by Manjit Kumar
Pauli may have been close to the Göttingen physicists, but he recognised the significance of what Schrödinger had done and was deeply impressed. Pauli had strained every ounce of grey matter he possessed as he successfully applied matrix mechanics to the hydrogen atom. Everyone was later amazed by the speed and virtuosity with which he had done so. Pauli sent his paper to the Zeitschrift für Physik on 17 January, only ten days before Schrödinger posted his first paper. When he saw the relative ease with which wave mechanics allowed Schrödinger to tackle the hydrogen atom, Pauli was astonished. ‘I believe that the work counts among the most significant recently written’, he told Pascual Jordan. ‘Read it carefully and with devotion.’27 Not long afterwards, in June, Born described wave mechanics ‘as the deepest form of the quantum laws’.28
Heisenberg was ‘not very pleased’, he told Jordan, by Born’s apparent defection to wave mechanics.29 Although he acknowledged that Schrödinger’s paper was ‘incredibly interesting’ with its use of more familiar mathematics, Heisenberg firmly believed that when it came to physics, his matrix mechanics was a better description of the way things were at the atomic level.30 ‘Heisenberg from the very beginning did not share my opinion that your wave mechanics is physically more significant than our quantum mechanics’, Born confided to Schrödinger in May 1927.31 By then it was hardly a secret. Nor did Heisenberg want it to be. There was too much at stake.
As spring had given way to summer in 1925 there was still no quantum mechanics, a theory that would do for atomic physics what Newtonian mechanics did for classical physics. A year later there were two competing theories that were as different as particles and waves. They both gave identical answers when applied to the same problems. What, if any, was the connection between matrix and wave mechanics? It was a question that Schrödinger began to ponder almost as soon as he finished his first ground-breaking paper. After two weeks of searching he found no link. ‘Consequently,’ Schrödinger wrote to Wilhelm Wien, ‘I have given up looking any further myself.’32 He was hardly disappointed, as he confessed that ‘matrix calculus was already unbearable to me long before I even distantly thought of my theory’.33 But he was unable to stop digging until he unearthed the connection at the beginning of March.
The two theories that appeared to be so different in form and content, one employing wave equations and the other matrix algebra, one describing waves and the other particles, were mathematically equivalent.34 No wonder they both gave exactly the same answers. The advantages of having two different but equivalent formalisms of quantum mechanics quickly became apparent. For most problems physicists encountered, Schrödinger’s wave mechanics provided the easiest route to the solution. Yet for others, such as those involving spin, it was Heisenberg’s matrix approach that proved its worth.
With any possible arguments about which of the two theories was correct smothered even before they could begin, attention turned from the mathematical formalism to the physical interpretation. The two theories might technically be equivalent, but the nature of physical reality that lay beyond the mathematics was altogether different: Schrödinger’s waves and continuity versus Heisenberg’s particles and discontinuity. Each man was convinced that his theory captured the true nature of physical reality. Both could not be right.
At the beginning there was no personal animosity between Schrödinger and Heisenberg as they began to question each other’s interpretation of quantum mechanics. But soon emotions began to run high. In public and in their papers both managed, on the whole, to rein in their true feelings. In their letters, however, there was no need for tact and restraint. When he initially tried but failed to prove the equivalence of wave and matrix mechanics, Schrödinger was somewhat relieved that there might be none, since ‘the mere thought makes me shudder, if I later had to present the matrix calculus to a young student as describing the true nature of the atom’.35 In his paper, ‘On the Relation Between Heisenberg-Born-Jordan quantum Mechanics and My Own’, Schrödinger was at pains to distance wave mechanics from matrix mechanics. ‘My theory was inspired by L. de Broglie and by brief but infinitely far-seeing remarks of A. Einstein’, he explained. ‘I was absolutely unaware of any genetic relationship with Heisenberg.’36 Schrödinger concluded that, ‘because of the lack of visualization’ in matrix mechanics, ‘I felt deterred by it, if not to say repelled’.37
Heisenberg was even less diplomatic about the continuity that Schrödinger was trying to restore to the atomic realm where, as far as he was concerned, discontinuity ruled. ‘The more I think about the physical portion of the Schrödinger theory, the more repulsive I find it’, he told Pauli in June.38 ‘What Schrödinger writes about the visualizability of his theory “is probably not quite right”, in other words it’s crap.’ Two months earlier, Heisenberg had appeared more conciliatory when he described wave mechanics as ‘incredibly interesting’.39 But those who knew Bohr recognised that Heisenberg was employing exactly the sort of language favoured by the Dane, who always called an idea or an argument ‘interesting’ when in fact he disagreed with it. Increasingly frustrated as more of his colleagues abandoned matrix mechanics for the easier-to-use wave mechanics, Heisenberg finally snapped. He could hardly believe it when Born, of all people, started using Schrödinger’s wave equation. In a fit of anger, Heisenberg called him a ‘traitor’.
He may have been envious of the growing popularity of Schrödinger’s alternative, but after its discovery it was Heisenberg who was responsible for the next great triumph of wave mechanics. He might have been annoyed at Born, but Heisenberg had also been seduced by the mathematical ease with which Schrödinger’s approach could be applied to atomic problems. In July 1926 he used wave mechanics to account for the line spectra of helium.40 Just in case anyone read too much into his adoption of the rival formulation, Heisenberg pointed out that it was nothing more than expediency. The fact that the two theories were mathematically equivalent meant he could use wave mechanics while ignoring the ‘intuitive pictures’ Schrödinger painted with it. However, even before Heisenberg posted his paper, Born had used Schrödinger’s palette to paint an entirely different picture on the same canvas when he discovered that probability lay at the heart of wave mechanics and quantum reality.
Schrödinger was not trying to paint a new picture, but attempting to restore an old one. For him there were no quantum jumps between different energy levels in an atom, but only smooth, continuous transitions from one standing wave into another, with the emission of radiation being the product of some exotic resonance phenomenon. He believed that wave mechanics allowed the restoration of a classical, ‘intuitive’ picture of physical reality, one of continuity, causality and determinism. Born disagreed. ‘Schrödinger’s achievement reduces itself to something purely mathematical,’ he told Einstein, ‘his physics is wretched.’41 Born used wave mechanics to paint a surreal picture of a reality with discontinuity, acausality and probability, instead of Schrödinger’s attempt at a Newtonian-inspired old master. These two pictures of reality hang on different interpretations of the so-called wave function, symbolised by the Greek letter psi, , in Schrödinger’s wave equation.
Schrödinger had known from the very beginning that there was a problem with his version of quantum mechanics. According to Newton’s laws of motion, if the position of an electron is known at a certain time together with its velocity, then it is theoretically possible to determine exactly where it will be at some later time. However, waves are much more difficult to pin down than a particle. Dropping a stone into a pond sends ripples of waves across its surface. Exactly where is the wave? Unlike a particle, a wave is not localised at a single place, but is a disturbance that carries energy through a medium. Like people taking part in a ‘Mexican wave’, a water wave is just individual water molecules bobbing up and down.
All waves, whatever their size and shape, can be described by an equation that mathematically maps their motion, just as Newton’s equations do for a particle. The wave function, , represents
the wave itself and describes its shape at a given time. The wave function of a wave rippling across the surface of a pond specifies the size of the disturbance, the so-called amplitude, of the water at any point x at time t. When Schrödinger discovered the wave equation for de Broglie’s matter waves, the wave function was the unknown part. Solving the equation for a particular physical situation, such as the hydrogen atom, would yield the wave function. However, there was a question that Schrödinger was finding difficult to answer: what was doing the waving?
In the case of water or sound waves, it was obvious: water or air molecules. Light had perplexed physicists in the nineteenth century. They had been forced to invoke the mysterious ‘ether’ as the necessary medium through which light travelled, until it was discovered that light was an electromagnetic wave with interlocked electric and magnetic fields doing the waving. Schrödinger believed that matter waves were as real as any of these more familiar types of waves. However, what was the medium through which an electron wave travelled? The question was akin to asking what does the wave function in Schrödinger’s wave equation represent? In the summer of 1926 a witty little ditty summed up the situation that confronted Schrödinger and his colleagues:
Erwin with his psi can do
Calculations quite a few.
But one thing has not been seen:
Just what does psi really mean?42
Schrödinger finally proposed that the wave function of an electron, for example, was intimately connected to the cloud-like distribution of its electric charge as it travelled through space. In wave mechanics the wave function was not a quantity that could be directly measured because it was what mathematicians call a complex number. 4+3i is one example of such a number, and it consists of two parts: one ‘real’ and the other ‘imaginary’. 4 is an ordinary number and is the ‘real’ part of the complex number 4+3i. The ‘imaginary’ part, 3i, has no physical meaning because i is the square root of –1. The square root of a number is just another number that multiplied by itself will give the original number. The square root of 4 is 2 since 2×2 equals 4. There is no number that multiplied by itself equals –1. While 1×1=1, –1×–1 is also equal to 1, since by the laws of algebra, a minus times a minus generates a plus.
The wave function was unobservable; it was something intangible that could not be measured. However, the square of a complex number gives a real number that is associated with something that can actually be measured in the laboratory.43 The square of 4+3i is 25.44 Schrödinger believed that the square of the wave function of an electron, , was a measure of the smeared-out density of electric charge at location x at time t.
As part of his interpretation of the wave function, Schrödinger introduced the concept of a ‘wave packet’ to represent the electron as he challenged the very idea that particles existed. He argued that an electron only ‘appeared’ to be particle-like but was not actually a particle, despite the overwhelming experimental evidence in favour of it being so. Schrödinger believed that a particle-like electron was an illusion. In reality there were only waves. Any manifestation of a particle electron was due to a group of matter waves being superimposed into a wave packet. An electron in motion would then be nothing more than a wave packet that moved like a pulse sent, with a flick of the wrist, travelling down the length of a taut rope tied at one end and held at the other. A wave packet that gave the appearance of a particle required a collection of waves of different wavelengths that interfered with one another in such a way that they cancelled each other out beyond the wave packet.
If giving up particles and reducing everything to waves rid physics of discontinuity and quantum jumps, then for Schrödinger it was a price worth paying. However, his interpretation soon ran into difficulties as it failed to make physical sense. Firstly, the wave packet representation of the electron began to unravel when it was discovered that the constituent waves would spread out across space to such a degree that they would have to travel faster than the speed of light if they were to be connected with the detection of a particle-like electron in an experiment.
Figure 11: A wave packet formed from the superposition of a group of waves
Try as he might, there was no way for Schrödinger to prevent this dispersal of the wave packet. Since it was made up of waves that varied in wavelength and frequency, as the wave packet travelled through space it would soon begin to spread out as individual waves moved at different velocities. An almost instantaneous coming together, a localisation at one point in space, would have to take place every time an electron was detected as a particle. Secondly, when attempts were made to apply the wave equation to helium and other atoms, Schrödinger’s vision of the reality that lay beneath his mathematics disappeared into an abstract, multi-dimensional space that was impossible to visualise.
The wave function of an electron encodes everything there is to know about its single three-dimensional wave. Yet the wave function for the two electrons of the helium atom could not be interpreted as two three-dimensional waves existing in ordinary three-dimensional space. Instead the mathematics pointed to a single wave inhabiting a strange six-dimensional space. In each move across the periodic table from one element to the next, the number of electrons increased by one and an additional three dimensions were required. If lithium, third in the table, required a nine-dimensional space, then uranium had to be accommodated in a space with 276 dimensions. The waves that occupied these abstract multi-dimensional spaces could not be the real, physical waves that Schrödinger hoped would restore continuity and eliminate the quantum jump.
Nor could Schrödinger’s interpretation account for the photoelectric and Compton effects. There were unanswered questions: how could a wave packet possess electric charge? Could wave mechanics incorporate quantum spin? If Schrödinger’s wave function did not represent real waves in everyday three-dimensional space, then what were they? It was Max Born who provided the answer.
Born was nearing the end of his five-month stay in America when Schrödinger’s first paper on wave mechanics appeared in March 1926. Reading it on his return to Göttingen in April, he was taken completely ‘by surprise’ as others had been.45 The terrain of quantum physics had dramatically changed during his absence. Almost out of nowhere, Born immediately recognised, Schrödinger had constructed a theory of ‘fascinating power and elegance’.46 He was quick to acknowledge the ‘superiority of wave mechanics as a mathematical tool’, as demonstrated by the relative ease with which it solved ‘the fundamental atomic problem’ – the hydrogen atom.47 After all, it had taken someone of Pauli’s prodigious talent to apply matrix mechanics to the hydrogen atom. Born might have been taken by surprise but he was already familiar with the idea of matter waves long before Schrödinger’s paper was published.
‘A letter from Einstein directed my attention to de Broglie’s thesis shortly after its publication, but I was too much involved in our speculations to study it carefully’, Born admitted more than half a century later.48 By July 1925 he had made time to study de Broglie’s work and wrote to Einstein that ‘the wave theory of matter could be of very great importance’.49 Enthused, he had already begun ‘speculating a little about de Broglie’s waves’, Born told Einstein.50 But just then he shoved de Broglie’s ideas aside to make sense of the strange multiplication rule in a paper given to him by Heisenberg. Now, almost a year later, Born solved some of the problems encountered by wave mechanics, but at a price far higher than Schrödinger demanded with his sacrifice of particles.
The rejection of particles and quantum jumps that Schrödinger advocated was too much for Born. He witnessed regularly in Göttingen what he called ‘the fertility of the particle concept’ in experiments on atomic collisions.51 Born accepted the richness of Schrödinger’s formalism but rejected the Austrian’s interpretation. ‘It is necessary,’ Born wrote late in 1926, ‘to drop completely the physical pictures of Schrödinger which aim at a revitalization of the classical continuum theory, to retain only the formalism and to fill that
with a new physical content.’52 Already convinced ‘that particles could not simply be abolished’, Born found a way to weave them together with waves using probability as he came up with a new interpretation of the wave function.53
Born had been working on applying matrix mechanics to atomic collisions while in America. Back in Germany with Schrödinger’s wave mechanics suddenly at his disposal, he returned to the subject and produced two seminal papers bearing the same title, ‘quantum mechanics of collision phenomena’. The first, only four pages long, was published on 10 July in Zeitschrift für Physik. Ten days later the second paper, more polished and refined than the first, was finished and in the post.54 While Schrödinger renounced the existence of particles, Born in his attempt to save them put forward an interpretation of the wave function that challenged a fundamental tenet of physics – determinism.
The Newtonian universe is purely deterministic with no room for chance. In it, a particle has a definite momentum and position at any given time. The forces that act on the particle determine the way its momentum and position vary in time. The only way that physicists such as James Clerk Maxwell and Ludwig Boltzmann could account for the properties of a gas that consists of many such particles was to use probability and settle for a statistical description. The forced retreat into a statistical analysis was due to the difficulties in tracking the motion of such an enormous number of particles. Probability was a consequence of human ignorance in a deterministic universe where everything unfolded according to the laws of nature. If the present state of any system and the forces acting upon it are known, then what happens to it in the future is already determined. In classical physics, determinism is bound by an umbilical cord to causality – the notion that every effect has a cause.