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Trace of density of states

  1. Jul 19, 2010 #1
    regarding the density of states:
    how I GET THE FOLLOWING EQUALITY?
    [tex]


    \langle E_n\mid \delta(E-\widehat{H}) \mid E_n \rangle = \sum_n \delta(E-E_n)
    [/tex]
     
    Last edited: Jul 19, 2010
  2. jcsd
  3. Jul 19, 2010 #2
    If

    [tex]
    H|E_n\rangle = E_n|E_n\rangle
    [/tex]

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

    [tex]
    f(H)|E_n\rangle = f(E_n)|E_n\rangle
    [/tex]

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

    [tex]
    \delta(E - H)|E_n\rangle = \delta(E-E_n)|E_n\rangle
    [/tex]

    In order to understand better what's going on, you should take a closer look at how you got the [itex]H[/itex] inside the delta function in the first place.
     
  4. Jul 19, 2010 #3
    yes but then you get:

    [tex]



    \sum_n \langle E_n\mid \delta(E-E_n) \mid E_n \rangle.

    [/tex]

    So how do you eliminate the bra and kets? [tex]
    \langle E_n| , |E_n\rangle [/tex]
     
  5. Jul 19, 2010 #4
    If you think that the delta function is like any function, then [itex]\delta(E - E_n)[/itex] is a number, and it can be taken out from between the brackets, by bilinearity of the inner product.

    [tex]
    \langle E_n|\delta(E - E_n)| E_n\rangle = \delta(E - E_n)\langle E_n| E_n\rangle
    [/tex]
     
  6. Jul 19, 2010 #5
    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
    [tex] \delta(E-\hat{H}), \newline n(E)\equiv Tr \delta(E-\hat{H}). [/tex]

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

    My question is how do I get the final expression:[tex] \sum_n \delta(E-E_n)? [/tex]
    According to what you said above I get: [tex] \sum_n \delta(E - E_n) \langle E_n| E_n\rangle [/tex]
    BUT HOW DO I ELIMINATE THE BRA AND KETS?
    Because finally I need to get [tex] \sum_n \delta(E-E_n) [/tex].
     
    Last edited: Jul 19, 2010
  7. Jul 19, 2010 #6
    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.....!!!! :devil: :rofl:
     
  8. Jul 19, 2010 #7

    Demystifier

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    Science Advisor

    What is [tex]\langle \psi | \psi \rangle [/tex] for any conventionally normalized state [tex]| \psi \rangle [/tex]?
     
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