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

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the quantum story

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coefﬁ cients, which became known as Pauli spin matrices, appeared to work

well. This wasn’t yet a full accounting for electron spin, but it was clearly

a step in the right direction.

Dirac’s problem was that he now had to ﬁ nd a way to handle four

square-root operators; three momentum operators, and a fourth arising

from a term which accounted for the variation of the mass of the elec-

tron with velocity, as demanded by the special theory of relativity. He

initially ﬁ gured that he just needed to ﬁ nd another two-by-two matrix

coefﬁ cient, but this just didn’t work. No such coefﬁ cient exists. Two-by-

two matrices were not the answer.

Dirac was a mathematician ﬁ rst, a physicist second. What confronted

him was a problem in mathematics. His principal concern was to solve

the mathematical problem, then worry about the physical interpretation

of the solution. As he played around with the equations, he was struck by

an insight: ‘I suddenly realised that there was no need to stick to quanti-

ties, which can be represented by matrices with only two rows and col-

umns,’ he later explained. ‘Why not go to four rows and columns?’

This did the trick. Using coefﬁ cients that were structured as four-by-four

matrices, he could linearize the square-root operators and go on to solve

the equation.

It became known as the Dirac equation. Although it predicted no

results that were not already given by earlier theories, it was a conceptual

triumph. He had made his discovery; something he could call his own.

The use of matrices with the same form as Pauli’s spin matrices meant

that the property of electron spin automatically emerged from Dirac’s

relativistic equation. The introduction of a four-dimensional space–time

into the equations of quantum theory had resulted in a fourth ‘degree

of freedom’ for the electron which, in turn, demanded a fourth quantum

number, as Goudsmit and Uhlenbeck had reasoned in November 1925.

But whatever it is, the property of electron spin does not correspond

in any way to the notion of an electron spinning on its axis. It is a purely

relativistic quantum property with no counterpart in classical physics.

In this sense it is very different from other properties of the electron.

For example, the orbital angular momentum of an electron in an atom is

related to the quantum number k multiplied by h and there is in principle

an absolute wonder

131

no restriction or upper bound on the value that k can take.

So, as we

push to the classical limit of orbital angular momentum by reducing h to

zero we can compensate by increasing k to inﬁ nity, with the result that

the orbital angular momentum tends towards its classical value.

However, the same is not true for electron spin. The spin quantum

number is constrained to one value for the electron and cannot be

increased to inﬁ nity as h tends to zero. There is no classical counterpart

for electron spin.

Although its interpretation is obscure, we do know that electron spin

produces effects that give rise to a small magnetic moment. This moment

can become aligned in the direction of the lines of force of an applied

magnetic ﬁ eld or against that direction. We have learned to think of these

as possibilities as ‘spin-up’ and ‘spin-down’. In a magnetic ﬁ eld, the two

possible orientations of the electron’s magnetic moment give rise to two

energy levels that are characterized by a magnetic spin quantum number,

giving rise to the ‘two-valuedness’ corresponding to the spin-up and spin-

down states. The two levels give rise to two lines in an atomic spectrum.

Reworking Dirac’s equation to describe an electron moving in an elec-

tromagnetic ﬁ eld produced results that reproduced the Zeeman effect, in

which the spin magnetic moment is twice the magnitude that would be

anticipated on the basis of classical mechanics. Dirac’s theory predicted

that the spin magnetic moment is related to the spin quantum number

of the electron multiplied by the Landé ‘g-factor’ for the electron. As in

Kronig’s original proposal, so in Dirac’s theory the g-factor is exactly 2.

Charles Darwin visited Cambridge just before Christmas 1927 and was

astonished to learn of Dirac’s result. He immediately wrote to Bohr:

‘[Dirac] has now got a completely new system of equations which does

the spin right in all cases and seems to be “the thing”. His equations are

ﬁ rst order, not second, differential equations!’

News spread quickly. Dirac wrote a letter to Max Born in Göttingen

ahead of publication of a paper setting out his approach. Léon Rosenfeld

described its reception: ‘It was immediately seen as the solution. It was

The quantum number k has been replaced in modern quantum theory by the orbital angu-

lar momentum (or azimuthal) quantum number l, which takes values l = 0, 1, 2, …, n.

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132

regarded really as an absolute wonder.’ Fowler communicated Dirac’s paper

on the relativistic quantum theory of the electron to the British journal

Proceedings of the Royal Society. The paper was received on 2 January 1928.

Dirac had been obliged to pay a price for his solution, however. His

use of four-by-four matrices meant that he had twice as many solutions

on his hands than he thought he had needed. The two possible orien-

tations of the electron spin angular momentum account for half of the

solutions available from Dirac’s equation. The remainder correspond to

electron states of negative energy and are an inevitable consequence of

using the correct relativistic expression for the total energy of a freely

moving particle.

The negative-energy solutions had some bizarre properties. Where the

‘ordinary’, positive-energy electrons would accelerate under an applied

force, the particles described by the negative-energy solutions actually

slowed down the harder they were pushed. The temptation of most

physicists when faced with such unreasonable results is to dismiss them

as ‘unphysical’ and continue only with the positive-energy solutions.

This might have been acceptable when considering problems in classical

physics, in which energy changes are assumed to be continuous and sys-

tems that start with positive energy cannot suddenly jump to negative-

energy states.

However, quantum mechanics admits exactly this kind of sudden,

discontinuous jump and it was perfectly feasible for a positive-energy

electron to jump to a negative-energy state. This would appear in the lab-

oratory as a jump from a ‘conventional’ state characterized by the familiar

negative electron charge, −e, to a state characterized by a very unfamiliar

positive electron charge, +e. No such transition had ever been observed.

This unusual behaviour left nagging doubts. Heisenberg wrote to

Bohr: ‘I am much more unhappy about the question of the relativistic

formulation and about the inconsistency of the Dirac theory. Dirac was

here [in Leipzig, June 1928] and gave a very ﬁ ne lecture about his ingen-

ious theory. But he has no more idea than we do about how to get rid of

the difﬁ culty e → −e.’

The extra solutions had to be taken seriously, and they were a worry.

The question was: what could they represent?

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Dirac wrestled with what became known as the ‘±’ problem for the next two years. In

December 1929 he outlined a proposed solution. Suppose, he said, that the universe is

ﬁ lled with a ‘sea’ of negative-energy states all occupied by spin-paired electrons. We would

have no way of knowing of the existence of such a sea because, when ﬁ lled, it wouldn’t

interact with anything and would merely serve as a kind of backdrop against which posi-

tive energy changes would be measured.

However, if an electron were to be promoted out of the sea—to become an observable,

positive-energy electron—it would leave behind a ‘hole’. The negative-energy hole would

behave exactly like a positive-energy, positively charged particle.

Dirac suggested that the positively charged particle created by the hole in the negative-

energy sea was, in fact, a proton. His logic was not without precedent. Rutherford had

referred to the nucleus of the hydrogen atom as a ‘positive electron’ for six years before

introducing the term ‘proton’ in 1920. And, making protons out of ‘electron holes’ had a

nice symmetry to it that fed Dirac’s desire to ﬁ nd a unitary description of the fundamental

constituents of matter.

As he reported to a meeting of the British Association for the Advancement of Science

in September 1930: ‘It has always been the dream of philosophers to have all matter built

from one fundamental kind of particle, so that it is not altogether satisfactory to have two

in our theory, the electron and the proton.

There are, however, reasons for believing that

The Photon Box

Brussels, October 1930

The neutron had not yet been discovered.

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the electron and the proton are really not independent, but are just two manifestations of

one elementary kind of particle.’

But Dirac had reached for the dream too soon. His proposal was roundly criticized on

all sides because, among other things, it demanded that the masses of the electron and

‘hole’-derived proton should be the same. It was already well known that there is a sub-

stantial difference in the masses of these particles, the proton heavier by a factor of almost

2000. The debate continued.

As the world’s most distinguished physicists gathered once again in Brussels for the

sixth Solvay conference on 20 October 1930, there was much to discuss. The subject of

the conference was magnetism, chaired by French physicist Paul Langevin following the

death of Lorentz. But the conference would be remembered not for the formal lectures that

were delivered on this subject, but for the debate on quite a different topic that took place

between the formal proceedings, as Einstein and Bohr resumed their contest.

In an article published in the German journal Die Naturwissenschaften the

year before, Bohr had expounded further on his theory of complementa-

rity, drawing parallels with Einstein’s theories of relativity.

The substan-

tial (though ﬁ nite) speed of light means that we can treat space and time

separately for objects moving with velocities considerably smaller than

that of light. Likewise, the very small (though ﬁ nite) magnitude of Planck’s

constant of action means that for classical, macroscopic objects it is pos-

sible to apply simultaneously both space–time and causal descriptions.

However, Bohr explained, when we consider the behaviour of objects

moving at speeds close to that of light, we cannot any longer ignore the

effects of relativity. Similarly, when we consider objects at the quan-

tum level, we cannot any longer ignore complementarity. For quantum

objects, the space–time and causal descriptions of nature cannot any

longer be applied simultaneously.

Einstein had rejected Newton’s absolute space and time because in

practice there is no such thing as absolute simultaneity. Why not just

accept that the uncertainty relations lead us to reject the simultaneous

validity of the classical concepts we are seeking to apply in the quantum

domain?

Bohr changed his terminology in this paper, dropping complementarity in favour of ‘reci-

procity’, a decision he latter regretted (see Pais, Niels Bohr’s Times, p. 426). He quickly reverted to

the term complementarity. Consequently, I will stick with complementarity here.

the photon box

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It was a red rag to a bull.

Shortly after publication of Bohr’s article, Austrian philosopher Philipp

Frank pointed out that Einstein himself had invented the logic of Bohr and

Heisenberg’s interpretation in his seminal 1905 paper on special relativity. Ein-

stein responded grumpily: ‘A good joke should not be repeated too often.’

Einstein had been preparing his next challenge. Irrespective of what

Bohr had written about the parallels between complementarity and rela-

tivity, this time he was conﬁ dent that he could use his special theory of

relativity to undermine the logical consistency of the energy–time uncer-

tainty relation, and with it the consistency of the Copenhagen interpre-

tation. Gathered once again in Brussels, Einstein described to Bohr his

latest and most ingenious gedankenexperiment.

‘At the next meeting with Einstein at the Solvay Conference in 1930,’

wrote Bohr some years later, ‘our discussions took quite a dramatic turn.’

Suppose, said Einstein, that we build an apparatus consisting of a box

which contains a clock connected to a shutter. The shutter covers a small

hole in the side of the box. We ﬁ ll the box with photons and weigh it. At

a predetermined and precisely known time, the clock triggers the shutter

to open for a very short time interval sufﬁ cient to allow a single photon

to escape from the box. The shutter closes. We reweigh the box and, from

the mass difference and Einstein’s theory of special relativity (E = mc

) we

determine the precise energy of the photon that escaped. By this means,

we have measured precisely the energy and the time of release of a photon

from the box, in contradiction to the energy–time uncertainty relation.

Bohr’s immediate reaction was described by Léon Rosenfeld:

It was quite a shock for Bohr . . . he did not see the solution at once. During

the whole evening he was extremely unhappy, going from one to the other

and trying to persuade them that it couldn’t be true, that it would be the end

of physics if Einstein were right; but he couldn’t produce any refutation. I

shall never forget the vision of the two antagonists leaving the club [of the

Fondation Universitaire]: Einstein a tall majestic ﬁ gure, walking quietly, with

a somewhat ironical smile, and Bohr trotting near him, very excited . . .

Bohr experienced a sleepless night, searching for the ﬂ aw in Einstein’s argu-

ment that he was convinced must exist. ‘This argument amounted to a serious

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challenge and gave rise to a thorough examination of the whole problem,’

Bohr wrote. By breakfast the following morning he had his answer.

On the blackboard Bohr produced another rough, pseudo-realistic

sketch of the apparatus that would be required to make the measurements

in the way Einstein had described them. In this sketch the whole box is

imagined to be suspended by a spring and ﬁ tted with a pointer so that its

position can be read on a scale afﬁ xed to the support. A small weight is

added to align the pointer with the zero reading on the scale. The clock

mechanism is shown inside the box, connected to the shutter.

After the release of one photon, the small weight is replaced by another,

slightly heavier weight so that the pointer is returned to the zero of the

scale. We suppose that the weight required to do this can be determined

independently with arbitrary precision. The difference in the two weights

required to balance the box gives the mass lost through the emission of

one photon, and hence the energy of the photon. So far, so good.

Bohr now drew his colleagues’ attention to the ﬁ rst weighing, before the

photon escapes. Obviously, the clock is set to trigger the shutter at some

predetermined time and the box is sealed. The actual reading of the clock

face is, of course, not possible since this would involve an exchange of

photons—and hence energy—between the box and the outside world.

To weigh the box, a weight must be selected that sets the pointer to the

zero of the scale. However, to make a precise position measurement, the

pointer and scale will need to be illuminated and, following Bohr’s earlier

arguments, this implies an uncontrollable transfer of momentum to the

box and hence an uncertainty in its momentum.

How does this affect the weighing? The uncontrollable transfer of

momentum to the box causes it to jump about unpredictably. Although

the box’s instantaneous position against the scale can be ﬁ xed, the size-

able interaction during the act of measurement means that the box will

not stay in that position. Bohr argued that we can increase the precision

of the measurement of the average position of the pointer by allowing

ourselves a long time interval in which to perform the whole balancing

procedure. This will give us the necessary precision in the weight of the

box. Since we can anticipate the need for this, the clock can be set so that

it opens the shutter after this balancing procedure has been completed.

Now comes Bohr’s coup de grâce.

the photon box

137

fig 7 The photon box experiment. This is the hypothetical instrument sketched by

Bohr to show how the measurements suggested by Einstein might be carried out.

Reprinted by permission of Open Court Publishing Company, a division of Carus

Publishing Company, Chicago, IL, from Albert Einstein: Philosopher-scientist, Vol. I,

Philosophers, Inc.

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According to Einstein’s own general theory of relativity, a clock moving

in a gravitational ﬁ eld is subject to time dilation effects. The very act of

weighing a clock effectively changes the way it keeps time. Because the box

is jumping about unpredictably in a gravitational ﬁ eld (owing to the act of

balancing the weight of the box by measuring the position of the pointer),

the rate of the clock is changed in a similarly unpredictable manner. This

introduces an uncertainty in the exact timing of the opening of the shutter

which depends on the length of time needed to complete the balancing

procedure. The longer we make this procedure (the greater the ultimate

precision in the measurement of the energy of the photon), the greater the

uncertainty in its exact moment of release.

Bohr was able to show that the product of the uncertainties in energy

and time for the photon box apparatus cannot be greater than Planck’s

constant h, in accordance with the uncertainty principle.

The discussion, so illustrative of the power and consistency of relativistic

arguments, thus emphasised once more the necessity of distinguishing, in

the study of atomic phenomena, between the proper measuring instru-

ments which serve to deﬁ ne the reference frame and those parts which are

to be regarded as objects under investigation and in the account of which

quantum effects cannot be neglected.

This response was hailed as a triumph for Bohr and for the Copenhagen

interpretation of quantum theory. Bohr had used Einstein’s own general

theory of relativity against him.

Einstein remained stubbornly unconvinced. In subsequent discussions

of the photon box experiment he conceded that it now appeared to be

‘free of contradictions’, but in his view it still contained ‘a certain unrea-

sonableness.’

Bohr had once again sought to defend the Copenhagen interpretation by

arguing that the inevitable and sizeable disturbance of the observed sys-

tem in any physical measurement effectively precludes the acquisition

of knowledge with a precision greater than that permitted by the uncer-

tainty principle. At ﬁ rst sight, there seems to be no counter-argument

against this position. It seems that a measurement of any kind will always

involve physical interaction on the same scale as the quantum system

the photon box

139

under investigation, and to talk about the properties of quantum systems

only in the absence of measurement is a pointless descent into naïvety.

Einstein had somehow to ﬁ nd a way around this.

He believed there were still some clues in the photon box experi-

ment. What if, he reasoned some months after the Solvay conference,

the experiment is used not to challenge the consistency of the uncertainty

relations but is used rather to derive a logical paradox arising from what

he saw to be the theory’s lack of completeness?

As before, the clock is set to trigger the release of one photon, but this

time it is synchronized with a second, external clock. The box is ﬁ lled

with photons. Einstein now accepted that his own general theory of rela-

tivity precluded precise knowledge of the moment of release, but per-

haps this was not really the point after all.

Now let the released photon travel to a ﬁ xed mirror placed a long distance

from the box, say half a light-year away. Whilst the photon is on its round-trip

journey of one year, we can now choose what measurement to make. We could

open the box and compare the two clocks. They will no longer be synchronous

because of the effect of the initial balancing procedure on the rate of the clock

inside, but we can now correct for this by reference to the external clock. We

can therefore draw a retrodictive conclusion as to the exact timing of release

of the photon. We know how long the photon will take on its journey and so

we can calculate its exact time of arrival back in the laboratory.

Alternatively, we could choose to keep the box sealed and rebalance it

with a heavier weight. We could take as long as we need over this second

balancing procedure and, as before, this would tell us the exact energy of

the released photon.

We now make a further, perhaps rather obvious or even trivial assump-

tion that a photon half a light-year distant cannot be affected by the choice

we make in the laboratory back on earth. In other words, we assume that

the photon is not in any way disturbed by measurements we choose to

make in a laboratory over three thousand billion miles away. As we can

choose to measure either the time of release or the energy of the photon,

we are left to conclude that the photon must possess simultaneously exact

values of both energy and time.

As there is nothing in quantum theory that tells us about the simulta-

neous exact values of these complementary observables, Einstein further