Free online quantum mechanics course from a top guru
In September, I mentioned that some folks were transforming the Feynman Lectures on Physics to free web pages with MathJax, the same \(\rm\LaTeX\)-based system to write elegant mathematical expressions that has been used on this blog for two years or so.
Only Volume I was available at that time.
The URLs are:
feynmanlectures.caltech.edu/It seems to me that all the errata have been incorporated. Recall that the third volume is dedicated to quantum mechanics. Volume II is coming soon.
...Volume III TOC main server (primary),
...Volume III TOC backup (delayed)
The volume has 21 chapters. You may want to be reminded about some of Feynman's educational philosophy.
He has believed that you will understand all the subtle conceptual issues of quantum mechanics if you think about the double slit experiment carefully enough. So that's where he starts in Chapter one. Interference and the postulates of quantum mechanics are discussed in the very first chapter, too.
The second chapter explains how the particle and wave interpretations co-exist. The last section of the chapter tells you about the philosophical implications – about uncertainty and the probabilistic character of predictions that was already there in classical (statistical) physics in almost all practical applications and quantum mechanics just turned this "practical status quo" into a fundamentally new way how the laws of physics have to be formulated. Feynman says that philosophical implications of physics are completely distorted when imported to other fields. So he confines the discussion to physics. In physics, quantum mechanics has reinforced the picture that we shouldn't talk about things we can't measure. I would formulate the statement more carefully: a theory (in this case quantum mechanics) is allowed (but not obliged!) to declare questions about the things we can't measure as meaningless questions.
The third chapter discusses the operations we do with the probability amplitudes, their addition and linearity, and prepares you for identical particles discussed in the fourth chapter. Chapters five and six discuss spin-1 and spin-1/2 particles in detail.
The time dependence is only introduced in the seventh chapter. In quantum mechanics, the time dependence is controlled by the Hamiltonian which is discussed in the eighth chapter. In some sense, this "delay" with which Feynman introduces the time dependence is one of the main differences of his attitude to quantum mechanics relatively to average courses that would love to start with the dynamical equations "immediately". For Feynman, the Hamiltonian is just another operator that can do things with the state. I think that this postponing of the time dependence and the Hamiltonian allows the students to learn the basic quantum framework more properly, instead of being thrown to the traps of false analogies between the classical field equations and Schrödinger's equation.
Two-state quantum systems play a prominent role in his teaching. The ammonia molecule is mentioned at the end of the eighth chapter, the ammonia maser is given a whole chapter, chapter nine, and the tenth chapter discusses many other examples of two-dimensional Hilbert spaces in quantum mechanics whose main essence is mathematically isomorphic to all the other examples but that play many seemingly very different roles in physics. The early focus on finite-dimensional Hilbert spaces is a specific of Feynman's approach, too, and I think that it is also a feature that leads the students to avoid false analogies with classical field theory.
The eleventh chapter adds more sophisticated examples of two-state systems, those from (then) modern particle physics. The twelfth chapter is about the hyperfine structure of hydrogen – all these things are about finite-dimensional Hilbert spaces. So Feynman diminishes the importance of "how the wave functions of such systems depend on space" and instead focuses on the description of physical phenomena using bases and operators that have to exist regardless of the detailed form of the wave functions. Again, I find it wise.
Propagation in crystal lattices is covered in the thirteenth chapter; semiconductors in the fourteenth one. Those are the essential chapters that establish basics for modern condensed matter physics and electronics, among related fields. Similarly, the fifteenth chapter on the independent particles approximation introduces you to some common-sense thinking that chemists prefer most of the time.
Chapter sixteen discusses the dependence of amplitudes on the position. This is a truly "delayed" topic relatively to average quantum mechanics courses, many of which discuss \(\psi(x,y,z)\) in the first minutes. But Feynman discussed these quantum states without assuming the knowledge of the explicit dependence on \((x,y,z)\) and he could have gotten very far, anyway. Nevertheless, he returns to this issue, the momentum operator in the position representation, and so on.
The seventeenth chapter exposes Noether-like links between symmetries and conservation laws, i.e. generators of the symmetries being equal to the conserved quantities. Rotation matrices get some special space here but the angular momentum is fully exploited in the following, eighteenth chapter.
Somewhere in the middle of the eighteenth chapter, Feynman also settles all the issues about entanglement and the so-called EPR paradox. He clearly explains why Einstein et al. were wrong and while he says that it is fascinating, this analysis is simply meant to close the story.
...Now many people who learn quantum mechanics in the usual (old-fashioned) way find this disturbing. They would like to think that once the photons are emitted it goes along as a wave with a definite character. They would think that since “any given photon” has some “amplitude” to be \(x\)-polarized or to be \(y\)-polarized, there should be some chance of picking it up in either the \(x\)- or \(y\)-counter and that this chance shouldn't depend on what some other person finds out about a completely different photon. They argue that “someone else making a measurement shouldn't be able to change the probability that I will find something.” Our quantum mechanics says, however, that by making a measurement on photon number one, you can predict precisely what the polarization of photon number two is going to be when it is detected. This point was never accepted by Einstein, and he worried about it a great deal—it became known as the “Einstein-Podolsky-Rosen paradox.” But when the situation is described as we have done it here, there doesn't seem to be any paradox at all; it comes out quite naturally that what is measured in one place is correlated with what is measured somewhere else. The argument that the result is paradoxical runs something like this:It's remarkable how many people have ignored the paragraph starting with "Do you still think" above and how much additional garbage about these completely settled points has been written in the half-century after these 1963-1964 lectures.
Nature apparently doesn't see the “paradox,” however, because experiment shows that the prediction in (6) is, in fact, true. We have already discussed the key to this “paradox” in our very first lecture on quantum mechanical behavior in Chapter 37, Vol. I.6 In the argument above, steps (1), (2), (4), and (6) are all correct, but (3), and its consequence (5), are wrong; they are not a true description of nature. Argument (3) says that by your measurement (seeing a RHC or a LHC photon) you can determine which of two alternative events occurs for him (seeing a RHC or a LHC photon), and that even if you do not make your measurement you can still say that his event will occur either by one alternative or the other. But it was precisely the point of Chapter 37, Vol. I, to point out right at the beginning that this is not so in Nature. Her way requires a description in terms of interfering amplitudes, one amplitude for each alternative. A measurement of which alternative actually occurs destroys the interference, but if a measurement is not made you cannot still say that “one alternative or the other is still occurring.”
- If you have a counter which tells you whether your photon is RHC or LHC [and Feynman meant neither the music band nor the collider], you can predict exactly what kind of a photon (RHC or LHC) he will find.
- The photons he receives must, therefore, each be purely RHC or purely LHC, some of one kind and some of the other.
- Surely you cannot alter the physical nature of his photons by changing the kind of observation you make on your photons. No matter what measurements you make on yours, his must still be either RHC or LHC.
- Now suppose he changes his apparatus to split his photons into two linearly polarized beams with a piece of calcite so that all of his photons go either into an \(x\)-polarized beam or into a \(y\)-polarized beam. There is absolutely no way, according to quantum mechanics, to tell into which beam any particular RHC photon will go. There is a 50% probability it will go into the \(x\)-beam and a 50% probability it will go into the \(y\)-beam. And the same goes for a LHC photon.
- Since each photon is RHC or LHC—according to (2) and (3)—each one must have a 50-50 chance of going into the \(x\)-beam or the \(y\)-beam and there is no way to predict which way it will go.
- Yet the theory predicts that if you see your photon go through an \(x\)-polarizer you can predict with certainty that his photon will go into his \(y\)-polarized beam. This is incontradiction to (5) so there is a paradox.
If you could determine for each one of your photons whether it was RHC and LHC, and also whether it was \(x\)-polarized (all for the same photon) there would indeed be a paradox. But you cannot do that—it is an example of the uncertainty principle.
Do you still think there is a “paradox”? Make sure that it is, in fact, a paradox about the behavior of Nature, by setting up an imaginary experiment for which the theory of quantum mechanics would predict inconsistent results via two different arguments. Otherwise the “paradox” is only a conflict between reality and your feeling of what reality “ought to be.”
Do you think that it is not a “paradox,” but that it is still very peculiar? On that we can all agree. It is what makes physics fascinating.
The hydrogen atom is solved in the nineteenth chapter; the implications for more complicated atoms and chemistry are mentioned towards the end of the chapter.
Chapter twenty talks about operators from a more unified algebraic viewpoint. It doesn't mean that he hasn't spoken about operators earlier but here the rules and properties and lists are summarized in a more abstract, less physical way. Links between the "pictures" are explained here.
The last Chapter 21 discusses the situation in which "something like a single-particle wave function" does get upgraded to a classical field – namely superconductivity, including the flux quantization and the Josephson junction.
A memorable part of the lectures is Feynman's epilogue. After a paragraph of platitudes and wishes that the students didn't have digestion problems, he discusses the value and purpose of science:
Finally, may I add that the main purpose of my teaching has not been to prepare you for some examination—it was not even to prepare you to serve industry or the military. I wanted most to give you some appreciation of the wonderful world and the physicist's way of looking at it, which, I believe, is a major part of the true culture of modern times. (There are probably professors of other subjects who would object, but I believe that they are completely wrong.)I have always found these comments touching and deep. Clearly, Feynman disagreed with the claims that a purely physics-based approach to the world makes one uncultural or "lopsided".
Perhaps you will not only have some appreciation of this culture; it is even possible that you may want to join in the greatest adventure that the human mind has ever begun.
Well, I haven't really tried to study a whole book in such a format but it looks pretty, allows you to exploit many advantages of the computers, and I feel that a similar format could or should become the default one for all science books.