- #1
ddd123
- 481
- 55
Homework Statement
Why is it that the microcanonical partition function is ##W = Tr\{\delta(E - \hat{H})\}##? As in, for example, Mattis page 62?
Moreover, what's the meaning of taking the Dirac delta of an operator like ##\hat{H}##?
Homework Equations
The density of states at fixed energy is the microcanonical partition function (using a truth-valued Kronecker delta):
##W(E, V, N) = \sum_{\text{all microstates i at V, N}} \delta_{E_i = E}##
The Dirac delta of an n-dimensional vector ##\vec{x}## (maybe it can be generalized if we see an operator as a matrix?):
##\delta (\vec{x}) = \delta(x_1) ... \delta(x_n)##
The Attempt at a Solution
Maybe going in reverse to make sense of the result, using all microstates ##| i \rangle## (if the ##|i\rangle##'s are normalized we obtain the original W):
##Tr\{\delta(E - \hat{H})\} = \sum_i \langle i | \delta(E - \hat{H})\ | i \rangle = \sum_i \langle i | \delta(E - E_i)\ | i \rangle##
I don't even know if the last step is legitimate because I don't know what this Dirac delta of an operator is. If we see it as a big product of deltas of each matrix entry the action on a ket is lost (besides: in what basis? I suppose it doesn't matter since we take the trace after, but the formal expression even outside of trace shouldn't change meaning with change of basis to be well-defined, so it seems ill-defined this way).
Actually I'm not even sure I did something meaningful since we're interested in a large amount of degenerate states with respect to the Hamiltonian: should its trace be taken to entries with same or different energies? Maybe I should've used the degenerate energy eigenkets ##| n \rangle## instead? But then I don't get the desired result.
Simply put I'm completely confused.