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Angular Momentum and ladder operators

  1. May 24, 2006 #1
    Hi,

    I have done most of the problem in this word document (attached). I have some trouble though. In my QM class, we assumed that the z component of angular momentum Lz satisfies, Lz Ylm = m hbar Ylm and the ladder operator L+ and L- were defined as L+_ = Lx +- iLy. We were able to find the eigen values of Lx and Ly using the ladder operators.

    In this problem initially they define Lx to satisfy Lx Ylm = m hbar Ylm and the continue to say tha Lx = 1/2 (L+ + L-). How is this possible? Further, in part C, they are asking to find the eigen values of Lz. I am not sure how to find this. I would like to know what is Lz in terms of L+ and L-. Please help me out if possible. Thank You,

    Gamma
     

    Attached Files:

  2. jcsd
  3. May 24, 2006 #2
    My question is: can we have

    Lx |l,m> = m hbar |l,m> and

    Lx = 1/2 (L+ + L-)

    The familier thing in QM is to have Lz |l,m> = m hbar |l,m> and define
    L+- as L+_ = Lx +_ i Ly.

    Any help and ideas would be greatly appreciated. Thannk You.

    Gamma
     
  4. May 24, 2006 #3

    George Jones

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    The question looks fine. Note that in the original statement of the problem, the spherical harmonics are not defined as eigenfunction of L_x. However, since x, y, and z are arbitrary labels, if simultaneous eigenfunction of L^2 and L_z can be found, so can simultaneous eigenfunctions of L^2 and L_x. The question denotes these latter eigenfunctions by \Phi.

    The answer in blue for (a) is incorrect. The correct answer is, roughly, that both l and l' label definite values of L^2, and if a state has a definite value of L^2, this value is unique, so l = l'.

    I haven't had a chance to look at the other answers, and I have to go do work now, but, if no one else steps in, I'll be back in 2 or 3 hours to give more help.

    Regards,
    George
     
  5. May 24, 2006 #4
    In part (a), what I was saying is since Ylm' are eigen functions of L2 and Lx, then superposition of Ylm' should also be an eigen function fo L2 and Lx where m' (= -l, .........,+l ).

    I agree. But we are expected to do part (b) of this prolem using Lx=1/2 (L+ + L-). L+ and L- are ladder operators. So it looks like L+- has been defined as L+_ = Lx +_ iLy or L+_ = Lx +_ iLz.

    I would expect L+_ = Ly +_ i Lz. This is where I am confused. Please clarify if possible. Thank You.

    Gamma.
     
  6. May 24, 2006 #5

    George Jones

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    But the Y_{lm'} are not eigenfunctions of L_x, the \Phi_{lm'} are.

    In blue, you write

    L_x Y_{lm} = mћ * Y_{lm}.

    This is not true.

    However, as you wrote in your last post, that an eigenfunction of L_x is a superposition of the Y_{lm'} for m = -l, ... , +l.

    Regards,
    George
     
  7. May 25, 2006 #6
    okay, let see the problem like this: as George mentioned x,y,z are arbit labels. We generally use the covariant notations 1,2,3 in place of x,y,z.
    So this way, there is no priviage for a,y,or, z.
    if a relation is true for L_z in a co-ordinate frame. It'll be true for L_x and L_y too, in a suitable frame. Physics is invariant under such choices.
    So u can use the same algebra.
     
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