantum story

But, still, why only two electrons per orbit?

Perhaps, argued a young American physicist named Ralph de Laer

Kronig, this is because the electron’s ‘two-valuedness’ is associated with

self-rotation. Kronig was born in Dresden in 1904 to American parents.

In January 1925 he had travelled back to Germany from America, where

he had recently completed a PhD at Columbia University in New York.

He had returned to Germany to work with Landé.

In late December 1924 Pauli advised Landé by postcard that he intended

to visit Tübingen on 9 January. He was particularly interested to discuss

the exclusion principle and to try to ﬁ nd any spectroscopic data among

Landé’s extensive collection that might tell against it.

Landé showed his new American assistant the correspondence he had

received from Pauli concerning the exclusion principle. Kronig was struck

by the application of the fourth quantum number to individual electrons

and immediately saw a potential explanation. Sommerfeld had ascribed the

fourth quantum number to a ‘hidden rotation’. What if this rotation were,

in fact, a real self-rotation of individual electrons? If the electron rotates

about its own axis, in much the same way that the earth rotates on its axis

as it orbits the sun, then this would generate a small, local magnetic ﬁ eld.

The electron in an atom would behave like a tiny bar magnet. It would pos-

sess a magnetic moment that can become aligned with or aligned against

the lines of force of an applied external magnetic ﬁ eld, giving two states of

different energy which would appear as a splitting of spectral lines.

In the absence of this splitting, there is only one state of the electron and

hence only one line in the spectrum. Kronig calculated that the angular

momentum of electron self-rotation was required to have the ﬁ xed value

of ½ ħ, where ħ, is a shorthand for Planck’s constant h divided by 2p. In

addition, he was able to show that the ratio of the magnetic moment and

the angular momentum due to self-rotation, a characteristic factor known

as the Landé ‘g-factor’ for the electron, had to have the value 2. This was

rather curious, as the ratio of the electron orbital magnetic moment to the

orbital angular momentum is 1, as predicted by classical mechanics.

There was a further problem. From Kronig’s hastily derived expressions

he calculated a splitting of spectral lines that was a factor of two larger

the self-rotating electron

than the splitting observed experimentally. This was not connected with

the assumption of g = 2, as this was needed to explain other experimental

observations. Electron self-rotation was able to resolve two out of three of

the problems with Landé’s core model, but could not yet resolve them all.

The next day Kronig went to collect Pauli from the railway station:

For some reason I had imagined him [Pauli] as being much older and as

having a beard. He looked quite different from what I had expected, but

I felt immediately the ﬁ eld of force emanating from his personality, an

effect fascinating and disquieting at the same time. At Landé’s institute a

discussion was soon started, and I also had occasion to put forward my

ideas. Pauli remarked: ‘Das ist ja ein ganz witziger Einfall,’ [this is indeed quite

a witty idea], but did not believe that the suggestion had any connection

with reality.

Pauli, noted as much for his biting wit as for his talents as a theoretical

physicist, completely dismissed Kronig’s suggestion. Kronig discussed

it further with Bohr, Kramers, and Heisenberg, who were similarly dis-

missive. He was in any case troubled by the factor of two discrepancy

between prediction and experimental observation, and concerned also

that the equator of a spinning sphere of charged matter would be required

to move ten times faster than the speed of light, forbidden by Einstein’s

special theory of relativity. He subsequently dropped the idea.

Ten months can be a long time in physics. When two young Dutch phys-

icists Samuel Goudsmit and George Uhlenbeck, based in Leiden in the

Netherlands, independently reached the same conclusion, the climate in

the physics community was to prove more temperate. They summarized

their arguments in favour of electron self-rotation in a paper they sub-

mitted to the journal Naturwissenschaften. After submitting the paper, they

talked to the esteemed physicist Hendrik Lorentz. He advised them that

their proposal was almost impossible in classical electron theory. Fearing

that they had made a signiﬁ cant error, they scrambled to withdraw their

paper before it could be published. But it was too late.

The paper was published in November 1925. Initially it provoked

the same concerns that Kronig’s proposal, now almost forgotten, had

the quantum story

attracted. Travelling to Leiden in December 1925, Bohr was met at the

railway station by Pauli and Stern and asked for his opinion about the

Dutch proposal. Bohr may have said that he found it ‘very interesting’,

Bohr-speak implying that it was probably wrong. On arrival in Leiden

he was met by Einstein and Austrian physicist Paul Ehrenfest, who asked

him the same question. When Einstein went on to explain how some of

his objections to the proposal could be overcome, Bohr began to have a

change of heart.

From Leiden Bohr travelled to Göttingen, where he was met by Heisen-

berg and Jordan, who asked the question again. This time Bohr was

enthusiastic, although Heisenberg vaguely recalled having heard a simi-

lar proposal some time before. On his way back to Copenhagen Bohr’s

train stopped in Berlin, where he was met by Pauli who had travelled

from Hamburg expressly to ask Bohr once again what he thought about

electron self-rotation. Bohr now said it represented a great advance. Pauli,

still unable to get past the fact that the classical picture of a spinning

bit of charged matter made absolutely no sense in the context of atomic

physics, called it ‘a new Copenhagen heresy’.

Bohr became a strong advocate of electron self-rotation and may have

been the ﬁ rst to use the term ‘electron spin’. This is a term that has stuck,

despite the fact that its meaning in quantum theory is considerably

far removed from its classical interpretation. As the idea gained wider

acceptance, it became clear that Pauli (and Bohr, Kramers, and Heisen-

berg) had discouraged Kronig from developing his own proposal further

and from therefore becoming the ‘discoverer’ of electron spin.

Kronig

tended to play down the matter, but could not help feeling some bitter-

ness: ‘I should not have mentioned the matter [to Kramers] at all,’ he

wrote later to Bohr, ‘if it were not to take a ﬂ ing at the physicists of the

preaching variety, who are always so damned sure of, and inﬂ ated with,

the correctness of their own opinion.’

As Bohr had anticipated, the problem of the mysterious factor of

two discrepancy between the predicted and observed splitting of spec-

A verse penned some time later summarized the situation: ‘Der Kronig hätt’ den Spin entdeckt,

hätt’ Pauli ihn nicht abgeschreckt.’ (Kronig would have discovered the spin if Pauli had not discour-

aged him.) See Enz, p. 117.

the self-rotating electron

tral lines was resolved satisfactorily. English physicist Llewellyn Hilleth

Thomas subsequently showed that re-casting the problem in the proper

rest frame of the electron changed the expression for the splitting, replac-

ing the appearance of g in the expression with g−1. Assuming g = 2 effec-

tively reduces the predicted splitting by half.

It was Paul Dirac who subsequently suggested that if the electron can be

considered to possess two possible ‘spin’ orientations then this, perhaps,

explains why each atomic orbit can accommodate only two electrons.

The two electrons must be of opposite spin to ‘ﬁ t’ in the same orbit. An

orbit can hold a maximum of two electrons provided their spins are

paired.

This was great progress, but there were still many, many puzzles. A clas-

sical spinning object is not constrained in principle to only two positions

of alignment of its magnetic moment. It was reasoned that this restriction

must be somehow due to the quantum nature of the electron.

It was just not clear how.