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not strictly homework as my course doesn't get going again for a couple of weeks yet, but suppose I have a system with quantum number l=1 in the angular momentum state

[tex] u = \frac{1}{\sqrt{2}} \left(\begin{array}{cc}1\\1\\0\end{array}\right) [/tex]

and I measure L_{z}, the angular momentum component along the z-axis. The problem I am attempting is to find out the possible results and their probabilities, also the expectation value.

So, I got the operator matrix [tex] L_z = \hbar \left(\begin{array}{ccc}1&0&0\\0&0&0\\0&0&-1\end{array}\right) [/tex]

and after using that in the eigenvalue equation, [tex] L_zu = \lambda u [/tex] I multiplied the matrices together to find that [tex] L_zu = \frac{\hbar}{\sqrt{2}}\left(\begin{array}{cc}1\\0\\0\end{array}\right) = \frac{\lambda}{\sqrt{2}}\left(\begin{array}{cc}1\\1\\0\end{array}\right) [/tex].

from this I see that the possible eigenvalues (ie. the possible results of measurement, right?) are [tex] \lambda = \hbar [/tex] and [tex] \lambda = 0 [/tex]. If I'm not mistaken, this is a sensible result given that the angular momentum quantum number l=1.

however, i'm not sure how to approach getting the probability of each outcome. I have this expression in my notes, [tex] P_a = |<a|u>|^2 =\left| \sum a_i^*u_i \right|^2 [/tex]. What I thought about this was maybe that the state |u> was supposed to be taken as a superposition of two basis states, one for each eigenvalue, 0 and [tex] \hbar [/tex]. So, if I named these basis states as a_{1}and a_{2}, when I again used the eigenvalue equation I found:

[tex] L_za_1 = \hbar \left(\begin{array}{ccc}1&0&0\\0&0&0\\0&0&-1\end{array}\right) \left(\begin{array}{c}a11\\a12\\a13\end{array}\right) = \hbar\left(\begin{array}{c}a11\\a12\\a13\end{array}\right) [/tex] which leads to [tex] a_1 = \left(\begin{array}{c}1\\0\\0\end{array}\right) [/tex]

When I do the same thing again I find that [tex] a_2 = \left(\begin{array}{c}0\\0\\0\end{array}\right) [/tex].

This doesn't seem sensible to me, because how can that be combined with the eigenstate a_{1}in a summation that produces the state u? it cannot, as every element is zero and u has two non-zero elements, so I have given up at this point. I was looking at the coefficient a_{i}in the probability expression above, but I wasn't sure how to actually work out what they were spposed to be.

Does anyone have any ideas what I am supposed to do here? Am I even remotely trying to do the right thing?

Thanks.

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# Simple quantum problem - find eigenvalues, probabilities, expectation value?

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