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QM Measurements - probability, expectation value

  1. May 22, 2010 #1
    1. The problem statement, all variables and given/known data

    What are the possible results and their probabilities for a system with l=1 in the angular momentum state u = [tex]\frac{1}{\sqrt{2}}[/tex](1 1 0)? What is the expectation value?
    ((1 1 0) is a vertical matrix but I can't see how to format that)

    2. Relevant equations


    3. The attempt at a solution

    [tex]L_{z} = \hbar[/tex](1,0,-1) for l=1 where (1,0,-1) represents the block diagonal... again, not sure how to do matrices on here :rolleyes:

    By saying [tex]L_{z}u = \lambda u[/tex] and just comparing I have results for lambda of +1, 0, -1.

    I know probability is the modulus of <a|u> squared where a is a corresponding eigenvector... but I'm getting a bit lost somehow. Normally I'm ok with these, but this time I'm just not sure on what to do next.
    Any hints would be greatly appreciated :smile:
     
  2. jcsd
  3. May 22, 2010 #2

    gabbagabbahey

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    I assume you are asked the possible outcomes and their probabilities for a measurement of [itex]L_z[/itex]? You haven't actually said which observable your measuring in this problem statement.

    There are several environments you can use to display matrices and column vector in [itex]\LaTeX[/itex] (see my sig). To see how to generate the following image, just click on it.

    [tex]u=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \\ 0\end{pmatrix}[/tex]


    Again, click on the following image:

    [tex]L_z=\hbar\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1\end{pmatrix}[/tex]

    Shouldn't your eigenvalues have units of angular momentum ?:wink:

    Well, what are the eigenvectors [itex]|a\rangle[/itex] of [itex]L_z[/itex]?
     
  4. May 23, 2010 #3
    Urgh sorry, yes I meant for a measurement of [itex]L_z[/itex]... Late night :frown:

    And yes, I meant 0, [tex]\pm\hbar[/tex].


    Right, I think they're [tex]\begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix}[/tex] for [tex]\lambda=\hbar[/tex], [tex]\frac{1}{\sqrt{3}}\begin{pmatrix} 1 \\ 1 \\ 1\end{pmatrix}[/tex] for [tex]\lambda=0[/tex], and [tex]\begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix}[/tex] for [tex]\lambda=-\hbar[/tex].

    Are those right?
     
  5. May 23, 2010 #4

    gabbagabbahey

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    I'd choose [tex]\begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}[/tex] for [itex]\lambda=0[/itex], so that your eigenvectors are an orthonormal set.

    What does that make the probability of measuring zero for [itex]L_z[/itex]? How about [itex]\hbar[/itex]? And [itex]-\hbar[/itex]?
     
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