Timeline for Unjustified claim in Kittel about Bloch functions [duplicate]
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2020 at 21:18 | comment | added | insomniac | @questionask : So, to sum up, yes, your simple argument works. Why it works is a more subtle matter. | |
| Mar 26, 2020 at 21:12 | comment | added | insomniac | @questionask : answer to first part : yes, we may write $\sum_k C_k e^{ikx} = \sum_{k \epsilon BZ_1} \sum_G C_{k+G} e^{i(k+G)x}$. This is simply partitioning the sum. However why this partition makes sense is a little more subtle. This needs to be motivated from the requirement that we are seeking simultaneous eigenstates of $\hat{H}$ and $\hat{P}$, which we call $\psi_k$. The argument you provide (that $|\psi_k|^2$ must be periodic with periodicity of the lattice, hence can only contain fourier component $k+G$) is a consequence of this requirement (IMO, of course). | |
| Mar 26, 2020 at 19:53 | comment | added | sondre | In thinking about it some more, I came up with this simple argument, which I hope is correct: If you expand the wave function in a fourier series, the sum may be separated into a sum over the $k$ vectors in the first BZ, and a sum over all of the reciprocal lattice vectors. This is the bloch function, only with a sum over all $k$ in the first BZ in front. Since the absolute square of $\psi$ has to have to same periodicity as the lattice, we cannot have more than one $k$ in the sum, since including more would give cross terms in $|\psi(x+R)|^2$ making it impossible for it to equal $|\psi(x)|^2$ | |
| Mar 26, 2020 at 19:11 | comment | added | insomniac | No worries, we are all new to this :) (at least I am). My peeve here is that IMO, the linked question does not really answer your question (see comments above). I tried to (maybe with too much detail). But closure prevents others from interacting with your question (and any answers provided). | |
| Mar 26, 2020 at 19:06 | comment | added | sondre | Sorry if I did anything wrong by flagging the question, I am new to this ! However, the point made in the linked question is precisely what I was after, namely a connection between the two equations and a justification of treating them as the same. I also liked the last comment you made regarding translational symmetry, although it was a bit too technical for me. | |
| Mar 26, 2020 at 18:17 | comment | added | insomniac | So, linking the other question is absolutely kosher (in fact, desirable ; afterall, we want any future student who chances on one of these questions to be exposed to both, and as many other complete answers on related things as possible) But closure seems not to make sense to me. | |
| Mar 26, 2020 at 18:15 | comment | added | insomniac | Hate to be a pedant, but the linked question (and the answer provided) is specifically about how to get the Central equation by plugging $\psi$ into the wave equation. Different from OP's question which (1) About the way this is described in a specific book and (2) whether the quoted statement needs to be justified in order to prove Bloch theorem. I would say that these are different questions | |
| Mar 26, 2020 at 18:03 | history | closed | CommunityBot | Duplicate of Deriving Bloch's Theorem | |
| Mar 26, 2020 at 15:50 | review | Close votes | |||
| Mar 26, 2020 at 18:05 | |||||
| Mar 26, 2020 at 15:33 | comment | added | sondre | Thank you ! that was a very concise answer. I should have looked more to find that post before adding this. | |
| Mar 26, 2020 at 13:44 | comment | added | Massimo Ortolano | You might find of interest this answer of mine to a similar question (from another book, though). | |
| Mar 26, 2020 at 12:13 | answer | added | insomniac | timeline score: 0 | |
| Mar 26, 2020 at 11:20 | comment | added | Jeff | Not an answer to your specific question, but I like the following intuitive picture of why the Bloch equation makes sense. A crystal has translational symmetry along any lattice vector. Thus, any observable within the crystal must have the same symmetry. As this also holds for the electron density, the magnitude of the electron wavefunction must have the lattice symmetry. Thus, under translation along a lattice vector, the wavefunction itself is constant except for an optional phase difference, which is Bloch's theorem. | |
| Mar 26, 2020 at 9:10 | comment | added | Roger V. | In the current notation the first and the second equations are not really for the same quantity ($\psi$ vs. $\psi_k$). I do share your pain - Kittel is notoriously hard to follow. However, in my experience, his derivations are usually correct... after long reflection. | |
| Mar 26, 2020 at 9:00 | review | First posts | |||
| Mar 26, 2020 at 12:13 | |||||
| Mar 26, 2020 at 8:58 | history | asked | sondre | CC BY-SA 4.0 |