# QM Measurements - probability, expectation value

1. May 22, 2010

### duckie

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 = $$\frac{1}{\sqrt{2}}$$(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

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

By saying $$L_{z}u = \lambda u$$ 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

2. May 22, 2010

### gabbagabbahey

I assume you are asked the possible outcomes and their probabilities for a measurement of $L_z$? 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 $\LaTeX$ (see my sig). To see how to generate the following image, just click on it.

$$u=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \\ 0\end{pmatrix}$$

Again, click on the following image:

$$L_z=\hbar\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1\end{pmatrix}$$

Shouldn't your eigenvalues have units of angular momentum ?

Well, what are the eigenvectors $|a\rangle$ of $L_z$?

3. May 23, 2010

### duckie

Urgh sorry, yes I meant for a measurement of $L_z$... Late night

And yes, I meant 0, $$\pm\hbar$$.

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

Are those right?

4. May 23, 2010

### gabbagabbahey

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

What does that make the probability of measuring zero for $L_z$? How about $\hbar$? And $-\hbar$?