(a) Consider a quantum system with Hamiltonian $H=H_{0}+V$, where $H_{0}$ is independent of time. Define the interaction picture corresponding to this Hamiltonian and derive an expression for the time derivative of an operator in the interaction picture, assuming it is independent of time in the Schrödinger picture.

(b) The Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ satisfy

$$\sigma_{i} \sigma_{j}=\delta_{i j}+i \epsilon_{i j k} \sigma_{k}$$

Explain briefly how these properties allow $\sigma$ to be used to describe a quantum system with spin $\frac{1}{2}$.

(c) A particle with spin $\frac{1}{2}$ has position and momentum operators $\hat{\mathbf{x}}=\left(\hat{x}_{1}, \hat{x}_{2}, \hat{x}_{3}\right)$ and $\hat{\mathbf{p}}=\left(\hat{p}_{1}, \hat{p}_{2}, \hat{p}_{3}\right)$. The unitary operator corresponding to a rotation through an angle $\theta$ about an axis $\mathbf{n}$ is $U=\exp (-i \theta \mathbf{n} \cdot \mathbf{J} / \hbar)$ where $\mathbf{J}$ is the total angular momentum. Check this statement by considering the effect of an infinitesimal rotation on $\hat{\mathbf{x}}, \hat{\mathbf{p}}$ and $\boldsymbol{\sigma}$.

(d) Suppose that the particle in part (c) has Hamiltonian $H=H_{0}+V$ with

$$H_{0}=\frac{1}{2 m} \hat{\mathbf{p}}^{2}+\alpha \mathbf{L} \cdot \boldsymbol{\sigma} \quad \text { and } \quad V=B \sigma_{3}$$

where $\mathbf{L}$ is the orbital angular momentum and $\alpha, B$ are constants. Show that all components of $\mathbf{J}$ are independent of time in the interaction picture. Is this true in the Heisenberg picture?

[You may quote commutation relations of $\mathbf{L}$ with $\hat{\mathbf{x}}$ and $\hat{\mathbf{p}}$.]