A Brief Guide to the Great Equations (36 page)

Yet, as Beller notes, ‘in the initial stages of the controversy over interpretation
nobody
had a clearly articulated position, let alone a handle on the ‘truth.’ ‘
²⁷
The restructured clash now forced the partisans to argue for the physical interpretations of their theories. Schrödinger had to argue that, at its most fundamental level, the world was full of continuities, and that to describe it one did not require Heisenberg’s awkward formal methods. Schrödinger also had to explain how wave packets could hold together, elaborate the meaning of the ψ-function, and demonstrate how the discontinuities of quantum phenomena arise from continuous wave processes.
Schrödinger, finally, had to admit the problem of a wave that existed in multidimensional configuration space. Heisenberg and his allies had to argue that the world was full of discontinuities, and that it was misleading to present it otherwise. They had to provide some way of connecting the formal symbolic terms of matrix mechanics to familiar properties, and show why, to the extent that wave mechanics was visualizable, it was false. As partisans are wont, they were not always consistent; Beller has shown how each side ‘cheated’ a little, incorporating aspects of the other in order to make the theories work. But this new and ferocious clash now set the stage for the emergence of both the uncertainty principle and the Copenhagen interpretation of its meaning.
²⁸

Schrödinger began sniping already in his paper proving the identity of the two approaches. While they were indeed identical, he said, he was ‘discouraged, if not repelled’ by the ‘very difficult’ mathematical methods of matrix mechanics and by its lack of visualizability.
²⁹
Later he said that it was ‘extraordinarily difficult’ to attack atomic issues such as the transition problem so long as one has to ‘repress intuition’ and ‘operate only with such abstract ideas as transition probabilities, energy levels, etc.’
³⁰
He wrote to Wien that the avowals about the necessity of restricting physics to observables ‘only glosses over our inability to guess the right pictures.’
³¹

Heisenberg’s language was at least as sharp; he described wave mechanics as ‘disgusting’ and as ‘garbage.’ To the extent that wave mechanics was visualizable it was false, he claimed; physicists who use matrix mechanics are less deluded and thus see deeper into nature.
³²
As Born once said, ‘Mathematics knows better than our intuition.’
³³

The conflict was soon fought in face-to-face encounters. In July 1926, Schrödinger and Heisenberg met for the first time at a conference in Munich, where Schrödinger had many supporters. Schrödinger gave two talks about wave mechanics, and Heisenberg stood up at the end to object that no theory relying on continuous processes could possibly explain the discontinuities of quantum
phenomena, such as Planck’s radiation law and the Compton effect. The audience appeared to be on Schrödinger’s side. Heisenberg seems to have influenced no one, and left feeling defeated. He went to Copenhagen, where he stayed for several months working with Bohr. The two disagreed – Bohr argued that we
must
use classical concepts to describe experiments, with Heisenberg disagreeing – but they honed their arguments why quantum discontinuities implied that space and time could not be defined, meaning that the quantum realm could neither be represented by theories involving continuous functions nor imagined by the human mind whose picturing ability requires a space-time container.

The next bout between Schrödinger and the matrix allies took place 3 months later, in October 1926. Bohr invited Schrödinger to visit Copenhagen, which was largely matrix territory (though the Copenhagen group had already begun to use some wave mechanics as a tool); Schrödinger, intellectually honest and seemingly riding the popular side, was happy to visit the opposition’s headquarters. But he was utterly unprepared for what followed. Bohr met Schrödinger at the train station, almost immediately began pressing his case, and continued arguing day and night for several days. Bohr had arranged for Schrödinger to stay at his house, so that every possible minute could be used. As Heisenberg recalled:

Bohr was an unusually considerate and obliging person, but in this kind of discussion, which concerned epistemological problems which he thought were of vital importance, he was capable of insisting – with a fanatic terrifying relentlessness – on complete clarity in all argument. Despite hours of struggle, he refused to give up until Schrödinger had admitted his interpretation was not enough, and could not even explain Planck’s law. Perhaps from the strain, Schrödinger got sick after a few days and had to stay in bed in Bohr’s home. Even here it was hard to push Bohr away from Schrödinger’s bedside: again and again, he would say, ‘But Schrödinger, you’ve got to at least
admit that…’ Once Schrödinger exploded in a kind of desperation, ‘If you have to have these damn quantum jumps then I wish I’d never started working on atomic theory!’
³⁴

With Heisenberg at his side, Bohr persuaded Schrödinger into making a (temporary) retraction. But it did not last, and the originator of wave mechanics was soon back writing papers about it. In November 1926, indeed, he assembled his six seminal papers on wave mechanics – the four part series ‘Quantization as a Problem of Proper Values’ that appeared in the
Annalen der Physik
, plus his papers on the boundary problem and on the identity between wave and matrix mechanics – and had them published as a book.

By this time, Born had contributed his novel interpretation of wave mechanics. Trying to understand collisions between an electron and an atom, Born had carefully examined Schrödinger’s claim that the ψ-function referred to the electron’s charge density, found it did not make sense, and concluded that it does not tell us about the state of an event but rather about its
probability
. Pauli then wrote his letter to Heisenberg in which he proposed that ψ
²
represented the probability, not of states, but of particles at particular positions. This amounted to a partial restoration of the space-time container and of visualizability. It did not entail that the orbits or paths of electrons from one place to another could be visualized, but that, however they got there, they did
have
positions.
³⁵
The classical properties do exist, and can be measured precisely. Still, it involved the bizarre notion that the strange function that Schrödinger said flowed through space was not a real thing but the probability that a real thing could be found at that spot. At the time, the philosophical novelty of this was not noticed. ‘We were so accustomed to making statistical considerations’, Born remarked later, that ‘to shift it one layer deeper seemed to us not so very important.’
³⁶

In the same letter of October 19 in which Pauli made his proposal about the interpretation of the wave function, he also noted implications for the vexing
pq – qp
issue. Heisenberg had been arguing
that neither of the conjugate variables – the noncommuting terms – referred to classical variables such as positions or momenta that could be measured with precision together. Pauli was now saying that one of the pair could be – but if so, the other was only known as a probability. This made the noncommutativity even stranger. ‘The physics of this is unclear to me from top to bottom’, Pauli told Heisenberg. ‘My first question is: why can only the p’s, and not
simultaneously
both the p’s and the q’s, be described with any degree of precision?’ He was baffled. ‘You can look at the world with p-eyes or with q-eyes, but open both eyes together and you go wrong.’
³⁷
What could this mean?

Heisenberg’s response was delayed because he had a hard time retrieving Pauli’s letter from his excited Copenhagen colleagues who were sharing it. Heisenberg finally sent a reply on October 28. He still did not buy the implied restoration of visualizability and classical variables, dismissing Born’s ‘rather dogmatic’ view as ‘only one of several possible interpretations.’ The
pq – qp
=
I
h
/2π
i
relation, he continued to insist, meant that individual
p
s and
q
s were meaningless. ‘Above all, I hope there will eventually be a solution of the following type (but don’t spread this around): That time and space are really only statistical concepts, something like, for instance, temperature, pressure, and so on, in a gas. It’s my opinion that spatial and temporal concepts are meaningless when speaking of a single particle, and that the more particles there are, the more meaning these concepts acquire. I often try to push this further, but so far with no success.’

And a few weeks later, on November 15, Heisenberg presented to Pauli what seemed a conclusive argument why the discontinuities of the quantum world made the very concept of individual
p
s and
q
s meaningless.
³⁸
Let’s say an object such as an electron is at a specific point. Its velocity is defined in terms of the rate at which it moves continuously through points vanishingly close to it – but if space-time is discontinuous, and electrons flit from one state to another, it must lack velocity by definition! A week later, Heisenberg
returned obsessively to the issue.
³⁹
Because the world is discontinuous, the ‘c-numbers’ (classical numbers) imply that we know way too much about what is happening. ‘What the word ‘wave’ or ‘corpuscle’ mean, one does not know any more.’

Pascual Jordan now stepped in to challenge Heisenberg. In effect, Jordan played contrarian to Heisenberg’s postulate-of-impotence claim that single electrons could not have positions and momenta. What was stopping experimenters from measuring them? Observing equipment is made of atoms, and atoms rattle about at room temperature due to their thermal motion, imposing a practical limit on accuracy. So what if we somehow set up the equipment to make a measurement at absolute zero where thermal motion stops; or, in what amounts to the same thing, what if we use highly energetic probes such as α particles, whose rattling is negligible and whose paths can be tracked?

Born and Pauli had considered the theoretical possibility of fixing one conjugate variable, and noted that the other could only be said to have a certain probability. Jordan was now pointing out experimental conditions in which physicists could indeed measure what was supposedly forbidden: the ‘probability of finding an electron in a certain place.’ It’s not unobservable theoretically, just difficult experimentally.

Jordan’s article troubled Heisenberg.
⁴⁰
The day after it appeared, on February 5, 1927, Heisenberg wrote to Pauli that he found Jordan’s paper ‘nice enough but not very exact in places’, because he still thought phrases like ‘probability of finding an electron in a certain place’ were conceptually meaningless. But if things such as the time and position of individual electrons made experimental sense, they had to make theoretical sense. If they made theoretical sense, his approach had to be wrong.

In all these discussions, there was never any question that the mathematics was correct. It was the interpretation that was at issue, and even the nature of interpretation. Bohr demanded more of interpretation than Heisenberg, and both demanded more than Schrödinger.

Heisenberg was still in Copenhagen, working at Bohr’s institute and living in the garret apartment of Bohr’s brother Harald. After supper, Bohr would come by, pipe in hand, and the two would argue about the state of quantum mechanics until the morning hours. The demanding conversation was beginning to wear on their relationship, and the two grew testy. Realizing this, Bohr left to go skiing. One evening while Bohr was gone, Heisenberg took a walk in Faelled Park, behind Bohr’s institute. He pondered
p
s and
q
s in theory and in experiment. He thought about Jordan’s microscope. He was as convinced as ever that something
had
to be wrong with Jordan’s example. Jordan had brought Heisenberg down to earth, derailing his single-minded focus on theoretical meaning, for it forced him to stop philosophizing about concepts and think operationally about what experimenters did. Suppose you looked at a particle at absolute zero; this would mean bouncing a photon off it and capturing the photon in the instrument’s lens. But that would disturb the electron’s position. If you wanted to avoid this, you would have to use a less energetic photon. But the longer the wavelength of the photon, the less precisely you knew its position! The problem might occur, Heisenberg excitedly realized, because of the interaction between the instrument and what you are seeking to measure – between the tools you were using to observe and the system observed.

Heisenberg then did what he often did when excited: he wrote a letter to Pauli. This one, dated February 23, was unusually long – fourteen pages. The shift in his thinking inaugurated by Jordan’s article is evident at the outset, for he describes several thought experiments involving measurements of
p
s and
q
s. Then he writes, ‘One will always find that all thought experiments have this property: When a quantity p is pinned down to within an accuracy characterized by the average error p, then…q can only be given at the same time to within an accuracy characterized by the average error q
₁
≈ h/p
₁
.’

This is the uncertainty principle. Like many other equations, its first appearance was not in the form in which it is now famous. Today it is usually written as an inequality: