Quantum Man: Richard Feynman's Life in Science (7 page)

The question that Feynman then asked is, Can quantum mechanics be framed in terms of the
paths
associated with probability amplitudes rather than the probability amplitudes themselves? It turned out that he was not the first one who had asked this question, though he was the first to derive the answer.

CHAPTER
5

Endings and Beginnings

Instead of putting the thing into the mind, or psychology, I put it into a number.

—R
ICHARD
F
EYNMAN

A
s he was struggling to come up with a way to formulate quantum mechanics to accommodate his strange theory with Wheeler, Richard Feynman attended what he later called a “beer party” at the Nassau tavern in Princeton. There he met the European physicist Herbert Jehle, who was visiting at the time and asked Richard what he was working on. Feynman said he was trying to come up with a way to develop quantum mechanics around an action principle. Jehle told him about a paper by one of the fathers of quantum mechanics, the remarkable physicist Paul Dirac, that might just hold the key. As he remembered it, Dirac had proposed how one might use the quantity from which the action is calculated (which, recall from chapter 1, is the Lagrangian—equal to the difference in kinetic and potential energies of a system of particles) in the context of quantum mechanics.

The very next day they went to the Princeton library to go through Dirac’s 1933 paper, suitably titled “The Lagrangian in Quantum Mechanics.” In that paper Dirac brilliantly and presciently suggested that “there are reasons for believing that the Lagrangian [approach] is more fundamental” than other approaches because (a) it is related to the action principle, and (b) (of vital importance for Feynman’s later work, but a fact he wasn’t thinking about at the time) the Lagrangian can more easily incorporate the results of Einstein’s special relativity. But while Dirac certainly had the key ideas in his mind, in his paper he merely developed a formalism that demonstrated useful correspondences and suggested a vague analogy between the action principle in classical mechanics and the more standard formulations of the evolution in time of a quantum mechanical wave function of a particle.

Feynman, being Feynman, decided right then and there to take some simple examples and see if the analogy could be made exact. At the time he was just doing what he thought a good physicist should do—namely, work out a detailed example to check what Dirac meant by what he said. But Jehle, who was watching this graduate student carry out his calculations in real time, faster than Jehle could follow, in that small room in the Princeton library, knew better. As he put it, “You Americans are always trying to find out how something can be used. That’s a good way to discover things.”

He realized that Feynman had carried Dirac’s work one stage further, and in the process had indeed made an important discovery. He had established explicitly how quantum mechanics could be formulated in terms of a Lagrangian. In so doing, Feynman had taken the first step in completely reformulating quantum theory.

I
ADMIT TO
being skeptical about whether Feynman really outdid Dirac that morning in Princeton. Certainly anyone who understands Dirac’s paper can see that almost all of the key ideas are there. Why Dirac didn’t take the next step to see if they could actually be implemented is something we will never know. Perhaps he was satisfied enough that he had demonstrated a possible correspondence but never felt it would be particularly useful for any practical purposes.

The only information we have that Dirac never actually proved to himself that his analogy was exact is Feynman’s later recollection of a conversation with Dirac at the 1946 bicentennial celebration at Princeton. Feynman describes asking Dirac if he knew that his “analogy” could be made exact by a simple constant of proportionality. Feynman’s recollection of the conversation goes as follows:

Feynman: “Did you know that they were proportional?”

Dirac: “Are they?”

Feynman: “Yes.”

Dirac: “Oh, that’s interesting.”

For Dirac, who was known to be both terse and literal in the extreme, this was a long conversation, and it probably speaks volumes. For example, Dirac married the sister of another famous physicist, Eugene Wigner. Whenever he introduced her to people, he introduced her as “Wigner’s Sister,” not as his wife, feeling apparently that the latter fact was superfluous (or perhaps merely demonstrating that he was as misogynistic as many of his colleagues at the time).

More relevant perhaps is a story I heard regarding the famous Danish physicist Niels Bohr, who apparently was complaining about this far-too-quiet postdoctoral researcher, Dirac, who the equally famous physicist Ernest Rutherford had sent him from England. Rutherford then told Bohr a story about a person who goes into a pet shop to buy a parrot. He is shown a very colorful bird and told that it speaks ten different words, and its price is $500. Then he is shown a more colorful bird, with a vocabularly of one hundred words, with a price of $5,000. He then sees a scruffy beast in the corner and asks how much that bird is. He is told $100,000. “Why?” he asks. “That bird is not very beautiful at all. How many words then does it speak?” None, he is told. Flabbergasted, he says to the clerk, “This bird here is beautiful, and speaks ten words and is $500. That bird over there speaks a hundred words and is $5,000. How can that scruffy little bird over there, who doesn’t speak a single word, be worth $100,000?” The clerk smiles and says, “That bird thinks.”

W
HAT DIRAC HAD
intuited in 1933, and what Feynman picked up immediately and explicitly (although it took him awhile to describe it in these terms), is that whereas in classical mechanics the Lagrangian and the action function determine the correct classical path by assigning simple probabilities to the different classical paths between
a
and
c—
ultimately assigning a probability of essentially unity for the path of least action and essentially zero for every other path—in quantum mechanics the Lagrangian and the action function can be used to calculate, not probabilities, but probability
amplitudes
for transitions between
a
and
c
. And that moreover in quantum mechanics many different paths can have nonzero probability amplitudes.

While working out this idea with a simple example Feynman discovered—before Jehle’s surprised eyes that morning in the library at Princeton—that if he tried to calculate probability amplitudes using this prescription for very short travel times he could obtain a result that was identical with the result obtained in traditional quantum mechanics from Schrödinger’s equation. What’s more, in the limit where systems get big, so that classical laws of motion govern the system and quantum mechanical effects tend to become insignificant, the formalism that Feynman developed would reduce to the classical principle of least action.

How this happens is relatively straightforward. If we consider all possible paths between
a
and
c
, we can assign a probability amplitude “weight” to each path that is proportional to the total action for that path. In quantum mechanics many different paths—perhaps an infinite number, even crazy paths that start and stop and instantly change speeds and so on—can have nonzero probability amplitudes. Now the “weight factor” that is assigned to each path is expressed in terms of the total action associated with that particular path. The total action for any path in quantum mechanics must be some multiple of a very small unit of action called Planck’s constant, the fundamental “quantum” of action in the quantum theory, which we earlier saw also gives a lower bound on uncertainties in measuring positions and momenta.

The quantum prescription of Feynman is then to add up all of the weights associated with the probability amplitudes for the separate paths, and the square of this quantity will determine the transition probability for moving from
a
to
c
after a time
t
.

The fact that the weights can be positive or negative accounts not only for the weird quantum behavior, but also for the reason why classical systems behave differently than quantum systems. For if the system is large, so that its total action for each path is then huge compared to Planck’s constant, a small change in path can change the action, expressed in units of Planck’s constant, by a huge amount. As a result, for different nearby paths the weight function can then vary wildly from positive to negative. In general, when the effects of these different paths are added together, the many different positive contributions will tend to cancel the very many negative contributions.

However, it turns out that the path of least action (the classically preferred path therefore) has the property that any small variation in the path produces almost no change in the action. Thus, paths near the path of least action will contribute the same weight to the sum, and will not cancel out. Hence, as the system becomes big, the contribution to the transition probability will be essentially completely dominated by paths very close to the classical trajectory, which will therefore effectively have a probability of order one, while all other paths will have a probability of order zero. The principle of least action will have been recovered.

W
HILE LYING IN
bed a few days later, Feynman imagined how he could extend the analysis he made for paths over very short time intervals to ones that were arbitrarily large, again by extending Dirac’s thinking. As important as it was to be able to show that the classical limit was sensible, and that the mathematics could be reduced to the standard Schrödinger equation for simple quantum systems, what was most exciting for Feynman is that he now had a mechanism to explore the quantum mechanics of more complex physical systems, like the system of electrodynamics he devised with Wheeler, which they could not describe by traditional methods.

While his motivation was to extend quantum mechanics to allow it to describe systems that couldn’t otherwise be described quantum mechanically, it is nevertheless true (as Feynman later emphasized) that for systems to which Dirac, Schrödinger, and Heisenberg’s more standard formulations could be applied, all the methods are completely equivalent. What is important, though, is that this
new
way of picturing physical processes gives a completely different “psychological” understanding of the quantum universe.

The use of the word
picturing
here is significant, because Feynman’s method allows a beautiful pictorial way of thinking about quantum mechanics. Developing this new way took awhile, even for Feynman, who did not explicitly talk about a “sum over paths” in his thesis. By the time he wrote up his thesis as an article six years later in
Reviews of Modern Physics
, the notion was central. That 1948 paper, titled “The Space-Time Approach to Non-Relativistic Quantum Mechanics,” begins with a probability argument along the lines that I have given here, and then immediately starts discussing space-time paths. Surprisingly, drawings are conspicuously absent from the paper. Maybe it was too expensive in those days to get an artist to draw them. No matter, they would come.

W
HILE FEYNMAN WAS
writing up these results to form the basis of his thesis, the world in 1942 was in a state of turmoil, embroiled in the second world war of the century. Amid all of his other concerns—completing his thesis, getting married, finding a job—one morning he was suddenly interrupted in his office by Robert Wilson, then an instructor in experimental physics at Princeton. He sat Feynman down and revealed what should have been top-secret information, though the specific information was so new as to not yet have been thoroughly classified as such.

The United States was about to embark on a project to build an atomic bomb, and a group at Princeton was going to work on one of the possible methods for making the raw material for the bomb, a light isotope of uranium called uranium 235 (with the number 235 representing the atomic mass—the number of protons plus neutrons in the nucleus). Nuclear physics calculations had shown that the dominant naturally occurring isotope of uranium, uranium 238, could not produce a bomb with practicalamounts of material. The question was, How can the rare isotope uranium 235, which could produce a bomb, be separated from the far more abundant uranium 238? Because isotopes of an element differ only in the number of neutrons in the nucleus, but otherwise are chemically identical since they contain the same number of protons and electrons, chemical separation techniques wouldn’t work. Physics had to be employed. Wilson was revealing this secret because he wanted to recruit Feynman to help with the theoretical work needed to see if his own proposed experimental methods would work.

This presented Feynman with a terrible dilemma. He desperately wanted to finish his thesis. He was enjoying the problem he was working on, and he wanted to continue to do the science he loved. He also wanted to graduate, as this was one of his own preconditions for marriage. Moreover, Wilson would want him to focus on problems that Feynman viewed as engineering, a field he had explored but left for physics while still an undergraduate.

His first response was to turn down the offer. At the same time, how could he turn down a possibility to help win the war? He had earlier considered enlisting in the army if he could work in the Signal Corps, but he was told there were no guarantees. Here was a possibility of doing something far more significant. Moreover, he realized that the nuclear physics involved was not a secret. As he later said, “The knowledge of science is universal, an international thing. . . . There was no monopoly of knowledge or skill at that time . . . so there was no reason why if we thought it was possible that they [the Germans] wouldn’t also think it was possible. They were just humans, with the same information. . . . The only way that I knew how to prevent that was to get there earlier so that we could prevent them from doing it, or defeat them.” He did think for a moment about whether making such a frightening weapon was the right thing to do, but in the end he put his thesis work in a desk drawer and went to the meeting Wilson had told him about.

From that moment on he became occupied not in the abstract world of quantum mechanics and electrons, but in the minutiae of electronics and materials science. He was well prepared, as always, by what he learned on his own and in some excellent courses in nuclear physics from Wheeler and in the properties of materials from Wigner. Still, it took some getting used to. He and another research assistant of Wheeler’s, Paul Olum, a Harvard graduate in mathematics, set to work as fast as they could doing calculations they were not certain about even as the experimentalists around them were building the device that the two of them had to determine was or was not workable.