The Hamiltonian for a particle of spin $\frac{1}{2}$ in a magnetic field $\mathbf{B}$ is

$$H=-\frac{1}{2} \hbar \gamma \mathbf{B} \cdot \boldsymbol{\sigma} \quad \text { where } \quad \sigma_{x}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{y}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{z}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

and $\gamma$ is a constant (the motion of the particle in space can be ignored). Consider a magnetic field which is independent of time. Writing $\mathbf{B}=B \mathbf{n}$, where $\mathbf{n}$ is a unit vector, calculate the time evolution operator and show that if the particle is initially in a state $|\chi\rangle$ the probability of measuring it to be in an orthogonal state $\left|\chi^{\prime}\right\rangle$ after a time $t$ is

$$\left|\left\langle\chi^{\prime}|\mathbf{n} \cdot \boldsymbol{\sigma}| \chi\right\rangle\right|^{2} \sin ^{2} \frac{\gamma B t}{2} .$$

Evaluate this to find the probability for a transition from a state of spin up along the $z$ direction to one of spin down along the $z$ direction when $\mathbf{B}=\left(B_{x}, 0, B_{z}\right)$.

Now consider a magnetic field whose $x$ and $y$ components are time-dependent but small:

$$\mathbf{B}=\left(A \cos \alpha t, A \sin \alpha t, B_{z}\right) .$$

Show that the probability for a transition from a spin-up state at time zero to a spin-down state at time $t$ (with spin measured along the $z$ direction, as before) is approximately

$$\left(\frac{\gamma A}{\gamma B_{z}+\alpha}\right)^{2} \sin ^{2} \frac{\left(\gamma B_{z}+\alpha\right) t}{2}$$

where you may assume $|A| \ll\left|B_{z}+\alpha \gamma^{-1}\right|$. Comment on how this compares, when $\alpha=0$, with the result for a time-independent field.

[The first-order transition amplitude due to a perturbation $V(t)$ is

$$-\frac{i}{\hbar} \int_{0}^{t} d t^{\prime} e^{i\left(E^{\prime}-E\right) t^{\prime} / \hbar}\left\langle\chi^{\prime}\left|V\left(t^{\prime}\right)\right| \chi\right\rangle$$

where $|\chi\rangle$ and $\left|\chi^{\prime}\right\rangle$ are orthogonal eigenstates of the unperturbed Hamiltonian with eigenvalues $E$ and $E^{\prime}$ respectively.]