I am learning Ashcroft's Solid State Physics. In the Electrons in a Weak Periodic Potential, I got some problems. 1. Ashcroft mentioned in the footnote: The reader familiar with stationary perturbation theory may think that if there is no exact degeneracy, we can always make all level differences large compared with U by considering sufficiently small U. That is indeed true for any given k. However, once we are given a definite U, no matter how small, we want a procedure valid for all k in the first Brillouin zone. We shall see that no matter how small U is we can always find some values of k for which the unperturbed levels are closer together than U. Therefore what we are doing is more subtle than conventional degenerate perturbation theory. I don't quite understand this footnote. I thought that what we are doing is just degenerate and non-degenerate stationary perturbation theory. 2. In the previous section, when Ashcroft was deducing Bloch's theorem, he assumed that the crystal has inversion symmetry: U(r)=U(-r). However, he mentioned in the footnote: The reader is invited to pursue the argument of this section without the assumption of inversion symmetry, which is made solely to avoid inessential complications in the notation. How far we can go without the assumption of inversion symmetry and why? Can we see this problem quantum mechanically? Thank you for your help!
I don't think you'll find the answers very satisfactory: 1. Your puzzlement suggests you are not familiar with stationary perturbation theory. Probably an example of over-explaining. To understand the note, you need to become familiar. 2. As suggested by the author - you should pursue the argument. Then you will have answered your own question. The main difference is, as the author suggests, more complicated notation so it is annoying to write and harder to see the physics. To understand this properly, you should do it yourself. How far can you go? All the way. Note: you are seeing the problem quantum mechanically. Sadly, you cannot learn this theory just from reading a book - you have to do it. It is easy to convince ourselves that we have understood a theory right up until we try to apply it or, better, explain it to someone else. I'd like to see you attempt this stuff yourself before providing suggestions.