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

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the quantum story

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Suppose that this molecule is broken apart in a process that pre-

serves the total angular momentum to produce two equivalent atomic

fragments. The hydrogen molecule is split into two hydrogen atoms.

These atoms move apart but the spin orientations of the electrons in the

individual atoms remain opposed—one spin-up and one spin-down.

The spins of the atoms are therefore correlated. Measurement of the spin

of one atom (say atom A) in some arbitrary laboratory frame allows us

to predict, with certainty, the direction of the spin of atom B in the same

frame. We might be tempted to conclude that the spins of the two atoms

are determined by the nature of the initial molecular quantum state and

the method by which the molecule is fragmented. The atoms move away

from each other with their spins ﬁ xed in unknown but opposite orien-

tations and the measurement merely tells us what these orientations are.

But this is not how quantum theory deals with the situation. The two

atoms are instead described by a single wavefunction until the moment

of measurement. The atoms are entangled. If we choose to measure the

component of the spin of atom A along the laboratory z-axis, for exam-

ple, our observation that the wavefunction collapses into a state in which

atom A has its spin orientation aligned in the +z-direction (say) means

that atom B must have its spin orientation aligned in the −z-direction.

However, what if we choose, instead, to measure the x- or y-components

of the spin of atom A? No matter which component is measured, the

physics demand that the spins of the atoms must still be correlated, and

so the opposite results must always be obtained for atom B. If we accept

the deﬁ nition of physical reality offered by Einstein, Podolsky, and Rosen,

then we must conclude that all components of the spin of atom B are ele-

ments of reality, since it appears that we can predict them with certainty

without in any way disturbing B.

However, the wavefunction speciﬁ es only one spin component, associated

with the magnetic spin quantum number m

. This is because the operators

corresponding to the three components of the spin orientation in Cartesian

(x, y, z) coordinates do not commute (the components are complementary

observables). So, either the wavefunction is incomplete, or Einstein, Podolsky ,

and Rosen’s deﬁ nition of physical reality is inapplicable.

The Copenhagen interpretation says that no spin component of atom

B ‘exists’ until a measurement is made on atom A. The result we obtain

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for B will depend on how we choose to set up our instrument to make the

measurements on A, no matter how far away B is at the time.

This was a singular re-imagining of Einstein, Podolsky, and Rosen’s

original argument. It made completely transparent the nature of quantum

entanglement and its implications for non-local, ‘spooky’ action at a dis-

tance. The measurement of the spin orientation of an atom is much more

practicable than the measurement of its position or momentum. It opened

the possibility that further elaborations of Bohm’s version of the gedankenex-

periment might be carried out in the laboratory, and not just in the mind.

Responding to Einstein’s suggestion, Bohm met with him at Princeton

sometime in the spring of 1951. The doubts over the interpretation of

quantum theory that had begun to creep into Bohm’s mind as he had

worked on his book now crystallized into a sharply deﬁ ned problem. As

Einstein explained the basis for his own misgivings, most probably in

terms clearer than those in the original Einstein, Podolsky, Rosen paper,

Bohm acknowledged the need to change his position.

‘This encounter had a strong effect on the direction of my research,’

Bohm later wrote, ‘Because I then became seriously interested in

whether a deterministic extension of quantum theory could be found.’

Bohm was more committed to notions of causality and determinism

than perhaps he had realized. These notions also lay at the heart of

his Marxist ideology. In Marx’s dialectical materialism, social change

is caused by competing social forces and cannot be left to chance. The

Copenhagen interpretation, by contrast, appears to leave everything to

the roll of a dice.

Spurred by his discussion with Einstein, Bohm now reﬂ ected more

deeply on the question of interpretation. At issue was the Copenhagen

school’s outright denial that individual quantum systems could be

described objectively. He wrote:

According to Basil Hiley, one of Bohm’s long-term collaborators, Bohm said of his meet-

ing with Einstein: ‘After I ﬁ nished [Quantum Theory] I felt strongly that there was something

seriously wrong. Quantum theory had no place in it for an adequate notion of an individual

actuality. My discussions with Einstein clariﬁ ed and reinforced my opinion and encouraged me

to look again.’ Quoted by Basil Hiley, personal communication to the author, 1 June 2009.

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The usual interpretation of the quantum theory is self-consistent, but it

involves an assumption that cannot be tested experimentally, viz., that the

most complete possible speciﬁ cation of an individual system is in terms of

a wave function that determines only probable results of actual measure-

ment processes. The only way of investigating the truth of this assump-

tion is by trying to ﬁ nd some other interpretation of the quantum theory

in terms of at present ‘hidden’ variables . . . the mere possibility of such an

interpretation proves that it is not necessary for us to give up a precise,

rational, and objective description of individual systems at a quantum

level of accuracy.

Bohm was not speciﬁ cally seeking a new theory or a return to simple

classical physics. Rather, he acknowledged that quantum theory was

constructed on a set of assumptions of which the most important, con-

cerning the ‘completeness’ of the theory, is not subject to experimental

test. The Copenhagen interpretation is founded on this completeness

postulate and, rather than accept it at face value as the Copenhagen

school demanded, Bohm wanted to explore the possibility that other

descriptions and hence other interpretations are conceivable in principle.

As it appears that the only things we can detect are whole particles,

which form diffraction or interference patterns only when many such par-

ticles have been detected, the suggestion that the particles are real entities

that follow precisely deﬁ ned trajectories is very compelling. Were there

other ways in which the particles’ motions might be predetermined?

Perhaps. If Schrödinger’s wavefunction was reinterpreted as describing

an objectively real wave-like ﬁ eld, this could serve to guide the motion of

objectively real particles. Bohm now reworked Schrödinger’s wave equa-

tion into a form resembling a fundamental dynamical equation in classi-

cal physics that is actually a statement of Newton’s second law of motion

and which is therefore much more closely associated with a particle inter-

pretation. Bohm simply assumed that the wavefunction of the ﬁ eld can

be written in a form containing real amplitude and phase functions. In

itself, the assumption of a speciﬁ c form for the wavefunction represents

no radical departure from conventional quantum theory. However, Bohm

now assumed the existence of a real particle, following a real trajectory

through space, its motion embedded in the ﬁ eld and tied to or guided by

the phase function through the imposition of a ‘guidance condition’.

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Every particle in every ﬁ eld therefore possesses a precisely deﬁ ned

position and a momentum, and follows trajectories determined by their

respective phase functions. The equation of motion is then found to

depend not only on the classical potential energy, but also on a second,

so-called quantum potential.

The quantum potential is intrinsically non-classical and is alone

responsible for the introduction of quantum effects in what would other-

wise be an entirely classical description. Take out the quantum potential

or allow it to decline to zero and Bohm’s equations revert to the classical

equations of Newtonian mechanics.

True to its nature, the quantum potential has some peculiar properties.

It can exert effects on the particle in regions of space where the classical

potential disappears. This contrasts markedly with the effects exerted

by classical potentials (such as a Newtonian gravitational ﬁ eld), which

tend to fall off with distance. A particle moving in a region of space in

which no classical potential is present can therefore still be inﬂ uenced

by the quantum potential and some of the cherished notions of classical

physics—such as straight-line motion in the absence of a (classical)

force—must be abandoned.

The particle position and its trajectory are ‘deﬁ ned’ at all times during

its motion, and it is therefore not necessary in principle to resort to prob-

abilities. When we consider a large number of particles all describable in

terms of the same wavefunction, the above reasoning can still be applied.

There is nothing in principle preventing us from following the trajecto-

ries of each particle. However, in practice we do not usually have access

to a complete speciﬁ cation of all the particle initial conditions and, just as

in Boltzmann’s statistical mechanics, we resort to classical probabilities

as a practical necessity.

This contrasts strongly with the notion of quantum probability. In con-

ventional quantum theory, the wavefunction is really a calculation tool for

probabilities interpreted as the relative frequencies of possible outcomes

of repeated measurements on a collection of identically prepared systems.

These outcomes are not determined until a measurement is made. In

Bohm’s theory, the particle motions are predetermined and we calculate

probabilities because we are ignorant of the initial conditions of all the par-

ticles in the collection. These probabilities refer to individual states of indi-

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vidual particles—their positions and their trajectories—not measurement

outcomes. Measurement therefore has no mystical role: the measurements

merely tell us the actual states or positions of the particles or their actual

trajectories through an apparatus, which are determined all along.

The probability is still related to the amplitude of the wavefunction,

as in Born’s original prescription, but this does not mean that the wave-

function has only a statistical signiﬁ cance. On the contrary, it is assumed

that the wavefunction has a very strong physical signiﬁ cance—it deter-

mines the shape of the quantum potential.

This is a hidden variable theory. The hidden variable is not the guiding

ﬁ eld—that is revealed in the properties and behaviour of the quantum

wavefunction. It is actually the particle positions that are hidden. Bohm’s

theory reintroduces the classical concept of causality—classical particles

are directed along classical trajectories dictated by guiding wave ﬁ elds.

But change the wave ﬁ eld simply by changing the measuring apparatus

in some way, and the particles are obliged to respond instantaneously.

The hidden variables are said to be ‘non-local’. In this sense, at least, there

is no conﬂ ict with Bohr’s insistence on the primacy of the measuring

apparatus: ‘In this point,’ Bohm wrote, ‘We are in agreement with Bohr,

who repeatedly stresses the fundamental role of the measuring appara-

tus as an inseparable part of the observed system.’

The theory restored causality and determinism, and eliminated the

need to invoke a collapse of the wavefunction. But it had not eliminated

non-local inﬂ uences and ‘spooky’ action at a distance, and so appeared

to be incompatible with special relativity.

In truth, Bohm had actually rediscovered and extended de Broglie’s

theory of the ‘double solution’.

He drafted two papers on his hidden

For this reason, Bohm’s redevelopment is often referred to as de Broglie–Bohm theory.

Nathan Rosen had also attempted a very similar approach in 1945 but had not pursued it

further. See Jammer, The Philosophy of Quantum Mechanics, p. 285. Note that this is a non-local

hidden variable theory which is capable, in principle, of making predictions which differ from

conventional quantum theory. Subsequent non-local hidden variable interpretations of quan-

tum theory make predictions which cannot be distinguished from those of conventional

quantum theory.

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variable theory and its application to the hydrogen atom and in July 1951

he submitted these to the journal Physical Review. His conversion had

been swift. His book, overtly supportive of the Copenhagen view, had

been published only four months previously.

He sent pre-prints to de Broglie, Bohr, Pauli, and Einstein. From de

Broglie he learned for the ﬁ rst time all about the double solution and

why de Broglie had abandoned it shortly after the ﬁ fth Solvay conference

in 1927. From Pauli he received objections concerning the implications

of the theory for multi-particle systems, but Bohm was conﬁ dent that he

could deal with these.

Unable to ﬁ nd a position in America or Britain, in October 1951 Bohm left

Princeton and went into exile at the University of São Paulo in Brazil. Both

Einstein and Oppenheimer had provided letters of recommendation and

Bohm received support from Abrahão de Moraes, the head of the physics

department. As his plane taxied towards the runway, it was announced

that the pilot had been asked to return to the terminal. An irregularity had

been discovered with the passport belonging to one of the passengers.

Bohm feared a second arrest, but it turned out that the irregularity con-

cerned a different passenger, who was removed from the ﬂ ight.

Bohm’s two papers were published in January 1952. He observed the

reaction from exile in Brazil. Feynman was supportive. But Oppenhe-

imer reﬂ ected the mood of the majority by declaring Bohm’s work to be

‘juvenile deviationism’, urging that ‘… if we cannot disprove Bohm, then

we must agree to ignore him.’

Not surprisingly, Einstein was not enamoured of the approach Bohm

had taken. In a letter to Born he wrote:

Have you noticed that Bohm believes (as de Broglie did, by the way, 25

years ago) that he is able to interpret the quantum theory in deterministic

terms? That way seems too cheap to me.

Bohm died in 1992, but his vision for a causal form for quantum theory lives on. For

example, Basil Hiley has recently extended Bohm’s original theoretical structure using a Clifford

algebra approach to accommodate relativistic effects.

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Bohm did not adapt well to exile. He grew restless and in 1955 relocated to the Technion,

the Israel Institute of Technology in Haifa.

It was here that he met Yakir Aharonov, an

outstanding 22-year old undergraduate student who had already established a reputation

as a maverick. Together they worked on a further elaboration of Bohm’s version of the

Einstein, Podolsky, Rosen gedankenexperiment which they submitted for publication

in May 1957.

Bohm also published another book—essentially a manifesto for his deterministic pro-

gramme—titled Causality and Chance in Modern Physics. De Broglie supplied a

foreword. In the summer of 1957 Bohm moved once more, to a research associate position

at Bristol University in England. From Bristol, he subsequently took a professorship at

the University of London’s Birkbeck College, where he remained for the rest of his life.

Whilst his was not exactly a lone voice, Bohm had recruited few followers to his cause.

Although Bohm’s deterministic theory yielded all the predictions of non-relativistic quan-

tum mechanics, the majority of the physics community had moved on. Quantum elec-

trodynamics had triumphed in 1949. Physicists had turned their attentions to the search

for a quantum theory of the nuclear forces. In 1954 Yang and Mills had published their

‘beautiful idea’. Gell-Mann and Zweig had introduced the idea of a triplet of fractionally

charged constituents of mesons and baryons—quarks or aces—in 1963.

Bertlmann’s Socks

Boston, September 1964

The American authorities had withdrawn Bohm’s passport. To travel to Israel he ﬁ rst had

to become a Brazilian citizen.

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There seemed little to be gained by raking over the ashes of old philosophical conun-

drums. Besides, hadn’t Bohr already set the record straight on the Copenhagen interpreta-

tion in 1935? Hadn’t von Neumann proved that all hidden variable theories are impossible

in principle?

But the effort to interpret quantum theory and what it had to say about the nature of

physical reality was about to take a surprising turn. Belfast-born CERN physicist John

Bell had read Bohm’s 1952 papers with great interest. The collapse of the wavefunction

and the rather arbitrary boundary drawn by Bohr between the quantum objects of meas-

urement and the classical measuring device seemed to Bell to be at best a conﬁ dence-trick,

at worst a fraud. ‘A theory founded in this way on arguments of manifestly approximate

character,’ he wrote some years later, ‘however good the approximation, is surely of pro-

visional nature.’

In 1952 he had conceived some ideas concerning hidden variable theories that were to

form the basis for a paper he wrote twelve years later, whilst on leave from CERN at the

Stanford Linear Accelerator Center.

In the intervening years Bell had developed deep

reservations about the relevance of von Neumann’s impossibility proof. In his paper he

argued that the proof hinged on a critical assumption that, whilst valid, applied to a situ-

ation in which two complementary physical quantities are measured simultaneously. But

such measurements require completely incompatible measuring devices (or incompatible

conﬁ gurations of a single measuring device), which means that such measurements just

cannot be made simultaneously. He concluded that the proof is, in fact, irrelevant.

This meant that all varieties of hidden variables theories—local and non-local—

were ‘fair game’ once again. But as Bell probed further, he made a discovery that was at

once both simple and profound. In Quantum Theory, Bohm had asserted that: ‘ . . . no

theory of mechanically determined hidden variables can lead to all of the results of the

quantum theory.’

Bell now discovered just how right this assertion was. What he found was that the

choice between conventional quantum theory and adaptations based on local hidden

variables was not after all just a matter of philosophical disposition. It was a matter of

correctness.

In their 1957 paper, Bohm and Aharonov had pushed the EPR gedankenexper-

iment even closer to practical realization. Indeed, the purpose of the paper

Due to a mix-up, this paper, published in Reviews of Modern Physics, did not appear until

1966.

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was to claim that experiments capable of measuring non-local correlations

between distant quantum particles had already been carried out.

After visiting SLAC in 1964, Bell spent some time at the University of

Wisconsin at Madison and thence moved on to Brandeis University near

Boston. It was during this visit that he had an insight that was completely

to transform questions about the nature of reality at the quantum level.

He derived what was to become known as Bell’s inequality: ‘Probably I got

that equation into my head and out on to paper within about one week-

end. But in the previous weeks I had been thinking intensely all around

these questions. And in the previous years it had been at the back of my

head continually.’

Bell built his derivation on the version of the EPR experiment as elabo-

rated by Bohm and Aharonov. After fragmenting the hydrogen molecule,

the two hydrogen atom fragments move apart in opposite directions.

Hydrogen atom A moves to the left, atom B moves to the right. Bell now

imagined that two magnets would be placed left and right, designed to

determine the spin orientation of each atom using the Stern–Gerlach

effect.

Atoms passed between the poles of the magnet are either deﬂ ected

upwards, in the direction of the north pole (spin-up) or downward in

the direction of the south pole (spin-down). Because of the correlation

established between atom A and atom B, if the ﬁ elds of both magnets

are aligned, then opposite results are expected. If A is found to be in a

spin-up orientation, then B will be found in a spin-down orientation, and

vice versa.

Bell then turned his attention to an extremely simple example of a local

hidden variable theory, one that would not only restore causality and

determinism and eliminate the collapse of the wavefunction, as Bohm

had done in 1952, but would eliminate spooky action at a distance as well.

Any such theory is characterized as locally real. The hydrogen atoms in the

We will examine these claims in the next chapter.

Neither Bohm and Aharonov nor Bell were speciﬁ c about the nature of the molecule

involved, specifying only that it possess a total spin of zero. I will nevertheless continue to refer

to the example of a hydrogen molecule.

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309

above example are assumed to possess some variable or variables which

predetermine the outcomes of the subsequent spin measurements. These

variables are not obliged to be ﬁ xed at the moment the hydrogen mol-

ecule is fragmented, although it helps us to understand what needs to

happen in this simple example if we assume that they are.

Imagine, then, that hidden within each atom there exists a tiny sub-

atomic dial. The dial has a pointer. We assume that this pointer can point

in any direction on the dial and, at the moment of fragmentation, the

pointer directions within each fragmented atom are randomly ﬁ xed, such

that they take up any direction in the entire 360° range on the face of the

dial. However, whatever pointer direction is (randomly) ﬁ xed for atom A,

the pointer direction for atom B must be ﬁ rmly ﬁ xed in the opposite (180°)

direction, and vice versa. We assume that the physics of the fragmentation

process demands this (we can think of it as a law of aligned pointers).

The atoms move towards the poles of their respective magnets. For

the sake of simplicity, we assume that if the pointer lies at any angle in

the top half of the dial face, this will determine that the spin is ‘up’ and the

atom is deﬂ ected upwards towards the north pole of the magnet. Like-

wise, if the pointer lies anywhere in the bottom half of the dial face, the

atom is detected in a spin-down orientation and is deﬂ ected downwards

towards the south pole.

This is a local hidden variable theory. Whatever they are and however

they are supposed to work, the pointers predetermine the outcomes of

the spin measurements in a way which eliminates the need for mysteri-

ous inﬂ uences travelling over long distances faster than light-speed. It is

also a relatively commonsense theory: the outcomes of the experiments

are predetermined as soon as the hydrogen molecule is fragmented. The

measurements simply tell us what these outcomes are.

What happens in the situation where both magnetic ﬁ elds are aligned,

both with their north poles lying in the same direction? If the pointer for

atom A lies anywhere in the top half of its dial face, the result is spin-up,

which we designate as a ‘+’ result. The pointer direction for B must lie in the

bottom half of its dial face, predetermining a spin-down result which we

It is actually the law of conservation of angular momentum.