Density of states with delta function

[Arnd Obert]

Hello,
I'm stuck with this exercise, so I hope anyone can help me.

It is to prove, that the density of states of an unknown, quantum mechanical Hamiltonian $\mathcal{H}$, which is defined by
$$\Omega(E)=\mathrm{Tr}\left[\delta(E1\!\!1-\boldsymbol{H})\right]$$
is also representable as
$$\Omega(E)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathrm{d}k\ e^{ikE}\mathrm{Tr}\left[e^{-ik\boldsymbol{H}}\right]$$
when using the definition of the Dirac delta function
$$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathrm{d}k\ e^{ikx}$$
I really don't know what to do. Is it necessary to change to a dirac notation or is this just a simple representation of the Trace, which i don't know yet?

It would be great if anyone can give me a hand with that.
Thanks a lot,
Arnd

 

[blue_leaf77]

If you replace $x$ in the definition of Dirac delta function with $EI-H$ you get $e^{ik(EI-H)}$. $EI$ commutes with $H$, thus $e^{ik(EI-H)}= e^{ikEI}e^{-ikH}$. Now simplify $e^{ikEI}$ such that the identity operator "goes down" out of the exponent.

 

[Arnd Obert]

When multiplying $$e^{ikE1} \cdot e^{-ikH}$$ you get $$e^{ikE} e^{-ikH}$$ because $e^{ikE1}$ is diagonal, isn't it? And then you can pull it out of the trace?

Thank you a lot, but now it got even worse:
This Hamiltonian should be inserted
$$H = \sum_{j=1}^{N}\hbar\omega(a_j^{\dagger}a_j + 1/2)$$
and result in the density of states
$$\Omega(E)=\frac{1}{2\pi}\int\mathrm{d}k\ e^{ikE}\left(\sum_{n=0}^{\infty}e^{-i\hbar k\omega(n+1/2)}\right)^N$$

I don't know how to handle the ladder operators. Is there a good lecture?
Thanks a lot.

 

[blue_leaf77]

because eikE1eikE1e^{ikE1} is diagonal, isn't it? And then you can pull it out of the trace?

Yes.

Note that
$$
e^{ikH} = \exp \left( ik\sum_{j=1}^{N}\hbar\omega(a_j^{\dagger}a_j + 1/2)\right) \\
= \exp \left( ik\hbar\omega(a_1^{\dagger}a_1 + 1/2)\right) \otimes \exp \left( ik\hbar\omega(a_2^{\dagger}a_2 + 1/2)\right) \otimes \ldots \otimes \exp \left( ik\hbar\omega(a_N^{\dagger}a_N + 1/2)\right)
$$
Then use the property $\textrm{Tr }(A\otimes B) = (\textrm{Tr }A)(\textrm{Tr }B)$.

 

[vanhees71]

What are the tensor products doing there? Isn't this just the usual calculation of the microcanonical partition function in terms of a Fourier transform of the canonical one? As far as I can see, at the end of #3 you already have the correct result.

 

[Arnd Obert]

The End of #3 is the correct result but I have to find the way to get there.
Thank you! I will try the tensor products.

 

[blue_leaf77]

What are the tensor products doing there?

I believe this is a multiparticle system where each particle obeys the ordinary harmonic oscillator Hamiltonian. A basis in the whole system is then a tensor product between the single particle state $|n_1\rangle \otimes |n_2\rangle \otimes \ldots \otimes |n_N\rangle$ and the corresponding symmetry operator must be a Kronecker product from each particle's symmetry operator. Moreover, the formal form of a symmetry generator of the system, such as the Hamiltonian, is
$$
H = H_1\otimes I_2 \otimes \ldots \otimes I_N + I_1\otimes H_2 \otimes \ldots \otimes I_N + \ldots + I_1\otimes I_2 \otimes \ldots \otimes H_N
$$
where $H_i = \hbar\omega(a_i^{\dagger}a_i + 1/2)$. Using a hand-waving notation omitting those jumbled identity operators, the total Hamiltonian is rewritten as
$$
H = \sum_{j=1}^{N}\hbar\omega(a_j^{\dagger}a_j + 1/2)
$$

As far as I can see, at the end of #3 you already have the correct result.

Yes, but the OP was trying to prove that.

 

[Arnd Obert]

This is great, thank you very much! You are completely right with your assumption about the many particle system.
Now the last part remains:

When evaluating the geometric series, I get a density of states which looks like
$$\Omega(\epsilon)=\frac{1}{2\pi}\int dk\exp\left(N\left[ik\epsilon-\ln \left(2i\sin(k\hbar\omega/2)\right)\right]\right)$$
with $\epsilon=E/N$

I have to prove, that the average entropy per oscillator looks like
$$s(\epsilon)=\lim_{N\rightarrow\infty}\frac{\ln\Omega(\epsilon)}{N}=\left(\frac{\epsilon}{\hbar\omega}+1/2\right)\ln\left(\frac{\epsilon}{\hbar\omega}+1/2\right)-\left(\frac{\epsilon}{\hbar\omega}-1/2\right)\ln\left(\frac{\epsilon}{\hbar\omega}-1/2\right)$$
by using the saddle point approximation in the limit $N\rightarrow\infty$.
I'm familiar with the saddle point approximation but don't know how to evaluate the ln, how to find the $x_0$ and how to get the final result.

I hope you have any advice. Thank you!