Did some more reading of Bohm’s book Bohm’s book this weekend.
A couple weeks ago I had initially been super impressed with his Wave Packets and De Boglie Waves chapter (ch. 3), where he motivates the free particle wave equation. Unlike so many other Schrodinger equation motivations I’d actually call this one a derivation. It is such a logical sequence of arguments showing how to build on ideas already familiar and produce the QM equation. All the prerequisite ideas are covered, and each one so well. A small example is his excellent coverage of what group velocity is mathematically (first order term in a Taylor expansion of ) … reading that and his subsequent gaussian pulse deformation example made things so much clearer than the fuzzy ideas I can remember from my study of French’s vibrations and waves in school.
Based on for the energy of a wave packet, and the requirement that the group velocity of the wave packet matches the classical particle velocity, he shows how the angular frequency of the wave packet must be . This defines the time dependent phase change of the wave packet. Based on a Fourier propagator expression of the wave packet based on the wave packet at an initial time, and the angular frequency just determined he shows how the potential free wave equation describes this wave packet. Looking back on this treatment, it is perhaps not the most natural thing to do if one hadn’t seen the wave equation before … in a sense he is working backwards from the answer. Everybody else does this too in their “derivations” of the wave equation, but this one is, by far, the best approach.
Now, would I have liked this as much if I wasn’t comfortable with Fourier transforms (I’ve been playing with them recently in 1, 3 and 4 dimensions in the context of calculating propagator solutions to Maxwell’s equations). No. The whole book justifiably requires Fourier theory as a prerequisite, so this would have not likely been satisfying back in school in our dinky engineering intro QM course since I took that well before my systems and signals course where we covered that in detail (or as much detail as you can do without covering distributions).
Now, after deriving the free particle wave equation, Bohm takes FIVE chapters to discuss this and some of the related ideas, mostly in words and minimal equations before moving on to the more general wave equation in chapter 9. I’ve never seen any physics text do anything like that (with the exception of Feynman’s Lectures, which aren’t a traditional text).
After these five chapters comes the masterpiece, his Wave functions and operators chapter (ch 9). Here we have an extension of the free particle treatment, and the construction of the mathematical toolbox required to get there.
There’s a fair amount in the toolbox that’s required, but Bohm’s treatment cuts to the heart of it. Rather than focus on the cute notational gimmicks that you’ll find treatments by those already comfortable by all the ideas (eg: Susskind’s QM lectures, or a text like Liboff‘s), this treatment is minimal. Average values are all written out in full using integrals, just probability weighted sums.
The operator form of momentum, , isn’t pulled out of a magic hat, but just shown to be a consequence of an inverse Fourier transform representation given the momentum space wave function. Interesting, he shows the same thing for the position operator when expressed in the momentum space (). One immediately sees the hints of the Hamiltonian/Poisson-bracket structure that Susskind talks about in his lectures. Bohm introduces the commutator, Hermitian operators, and Hermitian conjugate nicely. Like the averages, these are all expressed in full integral form, which is very clear … no notational gunk to get in the way. Having seen this the long way the value of the fancier notation becomes clear, but for a first time through the content (or a time through the content after struggling to make sense of things in the bra/ket formalism) this just makes so much sense.
Seeing the Hermitian conjuate expressed this way, I can see why there’s both a conjugate and Hermitian conjugate in QM (with distinct symbols). Soon I may be equipt to go back to Doran/Lasenby and make some sense of it.
I enjoyed seeing where the requirement for to be Hermitian came from (in order to conserve probability), and also seeing how naturally the Hamiltonian/Operator commutator arrives from looking at the time rate of change of an operator’s average value. This is the Poisson structure mentioned in Susskind’s classical lectures, and it’s not actually all that complex to get to it.
The finale of the chapter was using the idea of Ehrenfest’s theorem (average wave packet behaviour in classical correspondence with Newton’s laws) to complete the QM wave equation derivation is just masterful. Seeing it done this way without the obstruction of the bra/ket notation is really nice, and to simultaneously get the Schrodinger equation for free tops the cake!
All in all this book is a supreme pleasure to read! Other than the Student’s guide to Maxwell’s equations, I’m not sure I’ve seen a better crafted physics text.