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

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the quantum story

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that at the same time one localizes the particle during the propagation.

I think that Mr de Broglie is right to search in this direction. If one works

solely with the Schrödinger waves [assuming quantum theory describes

individual processes], |y|

implies to my mind a contradiction with the

postulate of relativity.’

There is an alternative description, however. What if the wavefunction

represents a probability amplitude not for a single quantum particle, but

for a large collection of identically prepared particles, called an ensemble?

According to this view, each individual particle passes through the slit

along a deﬁ ned, localized path, to arrive at the second screen. There are

many such paths possible, and the diffraction pattern thus reﬂ ects the

statistical distribution of large numbers of particles each following differ-

ent but deﬁ ned paths. This distribution is related to the modulus-square

of the wavefunction, which expresses the probability for many particles

rather than a probability for each individual particle.

We cannot discriminate between these possibilities by observing what

happens to an individual quantum particle. Both descriptions say that

one particle passes through the slit to arrive at one speciﬁ c location on

the second screen. In the ﬁ rst description, the point of arrival is deter-

mined at the moment the particle interacts with the photographic ﬁ lm,

with a probability given by the modulus-square of the wavefunction. In

the second description, the point of arrival is determined by the actual

path that the particle follows, which is in turn obtained from a statistical

probability given by the modulus-square of the wavefunction. In both

cases we see the diffraction pattern only when we have detected a large

number of particles.

Bohr wasn’t sure what to make of Einstein’s remarks. ‘I feel myself in

a very difﬁ cult position because I don’t understand what precisely is the

point which Einstein wants to [make],’ he said. ‘No doubt it is my fault.’

He went on to explain that: ‘I do not know what quantum mechanics

is. I think we are dealing with some mathematical methods which are ade-

quate for description of our experiments.’ In his view, Einstein was trying

to hang on to an essentially classical, space–time description, one that had

to be recognized as no longer tenable. ‘The whole foundation for causal

spacetime description,’ he continued, ‘is taken away by quantum theory,

for it is based on assumption of observations without interference.’

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121

Bohr had missed the point, but Einstein was not about to give up.

The discussion continued in the dining room of the Hôtel Britannique,

where the conference participants were staying. This was the scene of

the opening skirmishes in one of the most important scientiﬁ c debates

ever witnessed, as Einstein directly challenged Bohr over the meaning of

quantum theory. At stake was the way we attempt to understand the very

nature of physical reality.

Otto Stern described what happened:

Einstein came down to breakfast and expressed his misgivings about the

new quantum theory, every time [he] had invented some beautiful experi-

ment from which one saw that [the theory] did not work . . . Pauli and

Heisenberg, who were there, did not pay much attention, ‘ach was, das

stimmt schon, das stimmt schon’ [ah well, it will be all right, it will be all

right]. Bohr, on the other hand, reﬂ ected on it with care and in the evening,

at dinner, we were all together and he cleared up the matter in detail.

Einstein now chose to attack the Copenhagen interpretation by attempt-

ing to show that as a result of its incompleteness, one of the theory’s

principal foundations—the physical meaning of the uncertainty rela-

tions—is inconsistent. The debate took the form of a series of puzzles,

developed by Einstein as gedankenexperiments, not to be taken too literally

as practical experiments that could be carried out in the laboratory. It

was enough for Einstein that the experiments could be conceived and

carried out in principle.

Einstein asked the assembled audience what might happen if a quan-

tum particle passed through a slit in a screen under conditions where

the transfer of momentum between the particle and the screen is care-

fully controlled and observed. A particle hitting the screen as it passes

through the slit would be deﬂ ected, its path beyond being determined by

the conservation of momentum. We can tell in which direction the parti-

cle has been deﬂ ected by watching the deﬂ ection of the screen itself.

Now, he said, imagine that we insert a second screen—one with two

slits—between the ﬁ rst screen and a piece of photographic ﬁ lm. If we

are able to discover the direction in which the particle is deﬂ ected by

the ﬁ rst screen, we can further learn which of the two slits the particle

subsequently passed through. From the position on the ﬁ lm in which it

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122

is ultimately detected, we can trace the particle’s trajectory through the

entire apparatus.

We now leave the apparatus to detect a large number of particles—one

after the other—from which we expect to see a double-slit interference

pattern. Thus, Einstein concluded, we can demonstrate the particle-like

(deﬁ ned trajectory) and wave-like (interference) properties of quantum

particles simultaneously, in contradiction to Bohr’s notion of comple-

mentarity and the Copenhagen interpretation of the uncertainty rela-

tions. This proves that the Copenhagen interpretation is internally

inconsistent.

Bohr reﬂ ected very carefully on Einstein’s challenge. He took the

thought experiment a stage further, sketching out in a pseudo-realistic

style the kind of apparatus that would be needed to make the measure-

ments to which Einstein referred. His purpose was not to try to imagine

how the experiments could be done in practice, but primarily to focus on

what he saw to be ﬂ aws in Einstein’s arguments.

Movement of screen

allows trajectory of

particle to be followed

Particle detected here

Interference pattern

produced when many

particles detected

fig 5 Einstein’s variation on the classic double-slit experiment. By passing a particle

through a single, narrow slit in the ﬁ rst screen and observing which way the screen moves,

we can tell in which direction (and therefore towards which slit in the second screen) the

particle is deﬂ ected. By observing where the particle subsequently strikes the third screen

we can then trace its trajectory through the entire apparatus. If we now allow many parti-

cles to pass through the apparatus, one after the other, the double-slit interference pattern

should in principle become visible . This thought experiment seems to allow us to observe

particle-like properties (such as a trajectory) and wave-like properties (interference) simul-

taneously, in contradiction to Heisenberg’s uncertainty principle.

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123

Controlling and observing the transfer of momentum from the quan-

tum particle to the ﬁ rst screen requires that the screen be capable of

movement in the vertical plane. Observing the recoil of the screen in one

direction or the other as the particle passes through the slit would then

allow the experimenter to draw conclusions about the direction in which

the particle had been deﬂ ected, as Einstein claimed.

Bohr envisaged a screen suspended by two weak springs. A pointer

and scale inscribed on the screen allows the measurement of the amount

fig 6 The hypothetical apparatus sketched by Bohr to demonstrate how measure-

ment of the deﬂ ection of the ﬁ rst screen might be made. Reprinted by permission

of Open Court Publishing Company, a division of Carus Publishing Company, Chi-

cago, IL, from Albert Einstein: Philosopher-scientist, Vol. I, p. 220, edited by Paul Arthur

the quantum story

124

of movement of the screen, and hence the momentum imparted to it

by the particle. The fact that Bohr had in mind a macroscopic appara-

tus presents no problem, provided we assume that the apparatus is suf-

ﬁ ciently sensitive to allow observation of individual quantum events.

This sensitivity is important, as we will see.

Bohr had to demonstrate the consistency of the uncertainty principle,

and hence complementarity, when applied to the analysis of this kind

of thought experiment. Controlling the transfer of momentum to the

screen in the way Einstein suggested must imply a concomitant uncer-

tainty in the screen’s position in accordance with the uncertainty prin-

ciple. If we measure the screen’s momentum in the vertical plane with a

certain precision, then there must result an uncertainty in the position

of the screen such that the product of the uncertainties is not less than

Planck’s constant, h.

Why should this be? Bohr’s answer was that, in order to read the scale

inscribed on the ﬁ rst screen sufﬁ ciently accurately, it has to be illumi-

nated. This illumination involves the scattering of photons from the

screen and hence an uncontrollable transfer of momentum, or an uncon-

trollable ‘jolt’. ‘Since, however, any reading of the scale, in whatever way

performed, will involve an uncontrollable change in the momentum of

the [screen], there will always be, in conformity with the [uncertainty]

principle, a reciprocal relationship between our knowledge of the posi-

tion of the slit and the accuracy of the momentum control.’

Bohr was able to show that the resulting uncertainty in the position

of the slit in the ﬁ rst screen destroys the phase coherence of the particle’s

associated wave as it spreads out beyond the double slits in the second

screen. The interference pattern is therefore ‘washed out’. Controlling the

transfer of momentum from the particle to the ﬁ rst screen allows us to

follow the trajectory of the particle through the apparatus, but prevents

us from observing interference effects, in accordance with the comple-

mentary nature of the wave and particle descriptions.

He concluded:

. . . we are presented with a choice of either tracing the path of a particle

or observing interference effects, which allows us to escape from the

paradoxical necessity of concluding that the behaviour of an electron or

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125

a photon should depend on the presence of a slit in the [second screen]

through which it could be proved not to pass. We have here to do with

a typical example of how the complementary phenomena appear under

mutually exclusive experimental arrangements and are just faced with the

impossibility, in the analysis of quantum effects, of drawing any sharp

separation between an independent behaviour of atomic objects and their

interaction with the measuring instruments which serve to deﬁ ne the con-

ditions under which the phenomena occur.

Einstein presented further gedankenexperiments. He could not shake his

deeply felt misgivings about the Copenhagen interpretation and forced

Bohr to defend it. But Bohr rebuffed all his challenges. ‘On his side,

Einstein mockingly asked us whether we could really believe that the

providential authorities took recourse to dice-playing (“. . . ob der liebe Gott

würfelt”),’ Bohr wrote, ‘to which I replied by pointing at the great caution,

already called for by ancient thinkers, in ascribing attributes to Provi-

dence in every-day language.’

Bohr may have been satisﬁ ed with his robust defence of the Copen-

hagen position, but under pressure from Einstein’s insistent probing,

the basis of his arguments had undergone a subtle shift. Bohr had fallen

back on the notion of a ‘clumsy disturbance’, an uncontrollable transfer

of momentum, of precisely the kind that he had earlier criticized Heisen-

berg for.

Although Bohr’s counterarguments won the day in the eyes of the

majority of physicists assembled in Brussels, the seeds of a much more

substantial challenge had been sown. The ﬁ fth Solvay Conference ended

with Bohr having successfully argued for the logical consistency of the

Copenhagen interpretation.

Einstein remained unconvinced.

126

Pauli may have been pleased with the progress that had been made on the question of

interpretation. He was nevertheless frustrated by the lack of progress on the incorporation

of electron spin and the exclusion principle into the main structure of quantum theory. He

called it an ‘aesthetic failure’.

Furthermore, the equations that had been adopted as part of mainstream quan-

tum mechanics suffered from the limitation that they did not meet the requirements

of Einstein’s special theory of relativity. There was a growing realization that the problem

of electron spin was in some way connected with the problem of ﬁ nding a fully relativistic

expression of the equations of quantum theory.

Oskar Klein had independently rediscovered a relativistic version of Schrödinger’s wave

equation in the spring of 1926. With some further modiﬁ cations by Hamburg theorist

Walter Gordon it came to be known as the Klein–Gordon equation. But this was, in

essence, the relativistic equation that Schrödinger himself had discovered and subsequently

abandoned in January 1926 when he found that it did not yield predictions in agreement

with experiment.

Meanwhile, the endless debate about the philosophy and interpretation of quantum

theory at the Solvay conference in Brussels had left English physicist Paul Dirac cold.

He did not have a great deal of time for eloquent elaboration of the ﬁ ner points of inter-

pretation if this produced no new equations. He was also aware that although his major

contributions to the mathematical structure of the theory were worthy of international

recognition, he had on several occasions now been pipped to the post by his European

An Absolute Wonder

Cambridge, Christmas 1927

an absolute wonder

127

rivals. He felt strongly that he needed to discover something original, something that he

was ﬁ rst to report and could claim as his own.

Electron spin was an obvious target for his attention. Towards the end of 1926 Dirac

had agreed a bet with Heisenberg on how soon spin could be understood within the

framework of quantum theory. Heisenberg had bet on three years at the least. Dirac had

rashly bet three months. Three months went by, but the problem remained unresolved. But

Dirac, who had become completely absorbed by the theory of relativity when he had ﬁ rst

heard about it as an engineering student in 1919, now turned his attention to ﬁ nding a

fully relativistic version of quantum theory.

What he would ﬁ nd would simultaneously solve the riddle of electron spin. But it

would also yield much more than anyone had bargained for.

Dirac was an archetype, if not a caricature, of the dry, introverted English

academic. His personality had been shaped in childhood by his imposing

Swiss father, Charles, who had run away from home at the age of twenty.

Charles had settled in Bristol, in south-west England, in 1890. He became

a schoolteacher. His second child Paul was born in 1902.

Charles, it seemed, had had an unhappy childhood. Sadly, this experi-

ence had not encouraged in him a greater generosity towards his own

children. He was a tyrant. Seeing little value in social contact himself, he

forbade his children from what he saw to be unnecessary socializing out-

side the immediate family. They became social prisoners and, within this

prison, silence reigned. Charles, a native French speaker and now French

teacher, insisted that Paul should speak to him only in French:

He thought it would be good for me to learn French in that way. Since I

found I couldn’t express myself in French, it was better for me to stay silent

than to [talk in] English. So I became very silent at that time—that started

very early . . .

Paul compensated for his stunted social and emotional development by

pouring his energies into mathematics. However, unable to see how he

could forge a career in the subject he loved, and under the stern inﬂ uence

of his father, in 1918 he began an engineering degree at Bristol University.

A year later he was caught up in the general public excitement over

Einstein’s theory of relativity. The press reported Arthur Eddington’s

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128

conﬁ rmation of the bending of starlight, predicted by Einstein’s general

theory. It was a revelation for the young student, and he went on to study

deeply both the special and general theories. He became a master of the

theory’s mathematical structure.

Depression-era Britain was not a good time for a new engineering

graduate to join the job market and on graduating in 1921 he failed to

ﬁ nd a suitable position. When he was offered the opportunity of some

free tuition in mathematics at Bristol University, he gladly accepted. He

continued to study mathematics for the next two years.

Aided by a grant from the Department of Scientiﬁ c and Industrial

Research, in 1923 he ﬁ nally escaped the suffocating emotional vice of

his parental home and moved to Cambridge to study for a PhD.

was initially disappointed to be assigned a research project under Ralph

Fowler, as he had hoped to study relativity. Fowler’s research interests

were atomic and quantum physics. However, as Dirac soon discovered,

there was much of interest in this emerging ﬁ eld, and much to do. ‘The

atoms were always considered as very hypothetical things by me,’ he

wrote years later, ‘and here were people actually dealing with equations

concerned with the structure of the atom.’

Fowler had good contacts with Bohr and the schools in Copenhagen

and Göttingen. When he received a proof copy of Heisenberg’s ﬁ rst paper

on matrix mechanics in August 1925, he passed it to Dirac.

Dirac’s fascination with relativity and his already burgeoning reputation as

a quantum physicist made the search for a fully relativistic form of the new

quantum theory irresistible. He had made several abortive attempts over

the previous two years. At the Solvay conference in October 1927 he raised the

problem with Bohr, only to be told by Bohr that Klein had already solved it.

Dirac tried to explain why Klein’s equation was unsatisfactory but was cut

off when the lecture they had gathered to hear commenced.

Though short, the conversation with Bohr convinced Dirac that a

proper solution had to be found as a matter of urgency. He once again set

to work on the problem as soon as he got back from Brussels. He worked

‘Escape’ should be interpreted literally. Paul’s older brother Reginald suffered both from his

father’s emotionally retarding inﬂ uence and a feeling of inferiority when measured against his

younger brother’s intellectual achievements. Reginald committed suicide in 1924.

an absolute wonder

129

alone, in relative isolation, consulting nobody. It was an approach that

suited his personality.

Einstein’s special theory of relativity is in many ways all about the cor-

rect treatment of time as a kind of fourth dimension, on an equal foot-

ing with the three dimensions of ordinary space. The time-dependent

version of Schrödinger’s wave equation is ‘unbalanced’ in this regard, as

it is a second-order differential equation in the three spatial coordinates

but only a ﬁ rst-order differential equation in time.

In this sense, it actu-

ally looks more like an equation describing diffusion phenomena. This

lack of balance between space and time means that Schrödinger’s wave

equation is non-relativistic.

Schrödinger and, subsequently, Klein and Gordon had tried to even

things up by developing an equation that contains a second-order dif-

ferential in time. This achieved the required balance but suffered from

the problems that had led Schrödinger to abandon it. Dirac didn’t like it.

Neither did Heisenberg. ‘Herr Pauli,’ Hungarian physicist Johann Kudar

reported to Dirac in a letter dated 21 December 1926, ‘regards the relativ-

istic wave equation of second order with much suspicion.’ There remained

no sign of electron spin in the Klein–Gordon equation.

It was understood that the relativistic equation had to remain a ﬁ r s t -

order differential equation in time. There was by now too much at stake,

not least Born’s probabilistic interpretation of the wavefunction, which

could only continue to apply if the equation remained ﬁ rst-order in time.

The challenge, then, was to restructure the equation in such a way as to

make both its spatial and time components ﬁ rst-order differentials. This

could easily be done, but led to some rather ugly-looking square-root

operators which left the theorists scratching their heads. Dirac needed to

ﬁ nd a way to handle these.

When in May 1927 Pauli had been faced with a similar problem deal-

ing with square-root operators for momentum in three spatial coordi-

nates, he had had much success using coefﬁ cients for each operator that

were actually square matrices consisting of two rows and two columns.

Pauli was trying to account for the property of electron spin, and the

In a ﬁ rst-order differential equation, some function ( f, say), is differentiated once with

respect to some variable (x, say). This is written df/dx. In a second-order equation, the function

is differentiated with respect to the variable a second time, written d

f/dx