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 $-\hbar$?

 

[paranormal]

The possible results for the system with l=1 in the angular momentum state u = \frac{1}{\sqrt{2}}(1,1,0) are +\hbar, 0, and -\hbar, with respective probabilities of 1/2, 0, and 1/2. This can be determined by calculating the expectation value, which is the average of all possible results weighted by their respective probabilities. In this case, the expectation value is 0, as the probabilities of +\hbar and -\hbar are equal.

To find the expectation value, we can use the formula <u|L_{z}|u>, where L_{z} is the angular momentum operator and u is the state vector. Substituting in the values, we get:

<u|L_{z}|u> = \frac{1}{2}(1,1,0)|\hbar(1,0,-1) = \frac{\hbar}{2}(1,1,0)|(1,0,-1) = \frac{\hbar}{2}(1,0,-1)\cdot(1,1,0) = 0

Therefore, the expectation value is 0, indicating that the system has an equal chance of measuring +\hbar or -\hbar.

It is important to note that the probabilities and expectation value can only be calculated for observable quantities. In this case, the observable quantity is the angular momentum along the z-axis, represented by L_{z}. The state vector u represents the possible states of the system, and the eigenvalues of the operator L_{z} represent the possible outcomes of a measurement.

In summary, the possible results and their probabilities for the system with l=1 in the angular momentum state u = \frac{1}{\sqrt{2}}(1,1,0) are +\hbar (with a probability of 1/2), 0 (with a probability of 0), and -\hbar (with a probability of 1/2). The expectation value for this system is 0.