(a) Let $|i\rangle$ and $|j\rangle$ be two eigenstates of a time-independent Hamiltonian $H_{0}$, separated in energy by $\hbar \omega_{i j}$. At time $t=0$ the system is perturbed by a small, time independent operator $V$. The perturbation is turned off at time $t=T$. Show that if the system is initially in state $|i\rangle$, the probability of a transition to state $|j\rangle$ is approximately

$$P_{i j}=4|\langle i|V| j\rangle|^{2} \frac{\sin ^{2}\left(\omega_{i j} T / 2\right)}{\left(\hbar \omega_{i j}\right)^{2}}$$

(b) An uncharged particle with spin one-half and magnetic moment $\mu$ travels at speed $v$ through a region of uniform magnetic field $\mathbf{B}=(0,0, B)$. Over a length $L$ of its path, an additional perpendicular magnetic field $b$ is applied. The spin-dependent part of the Hamiltonian is

$$H(t)= \begin{cases}-\mu\left(B \sigma_{z}+b \sigma_{x}\right) & \text { while } 0<t<L / v \\ -\mu B \sigma_{z} & \text { otherwise }\end{cases}$$

where $\sigma_{z}$ and $\sigma_{x}$ are Pauli matrices. The particle initially has its spin aligned along the direction of $\mathbf{B}=(0,0, B)$. Find the probability that it makes a transition to the state with opposite spin

(i) by assuming $b \ll B$ and using your result from part (a),

(ii) by finding the exact evolution of the state.

[Hint: for any 3-vector $\mathbf{a}, e^{i \mathbf{a} \cdot \boldsymbol{\sigma}}=(\cos a) I+(i \sin a) \hat{\mathbf{a}} \cdot \boldsymbol{\sigma}$, where $I$ is the $2 \times 2$ unit matrix, $\boldsymbol{\sigma}=\left(\sigma_{x}, \sigma_{y}, \sigma_{z}\right), \quad a=|\mathbf{a}|$ and $\left.\hat{\mathbf{a}}=\mathbf{a} /|\mathbf{a}| .\right]$