A Brief Guide to the Great Equations (34 page)

DISCOVERER:
Werner Heisenberg

DATE
: 1927

Everyone understands uncertainty. Or thinks he does.

We owe many a debt to Werner Heisenberg. As one of the founders of quantum mechanics, he left a huge legacy to physics. As the inventor of the uncertainty principle, he also left a huge legacy outside physics. Albert Einstein may be more widely recognized by the public – and his theory of relativity often crops up in popular culture – but Heisenberg has had a similar far-reaching impact on public discourse and popular culture.

While most nonscientists may recognize Einstein’s equation
E
=
mc
²
, generally they are aware that its effects are noticeable only in certain restricted conditions, and that its meaning is truly clear only
to physicists. The same is not true of Heisenberg’s equation Δ × Δ Ρ ≥ – h/2, the Heisenberg uncertainty principle, which seems to have a spiritual meaning to the public that is at once profound and transparent. Browse through any bookshop’s new-age section, for instance, and you’ll find wild claims confidently asserted about the uncertainty principle, such as that its implications are ‘psychedelic’ and that it heralds ‘cultural revolution.’ Strange interpretations turn up even in academic circles. Consider the following conversation, published in
American Theatre
, between the well-known theatre director Anne Bogart and Kristin Linklater, the noted vocal coach:
¹

Linklater
: Some thinker has said that the greatest spiritual level is insecurity.

Bogart
: Heisenberg proved that. Mathematically.

Linklater
: There you are.

But where are we, exactly? And how did we get here? The uncertainty principle sprang from a purely mathematical approach to atomic physics, where it has a well-defined and highly restricted scope of applicability.

Werner Heisenberg, son of a professor of Greek at the University of Munich, had the character one often associates with poets: dashing good looks, a physical frailty including severe vulnerability to allergies, excellent musicianship, and a sensitive and often emotional responsiveness to the world around him.
²
He also had a sharp but imaginative intellect and a willingness to risk unconventional but mathematically rigorous means to fit theories to experimental data. And he had the terrific fortune to be reared amid one of the sharpest and most ruthlessly demanding scientific communities ever, whose members included Niels Bohr, Max Born, Pascual Jordan, Hendrik Kramers, and Wolfgang Pauli.

These theorists were largely distributed among three centres of research: Munich, Göttingen, and Copenhagen. Each had a distinctive character. Munich was experimentally oriented, Göttingen was a world-renowned centre of formal mathematics, and Copenhagen had a rigorous philosophical approach to the quantum world stemming from its founder and leader, Bohr. The intense and often brutally frank exchanges of this community of physicists – carried on in personal conversations as well as in letters, drafts of work in progress, and copies of published papers – kept anyone who dared participate to a high standard. Many times a thought initiated by one person was completed by another. Heisenberg, a central player, circulated among the three centres, and his insights, too, often arose in conversation.

Werner Heisenberg (1901–1976)

In July 1923, Heisenberg completed his doctoral exam at Munich, and had arranged to work under Max Born in Göttingen that fall. But Heisenberg, a supposed wunderkind, had nearly failed thanks to his almost total ignorance of experimental physics – he could not even explain how a storage battery worked – and passed only after aggressive intervention by one of his examiners. The day after the humiliating exam he showed up at Born’s door in Göttingen, unannounced and despondent, to confess to Born the embarrassing news and ask if Born still wanted him. Born was supportive, and Heisenberg left, reassured, for a summer trip he regularly took with a youth group.

This was just the time in quantum theory that historian Max Jammer described as an unruly mess, when those problems that could be solved at all were first analysed classically, then restricted by quantum conditions to obtain a few ‘allowed’ states of motion. Heisenberg, still only 21 years old, was determined to make it all rational.

He knew that classical physics had to be the starting point. ‘The concepts for quantum mechanics can only be explained by already knowing the Newtonian concepts’, he remarked much later. ‘That is, quantum theory is based upon the existence of classical physics. This is the point that Bohr emphasized so strongly – that we cannot talk about quantum physics without already having classical physics.’
³

In classical physics, all events take place inside a four-dimensional space-time stage or playing field. Everything is at a specific place at any and every time. When things move from one place to another, they do so in response to definite forces and take definite paths. Classical physics mainly concerns itself with what happens when things are disturbed, and tracks which forces produce which effects. The path of each thing can be followed – and predicted – like that of a smoothly flowing stream, with the thing moving continuously and smoothly from each point to the next. Physical properties, which can all be measured, propagate smoothly and continuously through space-time in a mechanical way. Classical physics thus provides a confident ontology, or vision of what the ultimate elements of the universe are and how they interact. This kind of event is therefore
anschaulich
.

But a quarter-century of attempts to devise classical models of quantum phenomena had failed. Stimulated by the debates swirling in his rich intellectual environment, young Heisenberg began to wonder if
that
were not the problem; if the effort to construct pictures of the world inside the atom – the positions and paths of electrons, and the dimensions and frequencies of their orbits – was doomed from the start. He had heard Pauli remark that models of atomic events had ‘only a symbolic sense’ and were classical ‘analogues’ of the quantum phenomena.
⁴
Wasn’t the lesson of Maxwell’s path to his equations that sometimes one had to abandon mechanical explanations to capture reality? Wasn’t it likely, then, that when theorists construct models based on what experimenters measure, these models are only symbols of a reality that humans cannot picture?
⁵
Progress in science usually involves sacrifice, Heisenberg once wrote, a sacrifice that is at the cost of our claim to understand nature. What had to go, this time, Heisenberg and his colleagues thought, might be visualizability.

Heisenberg therefore decided to make a virtue of necessity by junking the attempt to produce theories that picture how atomic events unfold on a space-time stage. Drawing on the appreciation for formal structures he had acquired at Göttingen, he would seek a purely mathematical description of what experimenters actually observed: the frequencies and amplitudes of the light emitted by electrons. These descriptions would need to respect only the correspondence principle – that large quantum numbers obeyed classical laws – and certain other constraints such as the conservation of energy. But there would be no need to have measurable properties or continuously propagating functions; indeed, discontinuity seemed to Heisenberg the principal distinctive feature of the quantum realm, and thus would characterize its theory.

This insight was momentous. It has been likened to Copernicus’s insight into the structure of the solar system. Both changed the viewpoint from which scientists had been used to regard the world, treating what had been naïvely assumed to be the image of objective reality as the more complex product of an interaction between the human observer and nature.

The step was revolutionary, but the way had been prepared not entirely by Heisenberg. First, he was utilizing theoretical tools acquired from Bohr, Born, and others, and abandoning the space-time stage only because that seemed the price one had to pay to use them. Second, Heisenberg had a fine precedent in the strategy Einstein used in 1905 to give birth to special relativity. Einstein had abandoned the traditional meaning of ‘simultaneous’ as ‘happening at the same space-time instant’, and redefined it in terms of what an observer could see. Heisenberg hoped to achieve a similar breakthrough by abandoning the traditional conception of ‘position’ and ‘momentum’ inside the atom – which were unobserved but inferred
quantities – and redefine these in terms of what experimenters saw from the outside: the frequencies and amplitudes of spectral lines. Finally, it was not such a radical step to give up trying to construct a theory that tried to picture what could not be observed given the utter failure of all the theories that had tried.

But like most revolutions, it had long-range consequences that would take years to become clear. If to be a ‘thing’ meant occupying a specific place at a specific time, this approach meant ‘eliminating the concept of a particle, or ‘thinghood’, from the atomic domain.’
⁶
This approach essentially replaced the Newtonian ontology of nature, in which its most fundamental pieces are all objectively present in a particular place at a particular time, with a new ontology involving, as one philosopher of science put it much later, ‘a subtle subjectivity at the very heart of the scientific enterprise.’
⁷
The subjectivity relates to the fact that our pictures of the atomic world are not an image of objective reality but are partly a function of the human mind constructing the pictures. The subtlety related to the fact that it was not yet clear what role the mind played.

But so much was not at all clear at the time. Heisenberg’s path unfolded in fits and starts over the first few months of 1925. He co-wrote a paper with Hendrik Kramers in Göttingen with equations containing no classical variables but only frequencies and amplitudes. Kramers’s contribution was an important clue, for he showed that only when these frequencies and amplitudes are associated with pairs of states does one get the correct matrices. Then Heisenberg injured himself skiing and spent several weeks in Munich recovering. He visited Copenhagen and Göttingen, took another trip to the mountains, and by the end of April returned to Born’s institute in Göttingen to prepare to teach a summer session. All of these visits prepared Heisenberg to try to rewrite Bohr’s quantum descriptions of electron momentum (
p
) and position (
q
) in purely mathematical terms. He did not tell his supervisor what he was up to, keeping the idea, as Born once put it, ‘dark and mysterious.’
⁸

Then, Heisenberg once recalled, ‘My work along these lines was
advanced rather than retarded by an unfortunate personal setback.’
⁹
In May, he was hit by an attack of hayfever so severe that he asked Born for 2 weeks off. Born agreed, and Heisenberg headed for Helgoland, an isolated, rocky island in the North Sea that is inhospitable to grass, weeds, and other allergen producers. The evening before his departure, the landlady of the
Gasthaus
who showed him his room was so horrified by his swollen face that she assumed he had been in a fight. On the island, once able to take up his work again, he tried to see if his ideas were consistent with the conservation of energy. When they were, he grew excited. He made mathematical errors, owing to his condition and fatigue, but caught them and continued working late into the night, finally sorting out everything by about 3
A.M
.

At first I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me. I was far too excited to sleep, and so, as a new day dawned, I made for the southern tip of the island, where I had been longing to climb a rock jutting out into the sea. I now did so without too much trouble, and waited for the sun to rise.
¹⁰

Heisenberg returned to Göttingen late in June, and was soon scheduled to leave to lecture in Cambridge. In a few days he dashed off a paper, ‘On the Quantum-Mechanical Reinterpretation of Kinematic and Mechanical Relations.’
¹¹
The word ‘reinterpretation’ (
Umdeutung
) reveals Heisenberg’s audacity: it was the manifesto of a new approach to atomic physics. The abstract boldly declared that the aim was ‘to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.’ We have not been able to ‘associate an electron with a point in space’ based on the experimental information, he continued, and ‘in this situation it seems
sensible to discard all hope of observing hitherto unobservable quantities, such as the position and period of the electron.’ Heisenberg was groping in the dark here; it would turn out that quantum mechanics would contain the possibility of measuring position and momentum to any degree of accuracy, just not simultaneously. The paper showed how to compile tables of amplitudes and frequencies associated with transitions between states – he called such tables ‘quantum-theoretical quantities’ – and how the tables could be related by a new kind of calculus, which he called ‘quantum-mechanical relations.’