QM Measurements - probability, expectation value

[duckie]

Homework Statement

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)

Homework Equations

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

 

[gabbagabbahey]

Homework Statement

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)

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.

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

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

Again, click on the following image:

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

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

Shouldn't your eigenvalues have units of angular momentum ?

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

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

 

[duckie]

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.

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

Shouldn't your eigenvalues have units of angular momentum ?

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

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

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?

 

[gabbagabbahey]

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?

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 $-\hbar$?