Chapter 14

Verify for each direction $$\mathbf{n}=\sin \theta \cos \varphi {\mathbf{e}}_x+\sin \theta \sin \varphi c_y+\cos \theta {\mathit{e}}_x$$ the spin operator $${\sigma }_n=\left(\begin{array}{cc}\cos \theta & \sin \theta {\mathrm{e}}^{-1\varphi }\\ \sin \theta {\mathrm{e}}^{i\varphi }& -\cos \theta \end{array}\right)$$ has cigenvalues $\pm 1$. Show that up to phase, the cigenvectors can be expressed as $$\mid +n)=\left(\begin{array}{c}\cos \frac{1}{2}\theta {\mathrm{c}}^{-t\varphi }\\ \sin \frac{1}{2}\theta \end{array}\right),\mid -n)=\left(\begin{array}{c}-\sin \frac{1}{2}\theta {\mathrm{e}}^{-1\varphi }\\ \cos \frac{1}{2}\theta \end{array}\right)$$ and compute the expectation values for spin in the dircction of the various axes $${⟨{\sigma }_1⟩}_{\text{tn }}=(\pm n\left|{\sigma }_1\right|\pm n)$$ For a bcam of particles in a pure state $\mid +n$ ) show that after a measurcment of spin in the $+x$ direction the probability that the spin is in this direction is $\frac{1}{2}(1+\sin \theta \cos \varphi )$.

In the Heisenberg picture show that the time evolution of the expection value of an operator $A$ is given by $$\frac{\mathrm{d}}{\mathrm{d}t}{(A^{\prime }⟩}_{\psi }^{\prime }=\frac{1}{\mathrm{s}h}{⟨[A^{\prime }\cdot H^{\prime }]⟩}_{\psi }+{⟨\frac{\partial A^{\prime }}{\partial t}⟩}_{\varphi }$$ Convert this to an equation in the Schrödinger picture for the time evolution of $(A{⟩}_{\psi }$.

Calculate the canonical partition function, mean encrgy $U$ and entropy $S$, for a system having just two energy levels 0 and $E$. If $E=E(a)$ for a paramcter $a$, calculate the force $A$ and verify the thermodynamic relation $\mathrm{dS}=\frac{1}{r}(\mathrm{d}U+A\mathrm{da})$.

A particle of mass $m$ is confined by an tnfinite potental barrier to rematn within a box $0\le x,y,z\le a$, so that the wave function vanishes on the boundary of the box. Shew that the energy levels are $$E=\frac{1}{2m}\frac{{\pi }^2{\hslash }^2}{a^2}(n_1^2+n_2^2+n_3^2)$$ where $n_1,n_2,n_3$ are positive intcecrs, and calculate the stationary wave functions ${\psi }_E(\mathrm{x})$. Verify that the lowest energy state is non-degenerate, but the next highest is triply degenerate.

Prove the following commutator idertities: $$\begin{array}{c}[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0\text{ (Jacobi identity) }\\ [AB,C]=A[B,C]+[A,C]B\\ [A,BC]=[A,B]C+B[A,C]\end{array}$$

Show that the time reversal of angular momentum $\mathrm{L}=Q\times \mathbb{P}$ is ${\mathrm{\Theta }}^{\ast }{L}_t\mathrm{\Theta }=-L_i$ and that the commutation relations $[L_i,L_j]=ah{ϵ}_{ij}{L}_k$ are only preserved if $\mathrm{\Theta }$ is anti-unitary:

A spin system consists of $N$ particles of magnctic momcnt $\mu$ in a magnctic field B. When $n$ partacles have spin up. $N-n$ spin down, the encrgy is $E_n=n\mu B-(N-n)\mu B=$ $(2n-N)\mu B$. Show that the canonical partition function is $$Z=\frac{\sinh ((N+1)\beta \mu B)}{\sinh \beta \mu B}$$ Evaluate the mean energy $U$ and entropy $S$, skctching their dependence on the variable $x=\beta \mu B$.

Show that the eigenvalues of the three-dimensional harmonic oscillater have the form $(n+\frac{3}{2})\hslash \omega$ where $n$ is a non-negative integer. Show that the dcpeneracy of the $n$th eigenvalue is $\frac{1}{2}(n^2+3n+2)$. Find the corresponding eigenfunctions.