First, note that the probabilities of things we measure must be positive (we would never say that there is a probability of minus 1 percent of finding something) and the square of a quantity is also always positive, so quantum mechanics predicts positive probabilities—which is a good thing. But it also implies that the wave function itself can be either positive or negative, since, say, −½ and +½ both yield the same number (+¼) when squared.
If it were the wave function that described the probability of finding some particle at some location x, then if I had two identical particles, the probability of finding either particle at location x would be the sum of the two individual (and each necessarily positive) wave functions. However, because the square of the wave function is what determines the probability of finding particles, and because the square of the sum of two numbers is not equal to the sum of the individual squares, things can get much more interesting in quantum mechanics.
Let’s say the value of the wave function that corresponds to finding particle A at position x is P1, and the value of the wave function that corresponds to finding particle B at position x is P2, then quantum mechanics tells us that the probability of finding either particle A or B at position x is now (P1 + P2)2. Let’s say P1 = ½ and P2 = −½. Then if we only had one particle, say particle A, the probability of finding it at position x would be (½)2 = ¼. Similarly the probability of finding particle B at position x would be (−½)2 = ¼. However, if there are two particles, the probability of finding either particle at position x is ((½) + (−½))2 = 0.
This phenomenon, which on the surface seems ridiculous, is in fact familiar for waves, say, sound waves. Such waves can interfere with each other so that, for example, waves on a string can interfere and produce locations on the string, called nodes, that do not move at all. Similarly, if sound waves are coming from two different speakers in a room, we might find, if we were to walk around the room, certain locations where the waves cancel each other out, or, as physicists say, negatively interfere with each other. (Acoustic experts design concert halls so that hopefully there are no such “dead spots.”)
What quantum mechanics, with probabilities being determined by the square of the wave function, tells us is that particles too can interfere with each other, so that if there are two particles in a box, the probability of finding either of them at a given location can end up being less than the probability of finding one where only a single particle is in the box.
When waves interfere, it is the height, or amplitude, of the resulting wave that is affected by this interference, and the amplitude can be positive or negative depending on whether one is at a peak or a trough in the wave. So another name for the wave function of a particle is its probability amplitude, which can be positive or negative.
And just as for regular amplitudes for sound waves, separate probability amplitudes for different particles can cancel each other out.
It is precisely this mathematics that is behind the behavior of electrons shot at a scintillating screen, as described in chapter 2. Here we find that an electron can actually interfere with itself because electrons have a nonzero probability of being in many different places at any one time.
Let’s first think about how to calculate probabilities in a sensible, classical world. Consider choosing to travel from town a to town c by taking some specific route through town b. Let the probability of choosing some route from a to b be given by P(ab), and then the probability of choosing some specific route from b to c be P(bc). Then, if we assume that what happens at b is completely independent of what happens at a and c, the probability of traveling from a to c along a specific route that goes through town b is simply given by the product of the two probabilities, P(abc) = P(ab) × P(bc). For example, say there is a 50 percent chance of taking some route from a to b, and then a 50 percent chance of taking some route from b to c. Then if we were to send four cars out, two will make it to b on the chosen route, and of those two, one will take the next chosen route from b to c. Thus there is a 25 percent (.5 × .5) probability of taking the required route all the way from beginning to end.
Now, say we don’t care which particular point b is visited between a and c. Then the probability of traveling from a to c, given by P(ac), is simply the sum of the probabilities P(abc) of choosing to go through any point b between a to c.
The reason this makes sense is that classically if we are going from a to c, and b represents the totality of different towns we can cross through, say, halfway from both a and c, then we have to go through one of them during our journey (see figure).
(Since this picture is reminiscent of the earlier pictures of light rays, then we could say that if the example in question involved light rays going from a to c, then we could use the principle of least time to determine that the probability of going through one of the routes, that of least time, was 100 percent, and the probability of taking any other route was zero.)
The problem is that things don’t work this way in quantum mechanics. Because probabilities are determined by the squares of probability amplitudes to go from one place to another, the probability to go from a to c is not given by the sum of the probabilities to go from a to c via any definite intermediate point b. This is because in quantum mechanics it is the separate probability amplitudes for each part of the route that multiply and not the probabilities themselves. Thus, the probability amplitude to go from a to c through some definite point b is given by multiplying the probability amplitude to go from a to b times the probability amplitude to go from b to c.
If we don’t specify which point b to travel through, the probability amplitude to go from a to c is again given by the sum of the product of probability amplitudes to go from a to b and from b to c, for all possible b’s. But this means that actual probability is now given as the square of the sum of these products. Since some terms in the sum can be negative, the crazy quantum behavior I discussed in chapter 2 for electrons hitting a screen can occur. Namely, if we don’t measure which of two points, say b and b', a particle traverses as it travels through one of two slits between a and c, then the probability of arriving at point c on the screen is determined by the sum of the squares of the probability amplitudes for the two different allowed paths. If we do measure which point, b or b', the particle traverses in between a and c, then the probability is simply the square of the probability amplitude for a single path. In the case of many electrons shot one at a time, the final pattern on the screen in the former case will be determined by adding the squares of the sum of probability amplitudes for each of the two possible paths for each particle, while in the latter case it will be determined by adding the squares of the probability amplitudes for each path separately taken by each electron. Again, because the square of a sum of numbers is different from the sum of squares of these numbers, the former probability can differ dramatically from the latter. And as we have seen, if the particles are electrons, the results are indeed different if we don’t measure the particle between beginning and end points compared to what happens if we do.
Quantum mechanics works, whether or not it makes sense.
IT IS PRECISELY this seemingly nonsensical aspect of quantum mechanics that Richard Feynman focused on. As he later put it, the classical picture is wrong if the statement that the position of the particle midway on its trip from a to c actually takes some specific value, b, is wrong. Quantum mechanics instead allows for all possible paths, with all values of b to be chosen at the same time.
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.
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nbsp; —RICHARD FEYNMAN
As 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.”
WHAT 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 va
riation 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.
WHILE 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.
Quantum Man: Richard Feynman's Life in Science Page 6