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Quantum Mechanics: Question on Angular Momentum

  1. Nov 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider a system that is initially in the state:

    [itex]\psi\left(\theta,\phi\right)=\frac{1}{\sqrt{5}}Y_{1,-1}\left(\theta,\phi\right) + \frac{\sqrt{3}}{5}Y_{1,0}\left(\theta,\phi\right)+\frac{1}{\sqrt{5}}Y_{1,1}\left(\theta,\phi\right)[/itex]

    Part 1: Find [itex]<\psi|L_{+}|\psi>[/itex]
    Part 2: If [itex]L_{z}[/itex] is measured, what values would one obtain and with what probabilities?


    2. Relevant equations

    • [itex]L_{z}|lm>=mh|lm>[/itex]
    • [itex]L_{+}|lm>=[l(l+1)-m(m+1)]^{\frac{1}{2}}h|lm + 1>[/itex]
    • [itex]Probability = \frac{|<\varphi|\psi>|^{2}}{<\psi|\psi>}[/itex]

    3. The attempt at a solution

    For Part 1:

    So I started off by putting the expression for [itex]\psi\left(\theta,\phi\right)[/itex] in Bra-Ket notation:

    [itex]|\psi> = \frac{1}{\sqrt{5}}|1,-1> + \frac{\sqrt{3}}{5}|1,0> + \frac{1}{\sqrt{5}}|1,1>[/itex]

    Then I applied [itex]L_{+}[/itex] to each individual component:
    • [itex]L_{+}|1,-1> = \frac{1}{\sqrt{5}}[1(1+1)-(-1)(-1+1)]^{\frac{1}{2}}h|1,0> = \sqrt{\frac{2}{5}}h|1,0>[/itex]
    • [itex]L_{+}|1,0> = \frac{\sqrt{3}}{5}[1(1+1)-0(0+1)]^{\frac{1}{2}}h|1,1> = \frac{\sqrt{6}}{2}h|1,1>[/itex]
    • [itex]L_{+}|1,1> = \frac{1}{\sqrt{5}}[1(1+1)-(-1)(-1+1)]^{\frac{1}{2}}h|1,2> = 0[/itex]

    So, [itex]L_{+}|\psi> = \sqrt{\frac{2}{5}}h|1,0>[/itex][itex]+ \frac{\sqrt{6}}{2}h|1,1>[/itex]

    And then,

    [itex]<\psi|L_{+}|\psi> = <1,0|\sqrt{\frac{2}{5}}h|1,0>[/itex][itex]+ <1,1|\frac{\sqrt{6}}{2}h|1,1>[/itex]
    As [itex]<1,1|1,1> = 1[/itex] and [itex]<1,0|1,0>=1[/itex]
    [itex]<\psi|L_{+}|\psi> = \sqrt{\frac{2}{5}}h[/itex][itex]+ \frac{\sqrt{6}}{2}h[/itex]

    I think I'm going wrong here somewhere. I think I'm using the wrong complex conjugate. Can someone verify if I am or not?

    For Part 2:

    I took [itex]\psi\left(\theta,\phi\right)[/itex] in Bra-Ket notation as before, i.e.,

    [itex]|\psi> = \frac{1}{\sqrt{5}}|1,-1> + \frac{\sqrt{3}}{5}|1,0> + \frac{1}{\sqrt{5}}|1,1>[/itex]

    And used [itex]L_{z}|lm>=mh|lm>[/itex] to try and obtain a value for [itex]L_{z}[/itex]. I used this on individual components as follows;

    • [itex]L_{z}|1,-1> = \frac{-h}{\sqrt{5}}|1,-1>[/itex]
    • [itex]L_{z}|1,0> = \frac{\sqrt{3}}{5}(0)h|1, 0> = 0[/itex]
    • [itex]L_{z}|1,1> = \frac{h}{\sqrt{5}}|1,1>[/itex]

    Then I multiplied by the complex conjugate, i.e.,

    • [itex]<1,-1|L_{z}|1,-1> = <1,-1|\frac{-h}{\sqrt{5}}|1,-1>[/itex]
    • [itex]<1,1|L_{z}|1,1> = <1,1|\frac{h}{\sqrt{5}}|1,1>[/itex]

    So, [itex]L_{z}=[/itex][itex]\frac{-h}{\sqrt{5}}[/itex][itex] + \frac{h}{\sqrt{5}} = 0[/itex]

    Again, I'm not too sure if I'm right or wrong here. If someone could verify if I am or not, I'd really appreciate it. If I know where I'm going with [itex]L_{z}[/itex] I can continue on with finding the probabilities, which I understand how to do.

    Thanks again in advance. Any help appreciated!!
     
    Last edited: Nov 14, 2012
  2. jcsd
  3. Nov 15, 2012 #2
    Anyone ... please?
     
  4. Nov 15, 2012 #3

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Generally, your work on part 1 looks ok, but I think there are some minor errors. You wrote the wavefunction as
    First, I suspect that the numerical coefficient of the Y1,0 term should have a denominator of √5 rather than 5.

    Also, I think you need to check the numerical factors in the following:
    Finally, when constructing ##<\psi|## you left out the numerical coefficients contained in ##|\psi>##

    For part 2, you have found the "expectation value" of ##L_z##. But that won't give you much information about what values are possible for individual measurements of ##L_z##. The only possible value that you can get for a measurement of an operator in QM is one of the eigenvalues of that operator. Note that your wavefunction is a superposition of three eigenstates of ##L_z##. Each eigenstate corresponds to a specific eigenvalue of ##L_z##.

    So, what are the possible values of a measurement of ##L_z## for your wavefunction?

    The numerical coefficients of each of the terms in the wavefunction have something to do with the probability of measuring a particular eigenvalue of ##L_z##.
     
    Last edited: Nov 15, 2012
  5. Nov 19, 2012 #4
    Thanks TSny, that helped a lot!
     
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