What is the Trace of Density of States?

[john0909]

regarding the density of states:
how I GET THE FOLLOWING EQUALITY?
\langle E_n\mid \delta(E-\widehat{H}) \mid E_n \rangle = \sum_n \delta(E-E_n)<br />

 

[jostpuur]

If

<br /> H|E_n\rangle = E_n|E_n\rangle<br />

and if f:\mathbb{R}\to\mathbb{R} is some function, then the operator f(H) is defined by using the eigenbasis of H, like this:

<br /> f(H)|E_n\rangle = f(E_n)|E_n\rangle<br />

Then, if you think that the delta function is like any function, you can do this:

<br /> \delta(E - H)|E_n\rangle = \delta(E-E_n)|E_n\rangle<br />

In order to understand better what's going on, you should take a closer look at how you got the H inside the delta function in the first place.

 

[john0909]

yes but then you get:

<br /> \sum_n \langle E_n\mid \delta(E-E_n) \mid E_n \rangle.<br /> <br />

So how do you eliminate the bra and kets? <br /> \langle E_n| , |E_n\rangle

 

[jostpuur]

If you think that the delta function is like any function, then \delta(E - E_n) is a number, and it can be taken out from between the brackets, by bilinearity of the inner product.

<br /> \langle E_n|\delta(E - E_n)| E_n\rangle = \delta(E - E_n)\langle E_n| E_n\rangle<br />

 

[john0909]

But you didn't answer my question:

let me explain you my problem:

The density of states n(E) is defined as the trace of the spectral operator
\delta(E-\hat{H}), \newline n(E)\equiv Tr \delta(E-\hat{H}).

this expression is equal = \sum_n \langle E_n|\delta(E- \hat{H})| E_n\rangle.

My question is how do I get the final expression:\sum_n \delta(E-E_n)?
According to what you said above I get: \sum_n \delta(E - E_n) \langle E_n| E_n\rangle
BUT HOW DO I ELIMINATE THE BRA AND KETS?
Because finally I need to get \sum_n \delta(E-E_n).

 

[jostpuur]

I just decided that I'm in a nasty mood, and I refuse to answer your final question, even though I know the answer. BUHAHAHahahahahh...!

 

[Demystifier]

According to what you said above I get: \sum_n \delta(E - E_n) \langle E_n| E_n\rangle
BUT HOW DO I ELIMINATE THE BRA AND KETS?
Because finally I need to get \sum_n \delta(E-E_n).

What is \langle \psi | \psi \rangle for any conventionally normalized state | \psi \rangle?