(i) The two states of a spin- $\frac{1}{2}$ particle corresponding to spin pointing along the $z$ axis are denoted by $|\uparrow\rangle$ and $|\downarrow\rangle$. Explain why the states

$$|\uparrow, \theta\rangle=\cos \frac{\theta}{2}|\uparrow\rangle+\sin \frac{\theta}{2}|\downarrow\rangle, \quad \quad|\downarrow, \theta\rangle=-\sin \frac{\theta}{2}|\uparrow\rangle+\cos \frac{\theta}{2}|\downarrow\rangle$$

correspond to the spins being aligned along a direction at an angle $\theta$ to the $z$ direction.

The spin- 0 state of two spin- $\frac{1}{2}$ particles is

$$|0\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle_{1}|\downarrow\rangle_{2}-|\downarrow\rangle_{1}|\uparrow\rangle_{2}\right)$$

Show that this is independent of the direction chosen to define $|\uparrow\rangle_{1,2},|\downarrow\rangle_{1,2}$. If the spin of particle 1 along some direction is measured to be $\frac{1}{2} \hbar$ show that the spin of particle 2 along the same direction is determined, giving its value.

[The Pauli matrices are given by

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

(ii) Starting from the commutation relation for angular momentum in the form

$$\left[J_{3}, J_{\pm}\right]=\pm \hbar J_{\pm}, \quad\left[J_{+}, J_{-}\right]=2 \hbar J_{3},$$

obtain the possible values of $j, m$, where $m \hbar$ are the eigenvalues of $J_{3}$ and $j(j+1) \hbar^{2}$ are the eigenvalues of $\mathbf{J}^{2}=\frac{1}{2}\left(J_{+} J_{-}+J_{-} J_{+}\right)+J_{3}^{2}$. Show that the corresponding normalized eigenvectors, $|j, m\rangle$, satisfy

$$J_{\pm}|j, m\rangle=\hbar((j \mp m)(j \pm m+1))^{1 / 2}|j, m \pm 1\rangle,$$

and that

$$\frac{1}{n !} J_{-}^{n}|j, j\rangle=\hbar^{n}\left(\frac{(2 j) !}{n !(2 j-n) !}\right)^{1 / 2}|j, j-n\rangle, \quad n \leq 2 j$$

The state $|w\rangle$ is defined by

$$|w\rangle=e^{w J_{-} / \hbar}|j, j\rangle$$

for any complex $w$. By expanding the exponential show that $\langle w \mid w\rangle=\left(1+|w|^{2}\right)^{2 j}$. Verify that

$$e^{-w J_{-} / \hbar} J_{3} e^{w J_{-} / \hbar}=J_{3}-w J_{-}$$

and hence show that

$$J_{3}|w\rangle=\hbar\left(j-w \frac{\partial}{\partial w}\right)|w\rangle$$

If $H=\alpha J_{3}$ verify that $\left|e^{i \alpha t}\right\rangle e^{-i j \alpha t}$ is a solution of the time-dependent Schrödinger equation.