Talk:Van Hove singularity

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    All the edits on this page (so far) are by me; I didn't realize that I had been logged out. Alison Chaiken 04:55, 8 January 2006 (UTC)

    [edit] Mathematical Errors

    I don't know the theory, but the mathematics in the article are a disaster.

    • Division by \vec{\nabla} E is improper
    • Density and total number seem to be mixed up. There are not V / 2π3 possible k vectors
    • E = E_0 + \vec{\nabla} E \cdot d\vec{k} does not imply d\vec{k} = dE/\vec{\nabla} E
    • I assume E is energy, but it is not defined
    • I assume m is mass, but it is not defined
    • Assuming E is energy, m is mass, then E = \hbar^2 k^2/2m is dimensionally incorrect. \hbar has units of action ML2 / T, E has units ML2 / T2 and k has units of 1 / L.
    Thank you for taking the time to check the article. If you don't like division by \vec{\nabla} E, you won't like van Hove's original paper. While I've cribbed together several versions of the derivation from textbooks, the answer is correct!
    The equation E = \hbar^2 k^2/2m is one of the best-known equations in all of physics and is not dimensionally incorrect. \hbar has units as you suggest, of energy*time = action. \hbar k = p so the expression just says that E = p2 / 2m = 1 / 2mv2. Let's check: \hbar k = ML/T so \hbar^2 k^2/2m has units ML2 / T2 = E. Where's the mistake? By the way, I do say "in energy space" when I introduce the symbol E although I agree that strictly speaking I should define m.
    The idea that "Density and total number seem to be mixed up" worries me too. That part of the article was changed by someone else and I don't have time to think it through before running off to a real-world event that will rudely interrupt my time thinking about math. Alison Chaiken 03:17, 10 January 2006 (UTC)

    You are right about the dimensons - I was asleep I guess. Anyway, I have done some more with the derivation. I have defined what amounts to the differential density of states as g(k) - I think its clear that this is what is being discussed. Also, we can use the chain rule to go directly to dE\vec{\nabla} E\cdot d\vec{k} instead of making a linear approximation. Finally, I am quite sure that the \vec{\nabla} E in the denominators should be |\vec{\nabla} E| but I will do that later. PAR 05:35, 10 January 2006 (UTC)</math>

    You're right about |\vec{\nabla} E|; I've made that change. It still bothers me that density of states (as opposed to *number* of states) should be extensive (proportional to volume) but I don't see an error in the derivation. But now it's that time of day when I stop messing with WP and do research! Alison Chaiken 15:50, 10 January 2006 (UTC)

    I have changed the derivation. I think the dE/|\nabla E| expression only holds for one dimension. It's more complicated in 2 and 3 dimensions, but the conclusions are the same. PAR 03:03, 11 January 2006 (UTC)