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Intro (you may skip this if you're an expert, I'm including this for completeness):

Say I have two bases for two systems,

The first is a spin-1/2 system $|+\rangle = \left(\begin{array}{c} 1\\0 \end{array}\right),|-\rangle=\left(\begin{array}{c} 0\\1 \end{array}\right)$

The second is a spin-1 system, with $|1_+\rangle=\left(\begin{array}{c} 1\\0\\0 \end{array}\right),|1_0\rangle=\left(\begin{array}{c} 0\\1\\0 \end{array}\right),|1_-\rangle=\left(\begin{array}{c} 0\\0\\1 \end{array}\right)$

Now for the first system, I can use the Pauli matrix

$$\hat{S_z}=\frac{1}{2}\hbar\hat{\sigma}_z = \left( \begin{array}{cc} -\frac{\hbar }{2} & 0 \\ 0 & \frac{\hbar }{2} \\ \end{array} \right)$$

in order to get the projection of my state on the z-axis. Likewise, I could use the projection matrix $$\hat{J}_z=\hbar\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \\ \end{array} \right)$$

To project the other state on the z-axis. Those operators will act on my basis in the following way:

$$\hat{S}_z|+\rangle=\frac{\hbar}{2}|+\rangle$$ $$\hat{J}_z|1_+\rangle=\hbar|1_+\rangle$$

Problem: (here comes the question)

So far everything is good! Now the problem comes when I introduce a space for the composite system, so I'm getting the basis

$$|S_z\rangle\otimes |J_z\rangle\rightarrow\left\{ |+,1_+\rangle,|+,1_0\rangle,|+,1_-\rangle,|-,1_+\rangle,|-,1_0\rangle,|-,1_-\rangle\right\}$$

Now the question is: how do I use the matrix formalism to have such operations just like I had them before in the single systems:

$$S_z|+,1_+\rangle=\frac{\hbar}{2}|+,1_+\rangle$$ $$S_z|+,1_-\rangle=\frac{\hbar}{2}|+,1_-\rangle$$ $$J_z|+,1_0\rangle=0|+,1_0\rangle$$ $$J_z|+,1_-\rangle=-\hbar|+,1_-\rangle$$

In other words, how do I write the state-kets and the operators in the composite system in matrix formalism (just like I showed in the beginning) to give results compatible with what I would expect in the examples?

Is this wrong in some way?

Every time I try to do this with Kronecker Product (like $\hat{S}_z \otimes \hat{J}_z$) I arrive at a mess, and I get terms proportional to $\hbar^2$, and I don't get the eigen-values I expect, and I'm not sure what I'm doing wrong. Could you please show me how to do this?

Thank you.

The Quantum Physicist
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1 Answers1

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Just a remark on the introduction. $\hat{S}_z$ does not "project" states on the $z$ direction in spin space. Indeed $\hat{S}_z ^2 = \hbar/2 \;\text{Id}_{2\times 2}\neq \hat{S}_z$, while a projector $P$ verifies $P^2=P$. Actually, $\hat{S}_z$ rotates states around the $z$ axis in spin space ($e^{i\theta \hat{S}_z}$ rotates states by an angle $\theta$ around the $z$ axis).

Back to the main problem, I have two equivalent answers

  • The matrix elements of $\hat{S}_z\otimes \hat{J}_z$ are given by \begin{equation} \langle \pm , 1_{+,0,-}|\hat{S}_z\otimes \hat{J}_z |\pm ', 1'_{+,0,-}\rangle =\langle \pm| \hat{S}_z|\pm '\rangle \langle 1_{+,0,-}|\hat{J}_z |1'_{+,0,-}\rangle \end{equation} where $\pm$, and $\pm'$ refers to the eigenstates of $S_z$, while $1_{+,0,-}$ and $1'_{+,0,-}$ are the eigenstates of $J_z$. Hence you can compute all the matrix elements separately. For instance, in your basis the top left element of the matrix is \begin{equation} \langle + , 1_+|\hat{S}_z\otimes \hat{J}_z |+, 1_+\rangle =\langle +| \hat{S}_z|+ \rangle \langle 1_+|\hat{J}_z |1_+\rangle = \frac{\hbar}{2}\times \hbar = \frac{1}{2}\hbar^2 \end{equation} and the other elements have to be computed in the same way (most of them are zero).

  • Here you deal with direct product operators, since $\hat{S}_\alpha$ does not act on $\hat{J}_\beta$ eigenstates (whatever $\alpha$ and $\beta$) and vice versa. Thus, you get block diagonal matrices like \begin{equation} \hat{S}_z \otimes \hat{J}_z = \begin{pmatrix} \frac{\hbar}{2}J_z & 0_{3\times 3} \\ 0_{3\times 3} & -\frac{\hbar}{2}J_z \end{pmatrix}\quad . \end{equation} where $0_{3\times 3}$ is the 3 by 3 matrix with all elements set to zero and $J_z$ is the 3 by 3 matrix given by \begin{equation} J_z = \hbar \begin{pmatrix} 1 & 0 & 0 \\ 0 &0 & 0\\ 0 & 0 & -1 \end{pmatrix} \end{equation} in your basis.

The other matrices your are interested in, like $S_x\otimes J_y$ (as far as I understood) can be similarly obtained by one of those two ways.

Hope this helps!

Wajsbrot
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