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Proof of ladder operator identity

  1. Dec 13, 2015 #1
    1. The problem statement, all variables and given/known data
    Define n=(x + iy)/(2)½L and ñ=(x - iy)/(2)½L.

    Also, ∂n = L(∂x - i ∂y)/(2)½ and ∂ñ = L(∂x + i ∂y)/(2)½.

    with ∂n=∂/∂n, ∂x=∂/∂x, ∂y=∂/∂y, and L being the magnetic length.

    Show that a=(1/2)ñ+∂n and a=(1/2)n -∂ñ

    a and a are the lowering and raising operators of quantum mechanics.
    2. Relevant equations

    3. The attempt at a solution

    Sorry but I really don't have any idea on how start. I just know that a=(1/ (2)½) (x/x0-ip/p0) and a=(1/ (2)½) (x/x0+ip/p0) with x0=(ħ/mω)½ and p0=(ħmω)½
  2. jcsd
  3. Dec 13, 2015 #2


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    Have you tried calculating the commutator of your a and ##a^\dagger##? The idea is to show that they are the same as for the raising and lowering operators.
  4. Dec 14, 2015 #3
    Wouldn't that just be verifying but not showing? I think the problem wants me to derive it.
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