hi,(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**