# Eigenvalues and eigenstates of a pair of spin-1/2 systems

I came across a problem in the context of degenerate perturbation theory.

$$\newcommand{\ket}{|{#1}\rangle} \newcommand{\bra}{\langle{#1}|} \newcommand{\braket}{\langle{#1}|{#2}\rangle} \newcommand{\acomm}{\left\{#1,#2\right\}}$$ Consider a pair of two spin-1/2 systems with orthonormal basis $$\{ \ket{\uparrow}, \ket{\downarrow} \}$$ for each system where $$S_z\ket{\uparrow} = \frac{1}{2}, \quad S_z\ket{\downarrow} = -\frac{1}{2}\ket{\downarrow}$$ Define unperturbed Hamiltonian $$H_0 = \hat{S_z}\otimes\hat{S_z}$$. What are the eigenstates $$\ket{\phi_n}$$ and eigenvalues $$E_n$$ of $$H_0$$?

I know I should manage to transform the tensor product $$H_0 = \hat{S_z}\otimes\hat{S_z}$$ to a matrix and solve for the eigenvalues and eigenstates of that matrix. However, I feel confused how to deal with the tensor product. Can someone help me?

## 1 Answer

Given the matrix representation of $$\hat S_z$$: $$\hat S_z\rightarrow \frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \tag{1}$$ we can write the tensor product as follows: \begin{align} \hat S_z\otimes \hat S_z &\rightarrow \frac{\hbar}{2} \begin{pmatrix} 1 \cdot\frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} & 0 \cdot\frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\ 0 \cdot\frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} & -1 \cdot\frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \end{pmatrix} \\&=\frac{\hbar^2}{4}\begin{pmatrix} 1 &0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \end{align} \tag{2} You can find the above formula here.

• @AndrewSteane Fixed, thank you! – Charlie Nov 20 '20 at 22:24