# Quantum Mechanics: Eigenstate

1. Feb 16, 2015

### Robben

1. The problem statement, all variables and given/known data
Calculate $\triangle S_x$ and $\triangle S_y$ for an eigenstate of $\hat{S}_z$ for a spin$-\frac12$ particle. Check to see if the uncertainty relation $\triangle S_x\triangle S_y\ge \hbar|\langle S_z\rangle|/2$ is satisfied.

2. Relevant equations
$S_x =\frac12(S_+ +S_-)$
$S_y = \frac{1}{2i}(S_+-S_-)$

3. The attempt at a solution

I am not sure what I have to do in this problem. For the matrix representation I have that:

$S_x = \frac{\hbar}{2}\left[\begin{array}{ c c }0 & 1 \\1 & 0\end{array} \right]$
$S_y = \frac{\hbar}{2i}\left[\begin{array}{ c c }0 & -1 \\1 & 0\end{array} \right].$

2. Feb 16, 2015

### Orodruin

Staff Emeritus
What is the definition of uncertainty? I suggest you start from that and see what you can find based on it.

3. Feb 16, 2015

### Robben

I was more concerned on the first part of the question.

4. Feb 16, 2015

### Orodruin

Staff Emeritus
Yes? The uncertainty is $\Delta S_x$, how is it defined? Later you can go on to the uncertainty relation, which as the name suggests is a relation of uncertainties.

5. Feb 16, 2015

### Robben

Ops, I thought you were talking about the uncertainty relation. But regarding $\triangle S_x$ the uncertainty is defined as $\sqrt{\langle S_x^2\rangle -\langle S_x\rangle^2}$, where $S_x = \frac{\hbar}{2}P_1 -\frac{\hbar}{2}P_2$.

So all I do is do matrix multiplication with $S_z$, i.e.

$(S_x) (S_z)=\frac{\hbar}{2}\left[\begin{array}{ c c }0 & 1 \\1 & 0\end{array} \right] \left[\begin{array}{ c c }1 & 0 \\0 & -1\end{array} \right].$

Last edited: Feb 16, 2015
6. Feb 16, 2015

### Orodruin

Staff Emeritus
So what do you get when you evaluate the expectation values for an eigenstate of $S_z$? How do you compute the expectation value of any operator in a given state?

7. Feb 16, 2015

### Robben

To compute the expectation value of any operator $\hat{\mathbb{O}}$ for a particle in the state $|\phi\rangle$ is defined as $\langle \hat{\mathbb{O}}\rangle = \langle \phi|\hat{\mathbb{O}}|\phi\rangle.$ But what will the eigenstate for $S_z$ be?

8. Feb 16, 2015

### Orodruin

Staff Emeritus
I suggest you take one of the ones required by the problem statement, i.e., any of the eigenstates of $S_z$.

9. Feb 16, 2015

### Robben

That doesn't help me understand.

10. Feb 17, 2015

### Orodruin

Staff Emeritus
So let us try it this way: What are the eigenstates of $S_z$?

Edit: Also, you should perhaps look for a different way of computing your expectation values. I suspect the one you quoted will not be very helpful. How would you compute $P_1$ and $P_2$ and how would you compute $\left< S_x^2 \right>$?

11. Feb 17, 2015

S_x = \frac{\hbar}{2}\left[\begin{array}{ c c }0 & 1 \\1 & 0\end{array} \right].$Thus, I need to compute$\langle\phi|S_x|\phi\rangle$. 14. Feb 17, 2015 ### Orodruin Staff Emeritus So, how do you represent$S_z$in matrix form? What are the eigenvectors of that matrix? (The eigenvectors of the matrix represent the eigenstates.) 15. Feb 17, 2015 ### Robben In matrix form$
(S_z)=\frac{\hbar}{2}