The New Physics
Sir Isaac Newton said that light consisted of particles because how else can one explain why we cannot see around objects. If they were waves, they would bend and then cast no shadows. His name carried so much weight and his mechanics were so successful, that few questioned his authority.
Thomas Young, however, thought differently. In 1802 his "double-slit" experiments showed that light interferes with itself producing a pattern of varying intensity. These "interference patterns" are the signature of waves. And so light was shown conclusively to be a wave phenomenon.
At the beginning of the twentieth century, Max Planck reluctantly introduced the quanta. This discontinuous packet of energy seemed like the only way to explain certain behavior involving the transfer of heat in solids. What this meant was that when energy invaded matter in the form of heat, it must be computed in discreet, whole number elements.
Albert Einstein then showed how problems of light radiation could be explained by using a quanta of light which became "photons". And when this was proven shortly thereafter, the famous wave-particle controversy was born.
The crux of the matter is this: the quantum is a particle of energy with discontinuous, whole number gradations (i.e., 1,2,3...), while a wave propagates continuously with a proportional but undulating frequency. It's like the difference between walking up a set of stairs and walking up a smooth ramp. But in the either-or but not both-world of science, this conflict of appearances was hard to accept.
The popular atomic model at the time was the Rutherford atom, a miniature solar system where minus-charged electrons orbited around a plus-charged nucleus. However, it was found to have at least one major flaw. The constantly radiating electrons should lose their energy (according to Maxwell's electro-dynamics) and spiral right down into the nucleus unable to overcome the opposite charge. In other words, why doesn't the world short-circuit out of existence?
Niels Bohr decided to apply the new quanta to Rutherford's atom. The first thing he did was to renounce Maxwell's continuous radiation. This let the electrons rotate in stationary orbits until they emitted or absorbed at least one quanta of energy. The electron would then have to jump into a corresponding lower or higher orbit. This brilliant picture of the atom gave expedient results and quantum theory was well on its way. And so was the first appearance of "quantum weirdness". For the mathematical formalism disallowed the electron to be anywhere except in its prescribed orbit. It had to appear instantly--allowing no travel between--in another orbit. As if by magic. Hence, the first of the famous "quantum jumps".
Louis de Broglie, a French prince, no less, was the first to ponder the existence of matter waves. He approached the quantum via the theory of relativity, which stated that mass and energy can create each other (e=mc2, or, m=e/c2). Now the quantum postulate is simply that energy and frequency are proportional. And frequency means a pulsation. Pulsations are energy and energy leads to mass. Banesh Hoffman provides this picturesque description of de Broglie's idea:
If we write down the usual mathematical expression for such a pulsation we can interpret it in two ways: either as a bottled up heartbeat or else a spread out pulsation... Thus he assumed [using both interpretations at once] that a particle at rest not only possessed a localized heartbeat but also was accompanied by a widespread pulsation forever in step with it and extending all over the universe. This pulsation was as if a whole ocean were rising and falling like the floor of some vast elevator; there were no waves in the ordinary sense, just a steady rise and fall. Is this fantastic? 1
These pulsations tied in perfectly with the electron's orbits in Bohr's atomic model.
But de Broglie wasn't the only one thinking of waves at the time. For Bohr's new quantum ideas had produced much discussion, criticism and even contradictory evidence. And if discreet corpuscles with their quantum jumps seemed weird to some, they were downright unacceptable to others. Among the latter was Irwin Schrödinger, a gifted Viennese physicist working in Zurich.
Because of his family background and training, Schrödinger approached the situation from the standpoint of music; in particular, the way musical notes are derived. He noticed a striking similarity between the discreet nodes of a vibrating string (like a violin or guitar string) and the discreet orbits in the Bohr atom. Only wavelengths of discreet whole numbers will produce a tone. What a coincidence of nature. Could this possibly be a new idea awaiting discovery?
After reading about de Broglie's idea, Schrödinger got the affirmation and impetus he needed. And in no time at all, he made his quantum connection:
Now the scales begin to fall from our eyes: our dear old atoms, corpuscles, particles are Planck's quanta. The carriers of those quanta are themselves quanta.3
"The medium is the message!" This meant that the entire concept of the point-like particle, from the atom down to the ultimate quanta itself, could be conceived of as a discontinuity created by a vibrating continuum: the standing wave! The same type of continuous vibrations that create a discreet musical note could also apply to the constitution of matter. And this in turn meant that the wave concept contained and explained the particle as an effect of the standing wave. The particle however, could not even admit the wave much less explain it. Which then, is the higher level of truth?
The ingenious but nevertheless somewhat artificial assumptions...[of Bohr's atomic model]...are replaced by a much more natural assumption in de Broglies wave phenomena. The wave phenomenon forms the real "body" of the atom. It replaces the individual punctiliform electrons, which in Bohr's model swarm around the nucleus.4
In fact, it was clear to Schrödinger that the atom couldn't be constructed in any other way:
Such point-like single particles are completely out of the question within the atom, and if one still thinks of the nucleus itself in this way one does so quite consciously for reasons of expediency.5
Schrödinger started off by giving a mathematical structure to the de Broglie waves. Within a matter of months he developed the famous equation that bears his name. And this equation was to become "the centerpiece" of the bizarre new science of quantum mechanics:
The Schrödinger equation has the same central importance to quantum mechanics as Newton's laws of motion have for the large scale phenomenon of classical mechanics.6
Schrödinger's equation now forms the basis on all atomic, molecular and solid-state physics and much of physical chemistry.7
The production of this equation is still talked about because it came so fast, worked so well and seemed to appear from nowhere "gobbling up" the competition. And its precedence was of such noble lineage:
Strictly speaking, the new theory is not new; it is a complete organic development, one might almost be tempted to say a more elaborate exposition, of the old [Hamilton wave] theory.8
The "organic development" Schrödinger hints at was a bombshell. It consisted of adding a complex scalar function (the square root of -1) which was variously called the "psi-oscillation", "psi-essence", or "psi-function". It was depicted by y, the 23rd letter of the Greek alphabet and was destined to become the center of much controversy. First, because nobody knew what it meant: "If we pause here to try to think of its meaning we are lost" (Hoffmann); and secondly, it predicted some rather astonishing results:
...It follows that there is a close relationship between the area of interference with which the nucleus surrounds itself and the wavelength, and that the two are of the same order of magnitude. What this [relationship] is, we have to leave open... It is no longer a matter of chance that the size of the atom and the wavelength are of the same order of magnitude: It is a matter of course.9 (emphasis added)
These waves are becoming powerful indeed. Not only could they replace the entire particle/quantum concept and the problems they contained, but they could, once determined, define the size of the atom. This is no small feat, "because 'size of the atom' is no clearly defined term" (ibid). But what is also important is the "relationship" between wave-length and atomic size. And this relationship is determined by an unknown constant which bears the most controversial aspect of Schrödinger's theory:
Thirdly, and lastly, we can remark that the constant remaining unknown, physically speaking, does not in fact have the dimension of a length, but of action, i.e., energy x time.10
Now "action", in relativistic terms, is "a four dimensional volume of space and time" (Eddington). What a coincidence. For the wave function (the solution to the wave equation) shows that E1< E2< E3; that is, each energy state of the system as it evolves over time gets “fatter and fatter.” And energy translates into mass.
We will hear more about this 4-d "action" later. For now, suffice it to say that the heart of Schrödinger's equation is beginning to disclose its meaning. And what it means is the heart of this book. What the wave equation presents; what the wave function predicts, and what some very astonished scientists obscured—if not concealed—was the possibility that the psi-essence was the essence of atomic growth:
De Broglie considered his waves as real properties of the electron, representing physical vibrations of the electron itself... Just like de Broglie, Schrödinger believed in the physical reality of his waves and assumed that electrons were not particles but vibrating clouds extending in all directions in space...11
But a few months later, after he realized the meaning of his psi-function:
Schrödinger proposed that the particle concept be entirely discarded and his concept of wave function be given all the physical reality, which meant that the electron was to be pictured as spread out continuously throughout space.12 (emphasis added)
To Schrödinger, psi was a wave representation of the electron. He thought of the electron as a kind of superposition of waves--a "wave-packet". The major difficulty with this view was that the wave-packet, though initially small and compact, eventually got fatter. Real electrons didn't do this.13
Schrödinger first supposed that the electron is actually spread out and distributed in space... 14
It should not be too difficult to appreciate the significance of what is being said here. The implication that nature's physical building blocks can get "fatter", and be "spread out and distributed in space" is so profound that it cannot be put adequately into words. It is anathema to a physicist whose very existence depends on the science of measurement. The implication is staggering: "Suppose that every length in the universe were doubled; nothing in our experience would be altered" (Eddington). Nothing would be altered because the standards of measurement would be doubled also. Size would be ultimately meaningless because it would be so radically relative. "...the length of a meter stick expands, the atom expands? The how can it make any sense to speak of expansion at all? Expansion relative to what? Expansion relative to nonsense."15 However, as Niels Bohr said, "Only nonsense stands some chance of being the truth."
Obviously, the idea is rejected immediately. It's not even considered remotely, much less seriously. Even Schrödinger tried to avoid the possibility, as Abraham Pais recalled (while quoting Schrödinger):
...he considered a superposition of linear harmonic oscillator wave functions and showed "our wave group holds permanently together...[and]...does not expand over an ever greater domain in the course of time..."
But while his mathematical description was correct, his hope was not. The wave packet does expand, "explode", "dissipate", become "fuzzy" (uncertain), get "fatter", "disperse", "smear" or "spread out":
Schrödinger's calculation was right; his anticipation was not. The case of the oscillator is very special: wave packets do almost always disperse.16
Many months after he introduced his theory [January, 1926], months during which he had been applying it with phenomenal success and even gobbling up rival theories, Schrödinger at last ventured on an interpretation of his Ψ. It was to measure how thickly the electron was spread out... He gave a specific mathematical formula for it. The interpretation was quickly superseded, the actual formula survives.17
In other words, the mathematics were incredibly correct but the picture they produced was just plain incredible. Too incredible it appears. And as fate would have it, a new application was discovered when Max Born squared the psi-wave (y2).
Max Born, at Gottingen, found it difficult to accept Schrödinger's wave packet interpretation. He could see the path of an electron in a chamber specially designed to display tracks of particles (called a Wilson cloud chamber after its discoverer), and it didn't expand with time. The more he thought about it, the more he became convinced that psi (or more exactly, the square of psi) gave only the probability of finding an electron at a given point. With this, quantum theory took on a new twist.18
What an under-statement: “No sooner do we half reconcile ourselves to waves of a smeared-out electron than we are asked to replace them by waves of probability” (Hoffmann).
Born wasn't just applying the probability wave successfully. The application immediately and unanimously (not to mention, 'curiously') became the interpretation. Almost overnight, psi, and the entire concept of matter waves became statistical probabilities. And there can be no doubt that this “reality crisis” was caused by the fact that an electron (as a standing wave) existed in more than three finite dimensions. Born explains it himself:
We have two possibilities. Either we use waves in spaces of more than three dimensions...or we remain in three dimensional space, but give up the simple picture of the wave amplitude [psi] as an ordinary physical magnitude, and replace it by a purely abstract mathematical concept...into which we cannot enter.19
Schrödinger, astounded by this impertinence, remarked:
You, Maxel, you know that I am very fond of you... but I must once and for all give you a basic scolding.20
The battle lines were drawn. Representing the so-called "Copenhagen Interpretation" was Max Born, Werner Heisenberg, Niels Bohr, Paul Dirac and a host of other contributors to this “new physics”. Holding down the “classical” point of view (the point of view that discovered these phenomena), was Erwin Schrödinger, Max Planck, Albert Einstein, Louis de Broglie, Max von Laue and their constituents and followers.
But this is misleading. There was no battle, only a minor skirmish. While Planck, Einstein, de Broglie and others were in partial agreement with Schrödinger, he was the only major physicist actively pursuing the wave packet model, a minority of minute proportions. Born however, knew what was at stake. He was well armed, well motivated and easily garnered many allies for the ensuing onslaught. And foremost among them was Heisenberg (Born's assistant at the time) and his newly discovered “uncertainty principle”.
This principle states that the momentum and position of an atomic particle cannot be known at the same time. The precise knowledge of one allows only the probability of finding the other. And the reason this is so is that the action involved in an observation or measurement changes—if not creates—the very behavior of what is observed.
For instance, to know the momentum of an electron, two measurements must take place (i.e., elapsed time). The first measurement requires at least one photon of light to fall upon (bombard) the electron. Such a collision may locate the position of the particle, but it makes the second measurement impossible. You would never be able to find it. It's been knocked out of its orbit. The best you can do is to calculate the probability for finding it elsewhere. And this is where Born's probability "function" fits into the scheme. Was this just coincidence? Because they seem to have been created in unison; almost as if to corroborate each other: “...the actual mathematical scheme of [Heisenberg's] matrix mechanics came from Born.”21
It started in 1925, perhaps when Heisenberg dropped a paper (soon to become famous) on Born's desk as he was leaving for vacation. It contained a strange new idea that galvanised Born: if an electron was knocked out of an atom by a measurement, then it was nonsense to talk about it's orbit at all. If it can't be directly observed or measured, then it can't be in the theory. There definitely was something there, but rather than assign some fictional orbit to it, he described it in all orbits at once. Heisenberg's method was inherently statistical. And statistics lead directly to probabilities.
Also in 1925, Born received a letter from America:
One day in 1925 I received a letter from C.J. Davisson giving some peculiar results on the reflection of electrons from metallic surfaces. I, and my colleague on the experimental side, James Franck, at once suspected that these curves of Davisson's were crystal-lattice spectra of de Broglie's electron waves, and we made one of our pupils, Elsasser, investigate the matter. His result provided the first preliminary confirmation of the idea of de Broglies, and this was later proved independently by Davisson and Germer and G.P. Thomson by systematic experiments.22
The cards were on the table. The wave-particle conflict was coming to a head. The de Broglie waves were becoming too real. And Born, in the midst of one of the biggest revolutions in science, had a dire need to defend one of his most cherished convictions: “I am emphatically in favor of the retention of the particle idea.”23 So when Schrödinger's theory was published, it only took Born a few feverish months to make his quantum connection.
For according to the uncertainty principle, as soon as Schrödinger gave his wave packet location, he lost information about its momentum. From the Heisenberg-Born point of view, the spreading electron could no longer exist except as a probability.
It was an impossible situation for the smeared-out electron. The only way out seemed to be to regard the Y as not so much describing the particular behavior of an individual electron as telling what the electron was liable to do on the average in very many collisions.
This is where the suspicions begin to mount. Spreading electrons may be an outrageous hypothesis. But spreading probabilities are downright impossible.
Schrödinger’s waves are waves of probability. That, at least, is the accepted interpretation to this day, and there is nothing to indicate it is likely to be superceded… Even so, it is a curious concept. Born must have found compelling reasons for adopting it. What can have induced him to abandon the idea of a smeared-out electron?
This is a difficult idea. Maybe we had better hurry on to other ideas which make this new interpretation more reasonable. (ibid)