A Brief Guide to the Great Equations (35 page)

The paper associated physical quantities with tables whose rows and columns were labeled with the ‘allowed’ quantum states that Bohr had postulated in his groundbreaking paper on the hydrogen spectrum. This had been done before (for example, Einstein’s
A
and
B
coefficients are ‘tables’ labeled with two states), but Heisenberg applied the idea to a more fundamental set of quantities and went a step further and found a rule for ‘multiplying’ two such tables to make formulas similar to those used in classical mechanics. This was new, and it opened the way to do quantum calculations far beyond the limited capacity of the previous attempts, such as Born’s, at a ‘quantum mechanics.’

Heisenberg then hit a snag. The tables and the multiplication rule he invented for them obeyed a new kind of algebra that mathematicians had discovered long before, but was unfamiliar to most physicists, himself included. Most strikingly, the rule did not follow the ‘commutative law’, the mathematical principle according to which the order in which one multiplies two numbers does not affect the result:
ab
=
ba
. When Heisenberg used his new calculus to multiply one quantum-theoretical table (let’s call it
A
) by another (
B
), the result depended on the order:
AB
≠
BA
. The feature ‘was very disagreeable to me’, he said later, and try as he might he could not rid his theory of it.
¹²
‘I felt this was the only point of difficulty in the whole scheme, otherwise I would be perfectly happy.’ Heisenberg then did what many people do when a nuisance threatens to spoil
a promising new invention: he swept it under the rug. He waved at it in a single sentence – ‘Whereas in classical theory [
AB
] is always equal to [
BA
], this is not necessarily the case in quantum theory’ – mentioned circumstances in which the difficulty does not arise, and then dropped the subject. Heisenberg concluded his paper with a disclaimer of the sort that is often seen in early papers in a field, wondering whether it was ‘satisfactory’ or ‘still too crude an approach’ to quantum mechanics. The answer, he declared, would have to await ‘a penetrating mathematical investigation.’
¹³

After he finished, around July 9, Heisenberg gave Born a copy of the paper, asking his supervisor to see whether it was worth publishing, and to see whether he could investigate the basic idea, which Heisenberg knew seemed awkward and even bizarre. Born so promised – but set the paper aside for a few days, exhausted after a semester teaching and from research he was doing with his other assistant, Pascual Jordan.

Born read the paper only after Heisenberg’s departure. Impressed, he sent it on to the Z
eitschrift für Physik
, and on July 15 wrote to Einstein that Heisenberg’s work appeared ‘very mysterious, but is surely correct and profound.’
¹⁴
But Born also had a nagging feeling about Heisenberg’s tables and the strange mathematical rules used to multiply them. It looked so familiar! After a restless week in which he could hardly sleep, it hit Born that he had seen this peculiar structure in his high school maths classes. His intrepid young assistant had reinvented the wheel. The tables were what mathematicians called matrices, arrangements of numbers (or variables) in rows and columns – though Heisenberg’s tables had
infinite
numbers of elements. And Heisenberg’s funny quantum-mechanical relations were actually the most natural way that mathematicians had discovered to ‘multiply’ matrices.

Born was overjoyed. The mathematics of matrices gave him a framework in which to investigate and systematize Heisenberg’s work. He knew that matrices can be noncommutative – the order in which one multiplied them mattered. This explained Heisenberg’s
embarrassing difficulty that, for instance, the matrix
p
associated with momentum and
q
with position did not commute; the matrix
pq
was not the same as
qp
(by convention, physicists often indicate matrices with bold symbols). But there was more. This pair of variables – known as canonically conjugate variables – was not commutative, but in a special way. Though Born could not prove it, the difference between
pq
and
qp
seemed to be a specific matrix proportional to Planck’s constant:
pq – qp
=
I
h
/2π
i
, where
I
is the unit matrix – ‘ones’ along the diagonal entries and zeros everywhere else. Born wrote later, ‘I was as excited by this result as a sailor would be who, after a long voyage, sees from afar, the longed-for land, and I felt regret that Heisenberg was not there.’
¹⁵

A few days later, on July 19, Born ran into Pauli on a train, excitedly explained how Heisenberg’s paper could be translated into matrix language, and asked his former assistant if he wanted to collaborate in investigating the topic. Pauli was dismissive, and sarcastically accused Born of trying to ‘spoil Heisenberg’s physical ideas’ with ‘futile mathematics’ and ‘tedious and complicated formalism.’ (Historians find this remark humorous, for Heisenberg’s ideas in this case were formal and even more tedious than conventional matrix analysis.) The next day, July 20, Born approached Jordan, who happened to be knowledgeable in the mathematics of matrices. Within a few days the two were able to show how to derive the relation
pq – qp
=
I
h
/2π
i
from Heisenberg’s work, and again Born was awestruck: ‘I shall never forget the thrill I experienced when I succeeded in condensing Heisenberg’s ideas on quantum conditions in the mysterious equation
pq – qp
=
I
h
/2π
i
, which is the centre of the new mechanics and was later found to imply the uncertainty relations.’
¹⁶

By the end of September they sent off a paper, ‘On Quantum Mechanics.’ It carried out the ‘penetrating mathematical investigation’ that Heisenberg had hoped for, and was the first formulation of what became known as matrix mechanics. The maths was unfamiliar – many physicists had to bone up on matrices to understand
the paper – and its methods were unwieldy, but it worked on the limited number of problems for which calculations could be carried through to completion. The authors sent a copy to Heisenberg, who by then had left Cambridge and was in Copenhagen. He showed the paper to Bohr, saying, ‘Here, I got a paper from Born, which I cannot understand at all. It is full of matrices, and I hardly know what they are.’
¹⁷
But after Heisenberg brushed up on matrices, he too, shared their excitement, and on September 18 wrote to Pauli that Born’s bright idea,
pq – qp
=
I
h
/2π
i
, was the foundation of the new mechanics. Heisenberg, Born, and Jordan began a feverish discussion by letter, and Heisenberg interrupted his stay in Copenhagen and returned to Göttingen so that the three of them could finish work on another paper generalizing the results of the Born-Jordan paper before Born departed for a long-scheduled trip to the U.S. in October.
¹⁸

The result was a paper written by Born, Heisenberg, and Jordan entitled ‘On Quantum Mechanics II’, known to historians of physics as ‘the three-man paper.’ Its central feature is what they called the ‘fundamental quantum-mechanical relation’, the strange equation
pq – qp
=
I
h
/2π
i
. The paper is a landmark in the history of physics, for it is the first map of the quantum domain. Around the same time, Pauli published a paper in which he had successfully – though with considerable difficulty – applied matrix mechanics to the test case of the hydrogen atom.

Yet few besides its creators fully recognized the importance of matrix mechanics, for the appreciation of its value was hampered by several obstacles. One was its complexity: while matrix mathematics was not intrinsically that difficult, Heisenberg’s application of it appeared to be horrendously complicated, and most physicists had to take matrix mechanics on faith while they struggled to master it. Typical was the reaction of George Uhlenbeck, then a student at the University of Leiden, who remarked much later, ‘Everything became these infinite numbers of equations that you had then to solve, and so nobody knew exactly how to do it.’
¹⁹
Others were put off by the
unanschaulichkeit
– by the fact that matrix mechanics deliberately refrained from providing a picture of the mechanics of the atomic domain, and that its fundamental terms, the matrices, were strictly speaking meaningless, just formal symbolic artifacts.
²⁰
Still others were bothered by the failure to explain the transition from the microworld to the macroworld – between the unvisualizable world without space and time to the familiar space-time container that we and our imaginations live in. Many scientists therefore took, as historian Mara Beller once wrote, a ‘wait and see’ attitude, and even its originators viewed it as but a first and imperfect step toward an adequate theory.
²¹

But shortly after the three-man paper appeared in February 1926, its authors had unwelcome company.

Schrödinger’s first and second papers on wave mechanics appeared in the
Annalen der Physik
in March and April 1926. Wave mechanics mapped the same terrain as matrix mechanics, but physicists found the map much easier to read. It lacked the obstacles of matrix mechanics. First, the maths was part of the bread-and-butter training of classical physicists, who had been using and solving wave equations since their high school days. Second, wave equations were visualizable. Physicists saw water, sound, and light waves, and their properties – frequency, amplitude, and wavelength – smoothly and continuously propagating around them daily. They had trained themselves to see other wave properties such as nodes and interference. There was the small matter of the ψ-function, which existed in multidimensional ‘configuration space’ – three dimensions for each particle in the system – but even that seemed somewhat visualizable, as something that traveled through space or stayed ‘perched’ in a standing-wave-like way when bound inside an atom. Third, wave mechanics provided a natural way of describing the transition from the microworld to the macroworld, as Schrödinger’s third paper that
year showed, as particle-like groupings or packets of waves moved along the classical paths, which were the rays perpendicular to the phase fronts of the ψ-function.
²²

Small wonder most physicists took to wave mechanics. Planck was awestruck, Einstein ecstatic. U.S. physicist Karl Darrow reported that wave mechanics ‘captivated the world of physics’ as it promised ‘a fulfillment of that long-baffled and insuppressible desire’ to return to classical physics, its comfortable and continuously propagating functions.
²³
A flood of papers used Schrödinger’s approach to tackle atomic issues. Allies of the Göttingen physicists reacted badly: Heisenberg called it ‘too good to be true’, Dirac reacted with ‘hostility’, Pauli called it ‘crazy.’
²⁴
But many of them soon fell under its spell. Pauli, who had just laboriously worked out the theory of the hydrogen spectrum using matrix mechanics, found wave mechanics much easier to use for the same purpose. Born wrote to Schrödinger that he grew so excited upon reading the first wave mechanics paper that he wanted ‘to defect…to continuum physics…[to] the crisp, clear conceptual foundations of classical physics’,
²⁵
though his ardor soon cooled.

At first the conflict played out in arguments about the scientific merit of the two approaches: Which approach did the better job? The hydrogen atom – which Pauli solved with both methods – was a first key test case. It was the drosophila fly or lab rat of atomic physicists, the problem that any model had to tackle first – for the hydrogen atom had been successfully analysed with the old quantum theory, in close agreement with experiment, meaning that formula could be compared. Another test case was to account for the transition between the quantum and the classical world, or how to get from its rungs to ours. Schrödinger had shown that wave mechanics had an answer to this, but none was apparent yet for matrix mechanics. Yet another key problem was how to handle collisions between things in the atomic world, which would require showing how a system evolved over time.

The answer to the question of which approach had more
scientific merit was soon resolved: In May, in his fourth paper of 1926, Schrödinger proved that, mathematically speaking, the two approaches were identical.
²⁶
Pauli reached the same conclusion. It was not yet clear how to handle all the test cases, but the demonstration of mathematical equivalence showed that neither approach had more or less mathematical merit than the other. Scientifically, though, wave mechanics could do more than matrix mechanics, for it was essential for analyzing the continuous part of the spectrum.

As Heisenberg’s biographer David Cassidy notes, however, this conclusion only restructured the conflict so it could begin in earnest. With the issue of mathematical equivalence settled, the partisans now were liberated to argue about the physical interpretations of the theories. These were dramatically opposed: wave mechanics – at least as interpreted somewhat hopefully by Schrödinger – portrayed the atomic world as woven out of continuous processes that are causally responsible for what seem to be discontinuous events, and as unfolding in space and time, while matrix mechanics portrayed the atomic realm as lacking continuous processes and causal relations, as not related to space and time, and as not being a world at all in any way humans can imagine. This conflict was less decidable and more emotional than the one about scientific merit, for it reflected the adversaries’ sense of what physics was all about, of what the world was, and of the most fundamental relationship between human beings and the world.