Einstein and the Quantum (41 page)

His final ideal gas paper, submitted December 15, just before he left for his Christmas holiday, was a last effort to digest Einstein's new quantum statistics; it is titled simply “On Einstein's Gas Theory.” He sets out to derive the same answers as Einstein without accepting the
strange state counting of Bose, which he thinks requires too great a “sacrificium intellectus.” The way out, he says, is “
nothing else but taking seriously
the De Broglie–Einstein wave theory of the moving particles, according to which the particles are nothing more than a kind of ‘white crest' on a background of wave radiation.” He then formulates the mathematical problem differently from Einstein but shows how to reach exactly the same final equations. From the perspective of a modern physicist
¹⁰
the two calculations are equivalent and have the same meaning, but Schrödinger felt that by interpreting the fundamental objects as waves instead of as particles the weirdness of indistinguishability was somehow made more palatable. He concludes by speaking of the particles as “signals” or “singularities” embedded in the wave, highly reminiscent of Einstein's failed ideas of 1909–1910 and his later idea of “ghost fields” guiding the particles developed in the early 1920s. Except that Schrödinger clearly now thinks that the particles are the “ghosts,” the ephemera, since by viewing waves as fundamental he has “explained” Bose-Einstein statistics in a natural way. Later he would say, “
wave mechanics was born
in statistics.”

A few days before Christmas 1925 Schrödinger set off to a familiar mountain lodge in the Swiss village of Arosa, determined to find a new equation to describe these matter waves. Although he took his skis (he was an expert alpinist) and was accompanied by an unnamed “old girlfriend,”
¹¹
it seems that this trip was really focused on wave equations. De Broglie, despite Einstein's praise, had not produced a new governing equation, similar to Maxwell's electromagnetic wave equation, that could predict or explain the remaining mysteries of the atom. He had produced some suggestive mathematical relations, which made contact with the old quantum theory of Bohr and Sommerfeld, but only at the most elementary level. The central mystery of quantum
theory,
quantization
, was not really resolved by de Broglie's work. Why wasn't nature continuous? Why are only certain energies allowed for electrons bound to atomic nuclei?

Schrödinger saw an answer. Classical waves, or vibrations, in a confined medium have certain natural constraints on their properties. Consider a violin string of a certain length,
L
, clamped down at each end. The notes it can play arise from vibrations of the strings; these vibrations are waves of sideways displacement in the string, but they cannot have a continuously varying wavelength as is possible for waves in an open (essentially infinite) medium. The longest wavelength,
λ
, they can have is twice the string length. Why is this the longest? First, consider that, for any wave on a string, the string's displacement must be zero at the points where it is clamped down. The simplest form of displacement the string can have away from these fixed points is for it to be displaced everywhere in the same sideways direction (left or right) at a given instant, so that the maximum displacement is in the middle and it decreases back to zero at both clamped ends. This displacement then oscillates back and forth, causing sound waves of a certain pitch. Since we measure wavelength by the distance between points in the medium that take us through a peak
and
a trough, this shape corresponds to half the full wavelength, so the wavelength is
λ
= 2
L
. This will determine the lowest note the violin can play (for a given string tension). The next-lowest note will have
λ
=
L
, implying that there will be no displacement at the center of the string (even though that point isn't clamped down). In general the only allowed wavelengths are
λ
= 2
L
/
n
, where
n
is a whole number (
n
= 1, 2, 3 …). That's the point: for confined waves, nature produces whole numbers automatically. De Broglie had hinted at this, but now Schrödinger realized that this was the key to getting the quantum into quantum theory.

The mathematics of the old quantum theory had not done this in a natural way. Bohr and his followers had taken mathematical expressions that are continuous (i.e., don't involve whole numbers exclusively) and had simply restricted them to whole numbers by fiat. Schrödinger, by contrast, was looking for an equation that simply
did
not have
continuous solutions, one in which each solution would be connected to a whole number organically.

As noted above, wave equations, through the quantization of the wavelength, have this property. They are differential equations, describing continuous change in space and time; but when the waves are confined, only certain wavelengths are physically possible. So Schrödinger was looking for a matter-wave differential equation to describe electrons: it would have to describe a matter field and an extended “disturbance” of the field that varied in space and time but was confined to the vicinity of the atomic nucleus by the electron's attraction to the nucleus.

Maxwell's electromagnetic wave equation was the only fundamental wave equation of physics at that time, but because photons are massless, it had no safe place for the introduction of Planck's constant (as Einstein had learned to his dismay fifteen years earlier). Schrödinger's challenge was to fashion a wave equation, modeled on Maxwell's, that included Planck's constant as well as the physical constants
e
and
m
, representing the charge and mass of the electron. Maxwell's wave equation does, however, contain the wavelength,
λ
, of the EM waves which then lead to quantized values and whole numbers, when it is written for “standing waves” confined to a specific region. So the key idea was to write an equation similar to the electromagnetic wave equation but for a new wave field, a “matter wave” described by a mathematical expression now known as the “wavefunction,” and to replace
λ
using de Broglie's relation
λ
=
h
/
p
=
h
/
m
v.

With this approach Schrödinger had an equation containing
h
, Planck's constant, and
m
, the electron mass. The electron velocity,
v
, can be eliminated from the equation in favor of the difference between its total energy and potential energy. The potential energy of the electron orbiting the nucleus of course depends on its charge,
e
. The resulting “time-independent Schrödinger equation” for hydrogen contains the “holy trinity,”
h
,
m
, and
e
. And now the moment of triumph: since only certain wavelengths are allowed, that implies that the only unknown in the equation, the total energy of the electron, can only take on certain allowed values. The energy of electrons in an
atom
is
quantized, not by fiat, but due to the fundamental properties of waves confined to a fixed region in space.
¹²

After returning from his “ski” trip, Schrödinger immediately subjected his new equation to the acid test: could it reproduce the energy levels, grouped into different “shells,” for the case of atomic hydrogen, which were known from spectral measurements for decades and “explained” in an ad hoc manner by the old quantum theory? The answer was a resounding yes. The details of the calculations took only a few weeks, and on January 27, 1926, the first of Schrödinger's seminal papers was received at the
Annalen der Physik
. It states the breakthrough thus: “
in this paper, I wish
to consider … the simple case of the hydrogen atom … and show that the customary quantum conditions can be replaced by another postulate, in which the notion of ‘whole numbers' … is not introduced…. The new conception is capable of generalization and strikes, I believe, very deeply at the true nature of the quantum rules.”

After presenting the detailed solution of his hydrogen equation, he briefly touches on its interpretation and its origins. “
It is, of course, strongly suggested
that we try to connect the [wavefunction] with some vibration process in the atom, which would more nearly approach reality than the electron orbits [of Bohr-Sommerfeld theory],” but he feels that this is premature, since the theory needs further development. However, “
Above all, I wish to mention
that I was led to these deliberations by the suggestive papers of M. Louis de Broglie … I have lately shown that the Einstein gas theory can be based on [such] considerations … the above reflections on the atom could have been represented as a generalization from those on the gas model.” Later Schrödinger would say, “
My theory was inspired
by L. de Broglie … and by brief, yet infinitely far-seeing remarks of A. Einstein.”

This first paper was followed in rapid succession by five more in just six months, in which Schrödinger, the consummate craftsman,
working alone, determined essentially all the known properties of atomic spectra from the solutions of his wave equation. It was a breathtaking display, about which even his competitor Born would later remark, “
what is more magnificent
in theoretical physics than Schrödinger's six papers on wave mechanics?” The normally reserved Sommerfeld called Schrödinger's equation “
the most astonishing
among all the astonishing discoveries of the twentieth century.” Hence, by June of 1926, physicists had uncovered most of the new laws and mathematical methods necessary to describe physics on the atomic scale; they just didn't yet know what they meant. However, an interpretation was soon to emerge, one that would challenge the philosophical principles that both Schrödinger and Einstein held dear.

 

¹
When Einstein held the theoretical physics position is was only at the level of an associate professorship (extraordinarius); it was subsequently upgraded to a full professorship (ordinarius).

²
Schrödinger's interest in philosophy was so great that in 1918, before the war ended, he had been planning to “
devote himself to philosophy
” more than physics, only to find that the chair he expected to receive, in Czernowitz, Ukraine, had disappeared along with Austrian control of the region.

³
While Schrödinger was discreet in this final public document, in 1933, to his diary, he confided that he never slept with a woman “
who did not wish, in consequence
, to live with me for all her life.” There is some evidence to back this up.

⁴
An English theoretical physicist, Paul Adrien Maurice Dirac, was the last of the trio of wunderkind to play a founding role in quantum mechanics, along with Pauli and Heisenberg. Born in 1902, he was even younger than Heisenberg, and upon hearing Heisenberg speak at Cambridge in July of 1925, he shortly afterward invented his own, mathematically elegant version of the quantum equations. A few years later he discovered the “Dirac Equation,” the quantum wave equation that takes into account the effects of relativity. However, he interacted little with Einstein during the period 1925–26 and so is not very relevant to our historical narrative; Einstein remarked of him in August 1926, “
I have trouble with Dirac
. This balancing on the dizzying path between genius and madness is awful.”

⁵
First in class.

⁶
“That is
the
Schrödinger.”

⁷
It is likely that it is at this time he studied and understood Einstein's 1917 reformulation of the Bohr-Sommerfeld theory, which he later praised so highly.

⁸
This is always a great compliment from one physicist to the other,
unless
the former is angling for priority.

⁹
For expert readers, this is the “division by
N
!” in the partition function (state counting), which is approximately correct at high temperature and is still used in modern texts.

¹⁰
Einstein immediately saw this equivalence, writing to Schrödinger after his paper appeared, “
I see no basic difference
between your work on the theory of the ideal gas and my own.”

¹¹
Hermann Weyl, a distinguished physicist and a friend of Schrödinger's, later famously commented that Schrödinger “
did his great work
during a late erotic outburst in his life.”

¹²
This simple argument, while certainly part of Schrödinger's reasoning, was not how he first presented the equation and omits his failed initial attempt to come up with an equation consistent with Einstein's relativity theory.

CHAPTER 28

CONFUSION AND THEN UNCERTAINTY

If we are still going to have to
put up with these damn quantum jumps, I am sorry that I ever had anything to do with quantum theory.

—SCHRÖDINGER TO BOHR, OCTOBER 1926

“
I am convinced that you have made a decisive advance
with your formulation of the quantum condition, just as I am equally convinced that the Heisenberg-Born route is off the track.” Thus Einstein wrote to Schrödinger in late April of 1926. The Heisenberg-Born route, a different approach to the “quantum conditions,” introduced the term “quantum mechanics” as a more rigorous replacement for the nebulous conceptual structure of “quantum theory.” This method had begun to bear fruit six months earlier than Schrödinger's, and unlike his work it arose independently of Einstein's recent successes with the quantum gas.

It represented the radical point of view that since atoms were, practically speaking, impossible to observe in space and time, one should stop attempting to describe them by space-time orbits as in classical mechanics. Instead one should develop a description in terms of observable atomic variables, which might not themselves be easily visualized, such as the absorption frequencies for light incident on the atom, and how strongly each frequency was absorbed. The first breakthrough using this approach had come from the twenty-three-year-old prodigy Werner Heisenberg, who formulated his method in July of
1925. By the time of Schrödinger's work, Einstein had been ambivalently struggling with this new framework for quite some time already, since Heisenberg was working in the research group of his close friend Max Born in Göttingen. Born had immediately informed him of Heisenberg's initial sighting of a New World of the atom, writing in a letter dated July 15, 1925, that Heisenberg's paper “
appears rather mystifying
, but is certainly true and profound.”