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

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As far as Schrödinger was concerned, the interpretation of the wavefunctions of his new

wave mechanics was simple and straightforward. Although de Broglie had developed

his ideas around the core concept of a duality of wave-like and particle-like behaviour,

Schrödinger was happy to dispense with particles altogether. He viewed the wavefunctions

as the very real manifestations of a completely undulatory material world. In his descrip-

tion, particle-like behaviour was an illusion created by the overlapping and reinforcing of

collections of ‘matter waves’.

What Schrödinger had in mind was a so-called ‘wave packet’ state. If a series of waves

with high amplitude gathered around a speciﬁ c point in space and time are added together,

one on top of the other, the resultant wave may have a large amplitude peak concentrated

at that point and little amplitude elsewhere. Such a wave packet would appear, to all

intents and purposes, like a concentrated bit of matter—a particle. If such a collection, or

‘superposition’ of waves, were then to move collectively through space and time, it would

carve out a path that would look like a particle trajectory.

Simple, perhaps, but it was in fact not so straightforward. Except for the special case

that Schrödinger had considered, wave packet states are not generally sustainable unless

their dimensions are much greater than the wavelengths of the waves that comprise them.

This was not behaviour that could be expected of wave packets constructed on atomic

scales. Dutch physicist Hendrik Lorentz argued that an electron wave packet would not

hold together. It would disperse, dissolving rapidly into nothingness as its constituent

waves went separate ways.

Ghost Field

Oxford, August 1926

the quantum story

Schrödinger began to have doubts. There were also troublesome phenomena, such as

the photoelectric effect, which he struggled to reconcile with his model of smooth, continu-

ous transitions between stable wave states.

Meanwhile, in Göttingen, Max Born took a distinctly different view of the meaning of

the wavefunctions. He came to reject Schrödinger’s attempts to bring a classical physical

perspective back into the heart of the atom. Although at the time Born’s interpretation

was not perceived as particularly radical by physicists of the Copenhagen and Göttingen

schools, it was to provoke a debate about the nature of reality at the quantum level that

continues to this day.

Max Born had studied mathematics at universities in Breslau, Heidelberg,

and Zurich before completing his doctorate in theoretical physics at the

University of Göttingen. At Göttingen he had come into contact with

the ‘mandarins’ of mathematics: Felix Klein, David Hilbert, and Hermann

Minkowski. He was quickly appointed as the note-taker for Hilbert’s

lectures, eventually becoming Hilbert’s unpaid assistant.

Born went on to become professor of theoretical physics at the Uni-

versity of Berlin in 1915. It was here that he met Einstein, and they became

close friends. Einstein had dragged Born from his sickbed in November

1918 to help him release the university’s rector and deans, who had been

imprisoned by student revolutionaries. In 1919 Born moved to Frankfurt,

before taking up an appointment as Director of the newly established

Institute for Theoretical Physics in Göttingen. The Institute became an

important centre for theoretical physics which, like Bohr’s Institute in

Copenhagen, attracted a continual stream of esteemed visitors and bright

students from around the world.

It had been Born who, in July 1925, had realized that Heisenberg’s odd

multiplication rule was, in fact, the rule for multiplying matrices. He was

therefore one of the founders of quantum mechanics. And yet when he

saw the new wave mechanics, he immediately recognized the utility of

Schrödinger’s approach and was initially enamoured of his attempt to

restore a classical space–time description to quantum mechanics. But he

was appalled by Schrödinger’s attempt to eliminate quantum jumps.

Born had left Göttingen in November 1925 for a lecture tour of America.

During a lecture at the Massachusetts Institute of Technology (MIT) in Bos-

ton, he emphasized the need for caution regarding the matrix approach:

ghost field

Only a further extension of the theory; which in all likelihood will be very

laborious, will show whether the principles [of matrix mechanics] are

really sufﬁ cient to explain atomic structure. Even if we are inclined to put

faith in this possibility, it must be remembered that this is only the ﬁ rst

step towards the solution of the riddles of the quantum theory.

On his return, he quickly reached for Schrödinger’s wave approach to

tackle problems concerning the nature of the interactions between

quantum particles such as electrons and atoms. Both matrix and wave

mechanics had been shown to be at least partially successful

in provid-

ing a framework within which the stable orbits of electrons bound in

atoms could be understood and the positions and intensities of spectral

lines could be predicted. These new quantum theories addressed the

issue of structure, but they did not address issues concerning transitions

(quantum jumps) between structures.

Born hoped that by formulating a quantum theory of collisions

between electrons and atoms he would be well on the way to formulating

a theory of the interaction between radiation (light quanta) and matter.

In other words, he would have a theory that would incorporate quantum

jumps into Schrödinger’s wave mechanics.

This was the reason he had chosen to abandon matrix mechanics.

Heisenberg’s theory had been designed to describe the stationary states

(the stable orbits) of electrons in atoms and allow predictions of the posi-

tions and intensities of spectral lines. It was not a theory that could be

easily stretched to include collisions. Born had tried to apply matrix meth-

ods, in vain. Wave mechanics, however, proved to be far more amenable.

He quickly ﬁ nalized a paper ‘Quantum Mechanics of Collision Phe-

nomena’ which he submitted to Zeitschrift für Physik in June 1926. Although

he had employed the methods of wave mechanics the paper contained

the seeds of a radical re-interpretation of the wavefunctions.

Born had interpreted the collision between an electron and an atom as

an interaction between a plane electron wave and an atom ‘vibrating’ at

I say partially successful, as neither matrix nor wave mechanics had yet been cast in forms

that were consistent with Einstein’s special relativity and there was still no accounting for Pauli’s

exclusion principle or the anomalous Zeeman effect.

the quantum story

a characteristic frequency determined by its state. The result would be a

complex vibration consisting of a superposition of these waves which

would then separate, the electron wave ‘scattering’ as a consequence of

the interaction. In the case of colliding billiard balls, the direction of scat-

tering of one from the other can be predicted from the masses, speeds,

and directions of the balls immediately prior to the collision. Born now

saw that the wave interpretation eliminated this kind of predictability.

He reasoned that the direct causal connection between the state of the

electron and atom before and after the collision is lost.

In the wave theory of light, the connection between the square of the

wave amplitude and the intensity of the light was well understood. In

his papers Schrödinger had tried to establish a connection—through a

‘heuristic hypothesis’—between the modulus-square of the amplitude of

the wavefunction for a single electron and the density of electric charge.

Now Born was saying that the wavefunctions represented the probabilities

that the electron wave will be scattered in certain directions: ‘. . . only one

interpretation is possible, [the wavefunction] gives the probability for

the electron, arriving from [a speciﬁ c initial] direction, to be thrown out

into [a ﬁ nal] direction.’ In the proofs to this hastily written paper Born

added a footnote: ‘A more precise consideration shows that the probabil-

ity is proportional to the square of [the wavefunction].’

Born subsequently claimed that he had been inﬂ uenced by a remark

that Einstein had made in one of his unpublished papers. In the context of

light quanta interpreted using de Broglie’s wave–particle ideas, Einstein

had suggested that the waves represented a Gespensterfeld, a kind of ‘ghost

ﬁ eld’, which determines the probability for the light quantum to follow a

speciﬁ c path. Born had therefore chosen to reject Schrödinger’s attempts

to provide a literal interpretation of the wavefunction as a real wave dis-

turbance and, following Einstein’s logic, instead regarded the wavefunc-

tion as a measure of the probability of realizing speciﬁ c outcomes in a

quantum transition, such as a collision.

But the reason that Einstein had not published his speculations is

that this probabilistic interpretation has profound implications for the

The modulus-square of the amplitude is the wavefunction amplitude multiplied by its com-

plex conjugate, written |y|

. If the wavefunction is not complex (i.e. if it does not contain i, the

square-root of −1), then the modulus-square of the wavefunction is simply its square, y

ghost field

notions of causality and determinism, notions that Einstein held very

dear. Born understood these implications only too well. In his June 1926

paper he wrote:

Schrödinger’s quantum mechanics therefore gives quite a deﬁ nite answer

to the question of the effect of the collision; but there is no question of any

causal description. One gets no answer to the question: ‘what is the state

after the collision,’ but only to the question, ‘how probable is a speciﬁ ed

outcome of the collision’ . . .

Here the whole problem of determinism comes up. From the standpoint

of our quantum mechanics there is no quantity which in any individual

case causally ﬁ xes the consequence of the collision; but also experimen-

tally we have so far no reason to believe that there are some inner prop-

erties of the atom which conditions a deﬁ nite outcome for the collision.

Ought we to hope later to discover such properties . . . and determine them

in individual cases? Or ought we to believe that the agreement of theory

and experiment—as to the impossibility of prescribing conditions for a

causal evolution—is a pre-established harmony founded on the nonexist-

ence of such conditions? I myself am inclined to give up determinism in

the world of atoms. But that is a philosophical question for which physical

arguments alone are not decisive.

These paragraphs frame a debate that would rage for decades. If the

wavefunctions carry only information about probabilities then they

are not ‘real’ in the sense that Schrödinger imagined them to be real.

If the only information available from quantum mechanics concerns

the probabilities of certain outcomes, then causality and determinism

are abandoned. In the realm of quantum transitions, we can no longer

say: ‘if we do this, then that will happen,’ we can only say: ‘if we do this,

then that will happen with a certain probability.’ It might be possible to

restore causality and determinism if some new, but currently ‘hidden’,

properties of quantum systems were to be uncovered that would allow a

cause to be traced directly through to an effect. Born did not feel the need

to invoke such hidden properties.

If the wavefunctions are not real then they are no longer obliged to

behave as real systems would be expected to behave. At a stroke Born

resolved many of the conceptual issues with Schrödinger’s wave mechan-

ics. There was now no need to invoke untenable wave packet states.

the quantum story

Schrödinger had also been troubled by the fact that his wavefunctions

could be complex, i.e. they contained ‘imaginary’ numbers based on the

quantity i, the square-root of minus one. For complicated systems con-

taining two or more electrons, the wavefunctions had to be written not

in three spatial coordinates representing three dimensions but in many

coordinates representing many dimensions. The wavefunction of a sys-

tem containing N particles depends on 3N position coordinates and is

a function in a 3N-dimensional conﬁ guration space or ‘phase space’. It

is difﬁ cult to visualize a reality comprising imaginary functions in an

abstract, multi-dimensional space. No difﬁ culty arises, however, if the

imaginary functions are not to be given a real interpretation.

In his hastily prepared June paper Born promised a more considered

perspective. This duly arrived a month later. In this second paper he con-

siderably sharpened and deepened his interpretation and acknowledged

the inspiration he had drawn from Einstein’s work. He considered a sys-

tem whose wavefunction, as a result of a transition of some kind, can be

represented as a superposition of two or more discrete eigenfunctions

of the system, each mixed together in speciﬁ c proportions. The propor-

tion or amplitude of each eigenfunction y

in the sum is determined by

a ‘mixing’ factor c

. Born now argued that the probability of the system

to be in the state characterized by y

after the transition is given by the

modulus-square of its amplitude, |c

, by deﬁ nition a number between

zero and one.

In his June paper he had talked about the probabilities of transitions

between states involving collisions between electrons and atoms. Now

he was talking about the probabilities of the speciﬁ c quantum states

themselves.

By the time Born came to deliver a lecture to a meeting of the Brit-

ish Association at Oxford, England, in August 1926, his understanding

was complete. His lecture was translated by American physicist J. Robert

Oppenheimer, who at that time was working with Born’s colleague James

Franck in Göttingen, and subsequently published in the British journal

Nature in 1927. In this lecture, for the ﬁ rst time Born drew a clear dis-

tinction between the statistical probabilities of classical physics and the

quantum probabilities associated with the wavefunctions. He wrote:

ghost field

In classical dynamics the knowledge of the state of a closed system (the

position and velocity of all its particles) at any instant determines unam-

biguously the future motion of the system; that is the form that the prin-

ciple of causality takes in physics . . . But besides these causal laws, classical

physics always made use of certain statistical considerations. As a matter

of fact, the occurrence of probabilities was justiﬁ ed by the fact that the ini-

tial state was never exactly known; so long as this was the case, statistical

methods might be, more or less provisionally, adopted.

We deal with probabilities in classical physics because we are often

ignorant of the states of large, complex systems. A good example is Boltz-

mann’s use of statistical methods to describe the properties of atomic

and molecular gases. In such a situation, we can perhaps be conﬁ dent

that the principles of causality and determinism hold at the microscopic

level for each individual particle undergoing a sequence of collisions, but

we are not in a position to follow these motions experimentally. Instead,

we deal with statistical averages.

Born now contrasted this situation with probability in quantum

mechanics:

The classical theory introduced the microscopic co-ordinates which deter-

mine the individual process, only to eliminate them because of ignorance by

averaging over their values; whereas the new [quantum] theory gets the same

results without introducing them at all. Of course, it is not forbidden to believe

in the existence of these co-ordinates; but they will only be of physical signiﬁ -

cance when methods have been devised for their experimental observation.

Born concluded his lecture with the observation: ‘. . . the fundamental idea

of waves of probability will probably persist in one form or another.’

This distinction between classical and quantum probability might seem

trivial, or irrelevant. To the physicists of the Göttingen and Copenhagen

schools, Born’s interpretation seemed intuitive and obvious. It was no

big deal. Consequently, when Heisenberg adopted the quantum prob-

ability interpretation in a paper submitted in November 1926, he did not

feel the need to cite Born’s June or July papers.

But this use of probability in quantum mechanics had removed a

fundamental building-block of physical theory. Quantum mechanics

the quantum story

appeared to provide a recipe for identifying the different possible out-

comes of a transition and for establishing their relative probabilities.

Applying the recipe was equivalent in many ways to stating the cause

of the transition. There was, however, nothing in the theory to enable

prediction of which of the various possible outcomes would be realized

in actuality. Having established the cause, and the range of possible out-

comes, the actual effect appeared to be left entirely to chance.

Some physicists were deeply troubled. Schrödinger was not persuaded

by Born’s arguments, as he explained in a letter to Wien:

From an offprint of Born’s last work in the Zeitsch. f. Phys. I know more or

less how he thinks of things: the waves must be strictly causally determined

through ﬁ eld laws, the wavefunctions on the other hand have only the mean-

ing of probabilities for the actual motions of light- or material-particles.

I believe that Born thereby overlooks that . . . it would depend on the taste

of the observer which he now wishes to regard as real, the particle or the

guiding ﬁ eld. There is certainly no criterion for reality if one does not want

to say: the real is only the complex of sense impressions, all the rest are

only pictures.

Most importantly, Schrödinger’s opposition to the idea of quantum

jumps remained unshaken.

Born had acknowledged his debt of inspiration to Einstein’s Gespenster-

feld in a letter to Einstein dated 30 November 1926. In his reply, Einstein

summarized the nature and extent of his doubts:

Quantum mechanics is very impressive. But an inner voice tells me that it

is not yet the real thing. The theory produces a good deal but hardly brings

us closer to the secret of the Old One. I am at all events convinced that He

does not play dice.

Einstein, whose genius and insight had established the foundations for

the construction of the new quantum theory, was rapidly transforming

into one of the theory’s most determined critics.

Born was dismayed by Einstein’s reaction. An intense debate about the

nature of reality at the quantum level now loomed.

Heisenberg was absolutely furious. The foundations of his new quantum mechanics were

at risk of being undermined by Schrödinger’s intuitively appealing and mathematically

more familiar and accessible wave mechanics. As more and more members of the physics

community began to defect—with great relief—to the wave approach, Heisenberg grew

increasingly dismayed. Matrix mechanics was losing out.

There was more at stake here than the question of interpretation; of Schrödinger’s stub-

born attempts to eliminate quantum jumps and return to a classical space–time perspec-

tive. If wave mechanics came to be adopted as the ‘deepest formulation of the quantum

laws’,

this not only tended to invalidate Heisenberg’s personal experience of discovery on

Helgoland but also thwarted his ambition to dominate the new ﬁ eld which he had helped

to create. Heisenberg’s future career was potentially at stake.

In April 1926 he had turned down the offer of an associate professorship in theoretical

physics at the University of Leipzig in favour of a return to Bohr’s Institute in Copen-

hagen, as a replacement for Hendrik Kramers, who had taken a professorial position in

Utrecht. The tradition in Germany was for a young, aspiring academic always to accept

the ﬁ rst offer of a professorial position. His father had advised him to accept. Virtually all

of the esteemed senior physicists he had consulted had advised him to go to Copenhagen.

He arrived back in Copenhagen in May 1926.

All This Damned

Quantum Jumping

Copenhagen, October 1926

In his June 1926 paper, Born had written: ‘Of the different forms of the theory only

Schrödinger’s has proved suitable for [the description of a collision] and for this reason I might

regard it as the deepest formulation of the quantum laws.’