- #1
Arnd Obert
- 4
- 0
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
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