Quantum Mechanics: Eigenstates of ##\hat{\mathbb{S}}_x##

[Robben]

Homework Statement

Determine the eigenstates of $\hat{\mathbb{S}}_x$ for a spin$-1$ particle in terms of the eigenstates $|1,1\rangle, \ |1,0\rangle,$ and $|1,-1\rangle$ of $\hat{\mathbb{S}}_z.$

Homework Equations

The Attempt at a Solution

Not sure exactly how to set this problem correctly.

We have for the matrix representation: $S_z= \hbar\left[\begin{array}{ c c }1&0& 0 \\0 & 0 &0\\0&0&-1\end{array} \right]$ and $S_x= \frac{\hbar}{\sqrt{2}}\left[\begin{array}{ c c }0&1& 0 \\1 & 0 &1\\0&1&0\end{array} \right].$ From this it asks to determine the eigenstates of $S_x$ so do I just find the eigenvalues for $S_x$ and then determine the eigenstates from those? Not sure exactly what is being asked.

 

[phyzguy]

The state space is a 3-dimensional vector space, so it can be spanned with any three linearly independent vectors. In particular, the three eigenvectors of Sx can span the space, or the three eigenvectors of Sz can span the space. The problem is asking you to find the three eigenvectors of Sx, and then transform those three vectors into the basis given by the three eigenvectors of Sz. Do you know how to transform a vector from one basis to another?

 

[Robben]

The state space is a 3-dimensional vector space, so it can be spanned with any three linearly independent vectors. In particular, the three eigenvectors of Sx can span the space, or the three eigenvectors of Sz can span the space. The problem is asking you to find the three eigenvectors of Sx, and then transform those three vectors into the basis given by the three eigenvectors of Sz.

So I need to find the eigenvalues of $S_x$, i.e., the $S_x$ matrix I have above? Once I find the eigenvalues I will then find the eigenvectors?

Do you know how to transform a vector from one basis to another?

Could you elaborate on how I can do that, please?

 

[Idoubt]

What you should understand is that when you write a matrix down for an operator such as you have done for $S_z$ and $S_x$, you have already chosen a basis to represent that operator in. In this case the basis you have written it in is $| 1 -1 \rangle$ , $| 1 0 \rangle$ and $| 11 \rangle$ . Can you see this?

The question is asking you to find the eigenvectors of $S_x$ and write it as a linear combination of these vectors.

 

[Robben]

What you should understand is that when you write a matrix down for an operator such as you have done for $S_z$ and $S_x$, you have already chosen a basis to represent that operator in. In this case the basis you have written it in is $| 1 -1 \rangle$ , $| 1 0 \rangle$ and $| 11 \rangle$ . Can you see this?.

I can't see it. Can you elaborate please?

The question is asking you to find the eigenvectors of $S_x$ and write it as a linear combination of these vectors.

In order to find the eigenvectors I must find the eigenvalues. Now from this, what exactly are we writing the eigenvectors as a linear combination of?

 

[Idoubt]

I can't see it. Can you elaborate please?

For example let us take $S_z$ and systematically write down it's representation in the $\{ |1,-1\rangle, |1,0\rangle, |1,1\rangle \}$ basis. First of all we need to know the action of the operator $S_z$ on this basis. Since these states are eigenstates of the $S_z$ operator this is simple to write down as,
$S_z |1,-1\rangle = -\hbar |1 -1\rangle$
$S_z |1,0\rangle = 0$
$S_z |1,1\rangle = \hbar |1 1\rangle$

Now that we know this we can construct the matrix representation of $S_z$ in this basis by making the following assignements

$|1,1\rangle = \left( \begin{array}{c} 1\\ 0 \\ 0 \end{array} \right), |1,0\rangle = \left( \begin{array}{c} 0\\ 1 \\ 0 \end{array} \right), |1,-1\rangle = \left( \begin{array}{c} 0\\ 0 \\ 1 \end{array} \right)$
and the matrix representation of the operator is

$S_z = \left( \begin{array}{ccc} \langle 1,1 | S_z | 1,1 \rangle &\langle 1,1 | S_z |1,0\rangle &\langle 1,1| S_z | 1,-1 \rangle \\ \langle 1,0 | S_z | 1,1 \rangle &\langle 1,0 | S_z |1,0\rangle &\langle 1,0| S_z | 1,-1 \rangle \\ \langle 1,-1 | S_z | 1,1 \rangle &\langle 1,-1 | S_z |1,0\rangle &\langle 1,-1| S_z | 1,-1 \rangle \end{array} \right)$

which you can verify is the definition that you have given.

In order to find the eigenvectors I must find the eigenvalues. Now from this, what exactly are we writing the eigenvectors as a linear combination of?

$S_x$ is also given in the same basis, so if you find the eigenvectors of $S_x$ you will get some vector

$|\phi\rangle = \left( \begin{array}{c} a\\ b \\ c \end{array} \right) = a\left( \begin{array}{c} 1\\ 0 \\ 0 \end{array} \right) + b\left( \begin{array}{c} 0\\ 1 \\ 0 \end{array} \right) + c\left( \begin{array}{c} 0\\ 0 \\ 1 \end{array} \right)$
or
$| \phi \rangle = a|1,1\rangle + b|1,0\rangle + c|1,-1\rangle$

So essentially you need to find the eigenvectors of the matrix representation of $S_x$ you have given.

 

[Robben]

I see now, thanks to your help. Thank you very much!