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

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the quantum story

Whilst he acknowledged the familiarity and utility of Schrödinger’s mathematical for-

malism, he argued in a letter to Pauli in June 1926 that these methods should be deployed

in the service of matrix mechanics, conﬁ ned to the calculation of individual matrix

elements. At the same time, he also sensed that the wave approach could provide some

elements of physical interpretation that had thus far been lacking in matrix mechanics.

All was in ﬂ ux. Heisenberg had to ﬁ nd a way to regain the initiative. He resolved to

force the issue in a direct debate with Schrödinger.

The ﬁ rst opportunity for such a debate arose towards the end of July

1926. Schrödinger had been invited by Wien and Sommerfeld to give two

lectures in Munich, one to the Bavarian section of the German Physi-

cal Society and the other to a select group of quantum physicists the

next day. Heisenberg, settling into his new post as university lector and

assistant to Niels Bohr in Copenhagen, had been busily familiarizing

himself with wave mechanics. He took a brief holiday in Norway before

arriving in Munich in July to stay with his parents. He attended both of

Schrödinger’s lectures armed with a raft of objections.

Heisenberg held his tongue as Schrödinger delivered his ﬁ rst lecture to

a packed auditorium. But, as Schrödinger concluded his second lecture on

the morning of Saturday, 24 July, Heisenberg stood to voice his objections.

Wave mechanics as conceived by Schrödinger, he argued, could not

explain some relatively fundamental physical phenomena. It could not

explain Planck’s radiation law, or the Compton effect, or the intensities

of spectral lines. These were all phenomena that could only be explained

using the ideas of discontinuity and quantum jumps. And these were

ideas that could not be accommodated in Schrödinger’s continuous

wave mechanics.

The response from the audience was not the one he might have hoped

for. Wien motioned for him to shut up and sit down. Heisenberg later

wrote that Wien:

. . . told me rather sharply that while he understood my regrets that [matrix]

mechanics was ﬁ nished, and with it all such nonsense as quantum jumps,

etc., the difﬁ culties I had mentioned would undoubtedly be solved by

Schrödinger in the very near future. Schrödinger himself was not quite so

certain in his own reply, but he, too, remained convinced that it was only

a question of time before my objections would be removed.

all this damned quantum jumping

These sentiments appeared to be shared by the majority of physicists

in the audience. The elderly experimentalist Wien may never have for-

given the young theoretician Heisenberg’s shirking of experiment and

the theory of experimental instruments as he had studied for his doctor-

ate in Munich three years before. Heisenberg’s poor doctoral grade was

testimony to the depth of Wien’s anger with the upstart theoretician. But

even Sommerfeld, who had defended Heisenberg’s ‘extraordinary abili-

ties’ in the ensuing row with Wien over the relative importance of theory

and experiment, now seemed to have succumbed to the simple elegance

of Schrödinger’s mathematics.

Heisenberg was bitterly disappointed. Shortly after the lecture he wrote

to Bohr about the unhappy outcome of his intervention. Bohr shared

Heisenberg’s misgivings, and decided to invite Schrödinger to Copenha-

gen to debate the issues further. Schrödinger agreed to visit Bohr towards

the end of September 1926. Heisenberg sped back to Denmark.

The stage was set for the ﬁ rst round in a dialogue that would shape the

future development of quantum theory, and with it our conception of

physical reality.

Towards the end of August 1926, Schrödinger took Anny on a three-week

holiday in the South Tirol. Anny then went to visit her parents in Salz-

burg while Schrödinger spent a few days with the Wiens before making

the journey to Copenhagen towards the end of September.

The debate began even as Bohr and Heisenberg collected Schrödinger

from the railway station in Copenhagen. Bohr and Schrödinger had not

met before, and Schrödinger must now have wondered just what he had let

himself in for. Bohr, normally considerate and friendly in his dealings with

visitors, confronted Schrödinger like some kind of remorseless fanatic. Bohr

would concede nothing. He and Heisenberg were right and Schrödinger was

wrong, and it was now Bohr’s mission to make Schrödinger see the light.

But Schrödinger, too, had deeply held convictions and an entrenched posi-

tion on how he believed his wave mechanics should be interpreted. Moreo-

ver, he had been greatly encouraged by the warmth with which the physics

community had received his approach. He was in no mood to back down.

‘Surely you realise,’ Schrödinger pleaded, ‘that the whole idea of quan-

tum jumps is bound to end in nonsense.’

the quantum story

He pointed to the difﬁ culties of understanding precisely how an elec-

tron moving inside an atom was meant to make such discontinuous

jumps. ‘Is this jump supposed to be gradual or sudden?’ he asked. ‘If it

is gradual, the orbital frequency and energy of the electron must change

gradually as well. But in that case, how do you explain the persistence of

ﬁ ne spectral lines? On the other hand, if the jump is sudden . . . then we

must ask ourselves precisely how the electron behaves during the jump.

Why does it not emit a continuous spectrum, as electromagnetic theory

demands? And what laws govern its motion during the jump? In other

words, the whole idea of quantum jumps is sheer fantasy.’

Bohr’s response established the grounds for the real argument,

which concerned the visualizability and understandability of physical

theories.

‘What you say is absolutely correct,’ Bohr replied, ‘But it does not prove

that there are no quantum jumps. It only proves that we cannot imagine

them, that the representational concepts with which we describe events

in daily life and experiments in classical physics are inadequate when it

comes to describing quantum jumps. Nor should we be surprised to ﬁ nd

it so, seeing that the processes involved are not the objects of direct expe-

rience.’

But Schrödinger did not want to be drawn into a philosophical debate

about the development and meaning of concepts. ‘I wish only to know

what happens inside an atom. I don’t really mind what language you

choose to discuss it,’ he said.

If electrons were assumed to be tiny particles orbiting the nucleus,

then Schrödinger expected that a physical theory should describe how

these particles move in their orbits and make transitions from one

orbit to another. But neither wave nor matrix mechanics could provide

such a rational description. Change the picture to one of matter waves,

Schrödinger argued, then everything suddenly looks very different, and

the unresolved contradictions disappear.

‘I beg to disagree,’ Bohr insisted. ‘The contradictions do not disap-

pear; they are simply pushed to one side.’ Bohr reiterated the problem

of explaining Planck’s radiation law using wave mechanics, and the fact

that Planck’s law demanded that an atom possess discrete energy values

all this damned quantum jumping

which change discontinuously. ‘You can’t seriously be trying to cast

doubt on the whole basis of quantum theory?’ he said.

Schrödinger had no choice but to concede the point. ‘I don’t for a

moment claim that all these relationships have been fully explained,’

he said. He suggested that the application of thermodynamics to wave

mechanics may eventually provide some answers.

No, Bohr countered, one cannot hope for that. ‘We have known what

Planck’s formula means for twenty-ﬁ ve years. And, quite apart from that,

we can see the inconstancies, the sudden jumps in atomic phenomena

quite directly, for instance when we watch sudden ﬂ ashes of light on a

scintillation screen or the sudden rush of an electron through a cloud

chamber. You cannot simply ignore these observations and behave as if

they did not exist at all.’

‘If all this damned quantum jumping were really here to stay,’ an exas-

perated Schrödinger exclaimed, ‘I should be sorry I ever got involved

with quantum theory.’

Bohr tried to calm the discussion. ‘But the rest of us are extremely

grateful that you did,’ he said, ‘Your wave mechanics has contributed so

much to mathematical clarity and simplicity that it represents a gigantic

advance over all previous forms of quantum mechanics.’

Schrödinger stayed at the Bohrs’ home and the debate raged day and

night. Perhaps as a result of the enormous intellectual effort demanded

of him, Schrödinger succumbed to a feverish cold. Bohr pursued him

even to his sick-bed. As Bohr’s wife Margrethe attempted to nurse him

back to health with tea and cake, Bohr sat at the edge of the bed and con-

tinued his harangue: ‘But you must surely admit that . . .’

But Schrödinger would admit nothing.

He was, nevertheless, deeply affected by the intensity of the debate and

recognized that it was going to be necessary somehow to accommodate

The cloud chamber was invented by Charles Wilson, a student of J.J. Thomson. As an ener-

getic particle passes through such a chamber, it dislodges electrons from atoms in the vapour

contained in the chamber, leaving charged ions in its wake. Water droplets condense around the

ions, revealing the particle trajectory.

the quantum story

both the wave and particle perspectives. Wave mechanics alone, he now

realized, couldn’t be the answer.

In a letter to Wien he described his ﬁ rst encounter with Bohr:

Bohr’s . . . approach to atomic problems . . . is really remarkable. He is com-

pletely convinced that any understanding in the usual sense of the word is

impossible. Therefore the conversation is almost immediately driven into

philosophical questions, and soon you no longer know whether you really

take the position he is attacking, or whether you really must attack the

position he is defending.

But there were no hard feelings. This had been a stimulating, if some-

what exhausting, intellectual debate about matters of profound impor-

tance to the future development of quantum theory. In the same letter,

Schrödinger continued:

In spite of all [our] theoretical points of dispute, the relationship with

Bohr, and especially Heisenberg, both of whom behaved towards me in

a touchingly kind, nice, caring and attentive manner, was totally, cloud-

lessly, amiable and cordial.

The simple truth was that nobody could yet offer a coherent interpretation

of quantum mechanics. However, Bohr and Heisenberg were convinced

it was they who were on the right track.

The debate with Schrödinger had a singularly powerful effect on Bohr.

Schrödinger’s grasping for the kinds of wave concepts associated with

classical physics and his relentless pursuit of a theory that provided some

kind of space–time visualizability brought home to Bohr the precise

nature of the problems they were all grappling with.

Schrödinger cherished the continuity and visualizability that was

afforded by wave concepts, and simply could not see how discontinuous

quantum jumps could ﬁ t with such a scheme. Bohr had argued that ‘. . .

the pictorial concepts we use to describe the events of everyday life and

the experiments of the old physics do not sufﬁ ce to represent also the

process of a quantum jump.’

Bohr and Heisenberg had abandoned any attempt to use classical space–

time concepts to describe an essentially unvisualizable physics. But the

all this damned quantum jumping

very fact that wave mechanics had struck such a chord with physicists now

suggested to Bohr that the use of classical concepts to describe decidedly

non-classical quantum phenomena was not going to be so readily aban-

doned.

Perhaps, Bohr reasoned, there was a sense in which both classical wave

and particle concepts were in some way equally valid; each helpful in

describing some aspect of an otherwise unfathomable quantum reality

depending on the speciﬁ c phenomena that were being observed. Each

was mutually exclusive to the other, yet both were needed for a complete

description of the inner workings of the atom.

Heisenberg, too, paused to reﬂ ect. Inevitably, whilst he was content to

embrace Born’s probability interpretation, he was rather unhappy that Born

had reached for wave mechanics as the ‘deepest formulation of the quantum

laws’. He felt that Born’s conclusion still left too much room for alternative

interpretations. Born, in the meantime, was beginning to regret the ‘victory’

of Schrödinger’s approach over matrix mechanics, as the approach contin-

ued to drag Schrödinger’s interpretation around closely behind it.

Heisenberg was now living in a small attic ﬂ at in Bohr’s Institute, with

slanting walls and windows which overlooked the entrance to Fælled

Park. Bohr would come to the ﬂ at late at night, and the two would discuss

their scientiﬁ c problems into the small hours.

But Heisenberg disliked the approach that Bohr was developing. It

seemed too vague, too philosophical, too arbitrary. He was uncomforta-

ble with a theory that could be this or that, one thing then another. What

he wanted was a theory that imposed a unique interpretation, if not of

physical, sub-atomic entities such as electrons, then at least of the magni-

tudes of their observable properties, such as energy and momentum.

Although they disagreed on many points, Heisenberg had good reason

to believe they were heading for the same result. He was astonished at

how simple phenomena—such as the observed trajectory of an electron

revealed in a cloud chamber—were proving so intractable. Indeed, the

very concept of a trajectory was absent from matrix mechanics. Wave

mechanics would have the electron matter wave spreading out, dispers-

ing or dissolving as it moved through the chamber. And yet it was difﬁ -

cult for anyone who had looked at the track left by an electron in a cloud

the quantum story

chamber not to be convinced of the reality of the electron’s particle-like

trajectory.

Heisenberg summarized this period of intense debate as follows:

I remember discussions with Bohr which went through many hours till

very late at night and ended almost in despair; and when at the end of the

discussion I went alone for a walk in the neighbouring park I repeated to

myself again and again the question: Can nature possibly be as absurd as

it seemed . . . ?

They reached no real conclusions. Their protracted debate left them

exhausted and somewhat tense. When Bohr decided to go on a skiing

holiday to Norway in February 1927, Heisenberg was greatly relieved.

He was left alone in Copenhagen, ‘ . . . where I could think about these

hopelessly complicated problems undisturbed.’

Born had eagerly embraced Schrödinger’s wave mechanics in June 1926, but by November

his opposition had hardened. He told Schrödinger: ‘It would have been beautiful if you

were right. Something that beautiful happens, unfortunately, seldom in this world.’ In

a letter to Einstein he explained that: ‘Schrödinger’s achievement reduces itself to some-

thing purely mathematical; his physics is quite wretched.’ Born would go on to champion

the matrix approach long after it had ceased to be fashionable.

But the Göttingen school remained a lone voice. Elsewhere, physicists began to use wave

mechanics to duplicate results that had already been obtained using matrix mechanics.

And, as Born had done, they also applied wave mechanics where matrix methods were

inapplicable or regarded as too cumbersome.

Schrödinger’s intransigence during their debate in Copenhagen in October 1926

left Bohr and Heisenberg more determined than ever to ﬁ nd a satisfactory resolution.

Heisenberg may have wanted simply to retreat to matrix mechanics, as his colleagues in

Göttingen had done, but his experiences in Munich and his long discussions with Bohr

convinced him that a fresh approach was needed. It was time to go back to the drawing-

board. It was time to think the unthinkable.

The Uncertainty Principle

Copenhagen, February 1927

Born and Pascual Jordan wrote a book on elementary quantum mechanics which was pub-

lished in 1930. In it they made no reference at all to wave mechanics. Pauli wrote a withering

review, ﬁ nding only one feature about which to be positive: ‘The setup of the book as far as

printing and paper are concerned is splendid,’ he wrote. Quoted in Beller, p. 38.

the quantum story

From the outset, Heisenberg had resolved to eliminate classical space–time pictures

involving particles and waves from the quantum mechanics of the atom. He had wanted

to focus instead on the properties that were actually observed and recorded in labora-

tory experiments, such as the positions and intensities of spectral lines. Left alone in

Copenhagen in February 1927, he now pondered on the signiﬁ cance and meaning of such

experimental observables. Somehow, he needed to introduce at least some form of ‘visu-

alizability’ into matrix mechanics.

This train of thought was to lead him to another fundamental discovery.

In explaining his opposition to wave mechanics many years later, Born

claimed that he: ‘ . . . was witnessing the fertility of the particle concept every

day in [ James] Franck’s brilliant experiments on atomic and molecular

collisions and was convinced that particles could not simply be abolished.’

Indeed, the track left by an electron in a cloud chamber seemed to pro-

vide unambiguous evidence of the electron’s particle-like properties. Yet

this was something that Schrödinger’s wave mechanics seemed unable to

rationalize except in terms of completely untenable ‘wave packets’. It was

here that wave mechanics was most vulnerable.

The situation was very confusing, however. What Heisenberg needed

to do was regain the ‘high ground’. He needed to ﬁ nd a way to use matrix

mechanics to formulate a speciﬁ cally particulate description, with the

attendant quantum jumping, thereby refuting Schrödinger’s exclusively

wave interpretation. This was not speciﬁ cally about matrix versus wave

mechanics, or about particles versus waves, as the matrix approach in any

case sought to eliminate such classical concepts. This was about making

a connection; it was about using the underlying quantum behaviour as

detailed in matrix mechanics to describe large-scale particle-like prop-

erties that could in principle be observed in a laboratory. It was about

doing what Schrödinger’s wave mechanics could not.

He drew inspiration from dialogues with Pascual Jordan in Göttingen

and with Dirac, who had arrived in Copenhagen in September 1926 for

a six-month period as a visiting fellow. Pauli, too, provided some signiﬁ -

cant clues. In a letter Heisenberg had received in October 1926, Pauli had

proposed to re-interpret the modulus-square of the wavefunction not as

a transition probability or as the probability for the system to be in a

speciﬁ c state, as Born had done, but as the probability of ‘ﬁ nding’ the

the uncertainty principle

electron at a speciﬁ c position in its orbit inside an atom.

This gives rise

to an image of a particulate electron which is ‘smeared’ through space

with, at any one time, different probabilities of being found in different

positions around the nucleus.

In essence, Pauli was re-admitting a space–time description, by fusing

together Schrödinger’s wavefunction and an adaptation of Born’s prob-

ability interpretation. Pauli, at least, was convinced of the need for such a

connection. He wrote: ‘I am now however, convinced with all the fervour

of my heart that the matrix elements must be connected with principally observ-

able kinematic (perhaps statistical) data of the particles in question in the stationary

states.’ This was a complete reversal of the position he had taken in his

January 1926 paper on the matrix mechanics of the hydrogen atom.

Pauli had gone on in his letter to make an important observation con-

cerning the relationship between position and momentum in quantum

mechanics. He had considered the situation where two electrons collide.

When the electrons are far apart they can be conveniently treated as

plane waves each with a clearly deﬁ ned position (q) and momentum ( p).

But as they come together they manifest what Pauli called a ‘dark point’,

at which things become fuzzy. If the positions are assumed to be control-

led, then the momenta are uncontrolled, and vice versa. ‘One may view the

world with the p-eye and one may view it with the q-eye, but if one opens

both eyes at the same time one becomes crazy,’ he wrote.

This was behaviour that could be traced back to the position–momentum

commutation relation.

The matrix elements must be connected with principally observable kin-

ematic data, Pauli had written, nearly four months earlier. Heisenberg

now tried to use matrix mechanics to do precisely this; to describe the

clearly visible path of an electron in a cloud chamber. He quickly ran into

trouble:

Strictly speaking, Pauli suggested that the probability of ﬁ nding the electron between posi-

tions q and q + dq, where dq is an inﬁ nitesimal increment along the q-coordinate, is given by

|y(q)|

dq.

This image remains with us today in the form of atomic ‘orbitals’, drawn as maps of elec-

tron density or as boundary surfaces within which there is a high probability (typically 90 per

cent or more) of ﬁ nding the electron.