bohm’s chapter 11, wave solutions for square potentials.
Posted by peeterjoot on May 16, 2009
I’ve finished working my way through chapter 11 of this book today.
I don’t recall covering anything but square potentials in the mini QM course we did back in engineering classes of University (we used French’s intro QM text). So I didn’t expect much more than I already knew from this chapter but was surprised by how much was covered here. This was more than the infinite square well and bound states, and actually starts with the “free particle” case, which is arguably a harder starting place. For free particles (modeled in a stream by non-normalizable wave functions), transmission, reflection, probability current conservation are all covered.
Also covered is the potential barrier and tunneling. I was surprised how such a scary sounding subject (quantum tunneling) actually turned out to be fairly straightforward. The algebra is messy as hell, but conceptually, not much more sophistication than matching the wave function and its derivative at the boundaries is required.
The optics analogy was also an interesting one. The sharp edges of the barriers have an impedance effect to the probability current flow. I’m now curious to go back and revisit the variational derivation of snell’s law from the principle of least time. That derivation (and exersize in Byrom and Fuller) specified that the speed of light in the material varied only continuously, which would rule out the sorts of discontinuous jumps in these QM problems. Was there something to that requirement for continuity?
Another interesting thing to potentially revisit is the more general case of non-square barriers. Is this going to be like diffraction where the interference pattern for arbitrary shaped gratings is a function of the Fourier transform of the grating?
Not covered in this chapter, is the momentum or wave number domain representation of the wave functions. If we do the Fourier transformation to switch from the position representation, what will the wave functions look like? I’d expect a few delta functions (one for each of the incident, reflected, and transmitted wave functions), and something else for the barrier contribution. What would that something be, and in what way would it be related to the shape of the barrier (this is where I’d guess the Fourier transformation of the barrier shape would come in).
I stopped doing the problems of this chapter after problem 5:
but can’t really count this chapter as done until I do the remainders. In particular, a few things I’ve been unclear about reading the text, and finishing these up may clarify them. What exactly means and the significance of the phase shift in relation to the resonances is one such example of something that isn’t clear to me just by reading (that confusion compounds since I then also have trouble for seeing the origin of equation 84 when the wave packet is treated).
Problem 8 on determining energy levels for the square well bound case will probably also help make that energy level determination content “more real”.
Overall it is ironic how so much of the math required for this chapter is only high school calculus, algebra, trig and complex numbers. I have found however that if I didn’t go through that algebra myself to verify each of the equations there was a level of suprising incomprehsibilty. Logically, I felt I ought to be able to say “sure, if I was to do the algebra I’d get that result … the details shouldn’t be important.” … but if I tried that at any point, I’d go blank and have trouble with whatever followed. Going through the tedious algebra in the end was well worth it. There’s not much physics in doing those parts, but without doing those “tedious routine bits” it is hard to be comfortable with the physics that remains.