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Introductory Quantum HW (Angular Momentum)

  1. Apr 17, 2014 #1
    1. The problem statement, all variables and given/known data
    Consider a three-dimensional system with wave function ψ. If ψ is in the l = 0 state, we already know that Lzψ=0. Show that Lxψ=0 and Lyψ=0 as well.


    2. Relevant equations

    [Lx,Ly]ψ = i*h-bar*Lzψ

    3. The attempt at a solution

    I'm having trouble figuring out where to start this. I think it should be clear and straight-forward, but for some reason I'm just not seeing how I can derive this. I tried using the above equation, to get

    [Lx,Ly]ψ=0.

    I assume from here I would prove that this is only true of Lxψ and Lyψ are 0. That is assuming, I'm on the right track.
     
  2. jcsd
  3. Apr 17, 2014 #2

    Simon Bridge

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    Have you tried that out to see what you get?
    i.e. expand out the commutator.

    Since you have ##\psi:\text{L}_z\psi=0## do you know what happens when you try to do ##\text{L}_x\psi##?
     
  4. Apr 17, 2014 #3
    The question states Lz ψ = 0, I don't have any specific function for ψ. I think the question makes this claim because Lz= m*h-bar, and m=0 if l=0.

    Could I possibly conclude that L^2 = 0 since l=0, and therefore L^2 = Lz^2+Lx^2+Ly^2 implies Lx = Ly =0?

    L^2 = l(l+1)h-bar
     
    Last edited: Apr 17, 2014
  5. Apr 17, 2014 #4

    Simon Bridge

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    You don't need a specific function - you have to exploit the relationships: using your understanding of QM.
    The question is telling you that the system is prepared in an eigenstate of the Lz operator.

    ... consider: can operators take values by themselves?
    (you will need to be careful about this to do the problem)
    What does the ##l## quantum number refer to in this case?

    Part of the reason these questions get set is to force you to do a long-winded exploration before finding the simple solution. Thus, you are best advised to settle on a direction for your inquiry and see it through rather than take random jumps around the theory.

    Until you settle on something, there's not much I can do to help.
    Did you try any of the other suggestions?
     
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