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Transition Radiation rates of Hamiltonian

  1. Apr 28, 2014 #1
    1. The problem statement, all variables and given/known data

    29p2edt.png

    Part (a): Show the Commutation relation [x, [H,x] ]
    Part (b): Show the expression by taking expectation value in kth state.
    Part (c): Show sum of oscillator strength is 1. What's the significance of radiative transition rates?


    2. Relevant equations



    3. The attempt at a solution

    Part (a)

    Manged to show.

    Part (b)

    [tex]\langle H \rangle = \langle k|\frac{p^2}{2m} + V|k\rangle[/tex]
    [tex]\frac{1}{2m}\langle k|p^2|k\rangle + \langle k|V|k\rangle[/tex]

    Not sure what to do at this point - it looks nothing like the answer.
     
  2. jcsd
  3. Apr 28, 2014 #2

    TSny

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    Instead of taking the expectation value of equation (2.2), take the expectation value of the commutation relation that you showed in part (a).
     
  4. Apr 28, 2014 #3
    I tried and that leads to nowhere..

    [tex]\langle \left[x,[H,x]\right] \rangle[/tex]
    [tex]= \langle k|\left[ x, [H,x] \right] |k\rangle[/tex]
    [tex] = \langle k | [x,Hx] - [x,xH]|k\rangle[/tex]
     
    Last edited: Apr 28, 2014
  5. Apr 28, 2014 #4

    TSny

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    Keep going. Write out [x,Hx] and [x,xH]. Then judiciously insert the identity operator in the form ##1 = \sum_n |n\rangle \langle n| ##
     
  6. Apr 28, 2014 #5
    [tex]= \langle k | [x,Hx] - [x,xH]|k\rangle[/tex]
    [tex] = \langle k | [x,H]x - x[x,H] |k\rangle[/tex]
    [tex] = \langle k | xHx - Hx^2 -x^2H + xHx|k\rangle[/tex]
     
  7. Apr 28, 2014 #6

    TSny

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    Take ##\langle k | xHx |k\rangle## and insert the identity: ##\langle k | x H \hat{1} x |k\rangle##
     
  8. Apr 29, 2014 #7
    Yeah got it!
     
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