A Brief Guide to the Great Equations

by Robert Crease

  Schrödinger, then thirty-eight, was ancient by physics standards. His approach covered much the same territory, but used a familiar tool of classical mechanics: a wave equation that he had developed, with continuous functions that described processes unfolding smoothly in space and time. For Schrödinger, the bottom rung was made of something quite anschaulich: waves.

  Erwin Schrödinger (1867–1961)

  Enter Schrödinger

  Erwin Schrödinger had arrived at the University of Zürich in 1921.12 New professors were asked to deliver a formal talk to a general audience, and Schrödinger’s was called ‘What Is a Natural Law?’ In it, he endorsed the possibility that ‘the laws of nature without exception have a statistical character.’13 True, Maxwell had introduced statistical laws into physics to describe the behaviour of systems, such as gases, consisting of large quantities of small things. But such laws were conveniences – approximations, cheats – used because in practice our knowledge is limited. In principle, we could track the behaviour of each and every molecule to predict the system’s behaviour; plug numbers for the forces and masses into Newton’s laws, turn the crank, and out would pop predictions of past and future behaviour. And while Einstein had used probabilities in his 1916 paper, he thought these would be temporary. Though Schrödinger could not know it, the work he was about to do would shortly be interpreted as implanting statistics permanently into nature’s laws, without a deeper underlying law.

  Illness slowed Schrödinger’s work for a few years, but by 1925 he was studying aspects of quantum theory and participating in joint colloquia between the university and the nearby technical school, the Eidgenössische Technische Hochschule (ETH). One day in the fall of 1925, one of the ETH’s organizers, the Dutch physicist Pieter Debye, asked Schrödinger to report on a recently published thesis by Louis de Broglie, now a newly graduated French physicist, who had indeed thrown himself into quantum theory after reading the proceedings of the Solvay conference. He introduced the notion of a wave process accompanying electrons, using Planck’s rule E = hv to connect the momenta of electrons with a wavelength. With this assumption, he was able to explain the quantization conditions of the old quantum theory. And so at one of the next ETH colloquia, Schrödinger dutifully explained the young Frenchman’s idea that the right orbits were obtained if one assumed electrons had integer wavelengths.

  Debye, sitting in the front row as was traditional for someone of his eminence, dismissed the idea as ‘rather childish.’ If something were a wave, he said, it needed a proper wave equation.

  What he meant seems to have been this: waves usually refer to something that is waving. Elsewhere in physics, waves are solutions of equations of motion for that ‘something.’ De Broglie had identified a wave associated with an electron, but he gave no clue as to what was waving or what its equation of motion should be.

  Schrödinger, unlike most people at the ETH colloquium, took Debye’s remark seriously – and was also stirred by Einstein’s remark about de Broglie’s work that an ‘undulatory field is connected with every motion.’14 Schrödinger finished a paper he was writing on the quantum theory of gases, and left on a skiing vacation in Arosa with an old girlfriend whose identity is a mystery, for his diary for 1925 is lost and obvious suspects have been ruled out. A colleague once commented that Schrödinger ‘did his great work during a late erotic outburst in his life’ – though the remark is puzzling, for while thirty-eight may be late for physics, it is not for eros. Schrödinger’s biographer writes that, ‘like the dark lady who inspired Shakespeare’s sonnets, the lady of Arosa may remain forever mysterious’, and that ‘whoever may have been his inspiration, the increase in Erwin’s powers was dramatic…he began a twelve-month period of sustained creativity activity that is without a parallel in the history of science.’15

  On December 27, Schrödinger wrote to Wien from Arosa:

  At the moment I am struggling with a new atomic theory. If only I only knew more mathematics! I am very optimistic about this thing and expect that if I can only…solve it, it will be very beautiful.16

  Schrödinger returned from Arosa to Zürich on January 9, seemingly still struggling. But shortly thereafter, he opened another colloquium talk with the words, ‘My colleague Debye suggested that one should have a wave equation; well, I have found one!’ And in a remarkable series of six papers published in 1926 – ‘Quantization as a Problem of Proper Values’, published in four parts and ‘undoubtedly one of the most influential contributions ever made in the history of science’,17 plus a paper on the transition between the quantum and the classical world, and another on the relation between wave mechanics and matrix mechanics – Schrödinger presented his wave equation and examined its implications.

  Schrödinger’s equation incorporated a wave function that he called ‘a new, unknown ψ’, and related (as de Broglie had) the wavelength to a momentum, and the frequency to an energy. The behaviour of the atomic realm, Schrödinger was proposing, is made up of waves of the ψ field – which was something like a charge density, a kind of particle fog, he initially felt – waves that added up, interfered, created nodes, and so forth. This ‘eminently visualizable’ picture, Schrödinger claimed in the first part of his multipart paper, allows us to picture such experimentally observable things as ‘how two colliding atoms or molecules rebound from one another, or how an electron or α-particle is diverted, when it is shot through an atom.’18 It gives rise, he said, to a picture of electron states in atoms as in effect standing waves – waves that, as in a violin string, retain their basic shape while oscillating. And, he remarked in the second part of the paper, he hoped to be able to demonstrate that his theory would show how wave groups or ‘packets’ form with ‘relatively small dimensions in every direction’ that ‘obey the same laws of motion as a single image point of the mechanical system;’ that is, that act like single particles.19

  It would not be that simple. The ψ field, after all, was a mathematical quantity whose properties would determine physical, observed properties only when supplemented with additional steps. When Schrödinger put in these steps, he found that they included the complex number i. He was initially disturbed by its presence in his wave theory and tried to get rid of it. He could not. He was disturbed because a complex number involves two components – a real part and an imaginary part – and its presence implied that the wave function had a phase that was not directly observable. A phase is like a clock, a cycling phenomenon, and the fact that the phase had an imaginary part meant that it had an aspect that could not be directly measured. It was oscillating in time in a way that could not be seen from the outside, from ‘reality.’ Schrödinger’s equation described something ‘waving’ in a multidimensional or ‘configuration’ space.

  What, then, was this ψ? In his first paper, Schrödinger wrote that originally he had hoped ‘to connect the function ψ with some vibration process in the atom, which would more nearly approach reality than the electronic orbits, the real existence of which is being very much questioned to-day.’ This vibrational process, that is, would create something like what physicists were calling a quantum state, a discrete, discontinuous thing built up out of continuous processes. ‘[T]o imagine that at a quantum transition the energy changes over from one form of vibration to another’, he commented at the end of his first paper, is far more satisfying than ‘to think of a jumping electron’, for ‘the changing of the vibration form can take place continuously in space and time’, and he contrasted his work with the notable failure of the wave theory of Bohr, Kramers, and Slater. Though for the moment, he continued, he would not pursue these thoughts and satisfy himself with presenting his ideas in ‘neutral mathematical form.’20

  Furthermore, he hoped that superpositions of waves could create a ‘wave packet’ that kept itself together – like a traveling wave in a pond with a stable, sharply defined peak – which would explain what was happening when this ψ field acted like a single particle.

  Schrödinger soon di
scovered that the straightforward ‘wave-packet’ intuition of this kind would not be possible. In the fourth and final part of his ‘Quantization’ paper series, he wrote, ‘the ψ-function itself cannot and may not be interpreted directly in terms of three-dimensional space.’ It was a wave only in a strange formal or ‘configuration space.’ Still, Schrödinger’s approach did what he wanted: it talked about the atomic world in terms of our world – space, time, waves, and so forth using equations with which physicists were familiar and knew how to use with relative ease. The picture he provided was intuitive, more or less. And interpreting the equation – relating it to our more commonsense notions of the world – was important, if only to guard against views that would give up on rational calculation by asserting that God or something supernatural were pushing the particle around.

  But others soon wrested Schrödinger’s intuitive interpretation away from him.

  Interpreting the Waves

  In the summer of 1926, Göttingen physicist Max Born – who was Heisenberg’s supervisor, and a developer of matrix mechanics – published his work on atomic collisions, as between electrons and atoms. Collisions, after all, were the central focus of classical physics, and Born regarded it as one of the key issues in understanding the atomic realm. He had struggled with the matrix approach, in vain, and had reached a startling conclusion. ‘[O]nly Schrödinger’s formalism proved itself appropriate for this purpose’, Born declared; ‘for this reason I am inclined to regard it as the most profound formulation of the quantum laws.’21 But he also had unwelcome news for Schrödinger. Born could not make any sense out of the claim that the ψ-function referred to the electron’s charge density. Schrödinger’s equation, Born concluded, does not tell us information about the state of an event, but rather about the probability of a state. The ψ function that Schrödinger’s equation described as continuously moving through space, interfering and interacting with potentials, was not some substantial field but probabilities. ‘We free forces of their classical duty of determining directly the motion of particles and allow them instead to determine the probability of states.’22 Born thus fashioned a strange hybrid with elements of both wave and matrix mechanics: it incorporated both continuity and causality on the one hand, and discontinuity and probability on the other. ‘The motion of particles conforms to the laws of probability, but the probability itself is propagated in accordance with the law of causality.’23

  A few months later, another interpretive step was taken by Wolf-gang Pauli, another former assistant of Born’s. While Born had interpreted ψ as about the probabilities of states, Pauli now said it was about the probability of particles – that ψ2 represented the probability of an electron being at a particular position. This was still further from Schrödinger’s interpretation of his function, for it completely stripped the ψ function of reality. The ψ function was about the possibility, not the actuality, of something – of the click of a counter, of the presence of a particle. The actuality had to be brought about by setting up equipment and having some subatomic performance be enacted – an interaction between whatever it is described by the wave function and the world.

  The Born-Pauli interpretation, a marriage of particle and wave theory, quickly became the interpretation adopted by most physicists. But the marriage had a strange cost, for features of each were lost. When Newton’s laws governed particles, the particles were observable and the laws were deterministic – one plugged in the initial states of the particles and turned the crank to get predictions. The same was true for Maxwell’s laws governing waves: waves were observable things, with fully measurable properties, and Maxwell’s laws were deterministic, describing how they governed over time. Both particle and wave theory, that is, were about predictables and observables. The Born-Pauli interpretation now married particle and wave theory together – but a part of each was destroyed. The Schrödinger wave waves in configuration space. The particles were observable but lost their predictability; the waves were predictable but lost their observability. Observing the position and momentum of something does not allow you to make predictions about where you will see it next. The wave is used to predict the probability of another event, but after the event is observed, the wave has no more value and has to be discarded or ‘reset’, modified to incorporate new information.

  Nowadays, this interpretation is often presented in a misleading way. Instead of saying that the wave function is discarded or reset when a measurement is made, one often hears that the wave function ‘collapses.’ The imagery captures the idea that, before the event happened – before you detected the particle, say – it could be anywhere, so one thinks that the event or particle is everywhere. The image this conjures up is of a structure, extended through space, suddenly getting sucked up at a single point. It’s a vivid but deceptive image. The wave is just a probability, not a ‘thing.’ (It is the one merit of what is called the pilot wave idea that nothing collapses; the wave merely deposits the particle already in it.) The wave function, whose purpose is only to give probabilities, has flowed along – predictably, deterministically – but once an event happens the function has exhausted its purpose, and must be reset.

  Refashioning John Wheeler’s famous remark about Einstein’s theory of general relativity, Stony Brook physicist Alfred S. Gold-haber says the following about Schrödinger’s equation: ‘The wave tells the particle where to go, the particle tells the wave where to start and stop.’

  It is a great irony that this approach, devised to be intuitive, often suffers thanks to a false picture.

  The Schrödinger equation implied a radical change in the events on the world-stage. No longer could it be assumed that, eventually, you could plug in numbers, turn the crank, and get predictions. Instead, you plugged in numbers, turned the crank, and got…probabilities. You got the likelihood of an event occurring in a particular place. And nothing could help you more than that: not more information, not adding more pieces to the machinery. No flip book could be constructed, for if you replayed an event over and over, a particle would wind up in different spots on the page, its location a question of averages.

  Schrödinger himself never warmed to this interpretation, calling it a ‘resignation.’24 It is ‘convenient’, he wrote, but we cannot allow ourselves to ‘get off so easily.’ We should keep trying to work out the interpretation, the causal mechanism, he insisted. He pointed to some seeming implications that he found ‘quite ridiculous’, including the image of the now-famous cat locked in a box along with a diabolical device that kills the cat upon the decay of a radioactive nucleus. Since the decay is defined by ψ, it appears that whether the cat is alive or dead is so defined as well, seemingly leading to the conclusion that the cat’s existence, too, is superimposed, and is half-alive and half-dead. While the conclusion is false, the image is a brilliant demonstration of the flaws of extending the theories of the microworld to those of the macroworld.

  But while Schrödinger’s wave methods are used by all the workers in the field, his premises about the wave structure of reality have been ignored; the interpretation worked out by Heisenberg’s Göttingen allies came to be the favored one. ‘Schrödinger’s methods proved indispensable, writes historian Mara Beller. ‘His philosophy did not.’25

  Born’s interpretation of Schrödinger’s equation changes the notion of what a complete theory of the world consists of. Conventionally, we expect a complete theory to tell us something about reality. Most theories that physicists teach and use fall short by deliberately cutting corners. These theories provide not a complete picture, but a model that is an idealized version of any real situation. For example, the ideal-gas law ignores well understood things like van der Waals forces and hard-core repulsion, but we do not mind because talking about an ideal, rather than a real, object buys us something: a tremendous ease of application. This is what one of my colleagues calls a ‘harmlessly fudging theory.’ Such theories are harmless because these limitations do not threaten normal assumptions about th
e world. We are cutting corners, know that we are cutting corners, and know that the absence of the corners we are cutting does not affect our impression of the world.

  Yet Schrödinger’s equation, as interpreted by Born, is different. It makes us aware that how we interact with the world affects what we bring into it. It makes us explicitly aware, that is, of ourselves as interacting with the world in doing something like taking a measurement. The idea that we are agents comes explicitly into view. We are not watching something happen on a stage and then measuring it, we are making the stage even as we do so. The prevailing interpretation of Schrödinger’s equation brings this to the fore in a way classical physics does not. It is our own expectations that determine whether we find this a sacrifice or an advance. The giant leap forward of quantum mechanics does require a substantial rethinking of what it means to ‘understand’ nature, and how to characterize ‘reality’, in a way that many scientists at the time experienced as what Heisenberg called a sacrifice – so painful a one, in fact, that they struggled mightily against having to make it. Many still do.

  Interlude

  THE DOUBLE CONSCIOUSNESS OF SCIENTISTS

  …a world which yields him no true self-consciousness, but only lets him see himself through the revelation of the other world. It is a peculiar sensation, this double-consciousness, this sense of always looking at one’s self through the eyes of others, of measuring one’s soul by the tape of a world that looks on in amused contempt and pity.