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Quantum Mechanics - Lowering Operator

  1. Jun 26, 2015 #1
    1. The problem statement, all variables and given/known data
    let A be a lowering operator.

    2. Relevant equations
    Show that A is a derivative respects to raising operator, A†,

    A=d/dA†

    3. The attempt at a solution
    I start by defining a function in term of A†, which is f(A†) and solve it using [A , f(A†)] but i get stuck after that. Can someone please explain briefly what i should do in the next step?
     
  2. jcsd
  3. Jun 26, 2015 #2
    ##A_+|n> = |n-1> \, , \, A_-|n+1> = |n> ## so
    $$ A_+\,A_-\, |n> = |n> \,\Rightarrow\,\left(A_+A_- + [A_+,A_-]\right)|n> = |n> $$
     
  4. Jun 26, 2015 #3
    sorry, i dont quite understand. Mind to explain it?
     
  5. Jun 26, 2015 #4
    ## |n> = \Psi_n ## my English is very poor for this theme. ## \frac{d}{dA_+}\left(A_+A_-+1\right) ## is that you need.
     
  6. Jun 26, 2015 #5

    strangerep

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    Science Advisor

    You didn't write out the commutation relations in your initial post. I.e., ##~[A, A^\dagger] = \dots ~?##

    Start by working out ##[A, (A^\dagger)^2]##, and ##[A, (A^\dagger)^3]## using the Leibniz product rule for commutators. This should help you see the pattern. Then try to prove ##[A, (A^\dagger)^k] = k (A^\dagger)^{k-1}## by induction on ##k##.
     
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