(i) A quantum mechanical system consists of two identical non-interacting particles with associated single-particle wave functions $\psi_{i}(x)$ and energies $E_{i}, i=1,2, \ldots$, where $E_{1}<E_{2}<\ldots$ Show how the states for the two lowest energy levels of the system are constructed and discuss their degeneracy when the particles have (a) spin 0 , (b) spin $1 / 2$.

(ii) The Pauli matrices are defined to be

$$\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

State how the spin operators $s_{1}, s_{2}, s_{3}$ may be expressed in terms of the Pauli matrices, and show that they describe states with total angular momentum $\frac{1}{2} \hbar$.

An electron is at rest in the presence of a magnetic field $\mathbf{B}=(B, 0,0)$, and experiences an interaction potential $-\mu \boldsymbol{\sigma} \cdot \mathbf{B}$. At $t=0$ the state of the electron is the eigenstate of $s_{3}$ with eigenvalue $\frac{1}{2} \hbar$. Calculate the probability that at later time $t$ the electron will be measured to be in the eigenstate of $s_{3}$ with eigenvalue $\frac{1}{2} \hbar$.

Part II 2004