Sample Test Problems

1. Calculate the commutator $[L_x,L_z]$ where $L_x=yp_z-zp_y$ and $L_z=xp_y-yp_x$. State the uncertainty principle for $L_x$ and $L_z$.
   Answer
   
   $$\begin{eqnarray*}[L_x,L_z]&=&[yp_z-zp_y,xp_y-yp_x]=x[y,p_y]p_z+z[p_y,y]p_x \\ ... ...(-i\hbar)\langle L_y\rangle={\hbar\over 2}\langle L_y\rangle \\ \end{eqnarray*}$$
   
   
   
   

2. * A particle of mass $m$ is in a 1 dimensional potential $V(x)$. Calculate the rate of change of the expected values of $x$ and $p$, ( ${d\langle x\rangle\over dt}$ and ${d\langle p\rangle\over dt}$). Your answer will obviously depend on the state of the particle and on the potential.
   Answer
   
   $$\begin{eqnarray*} {d\langle A\rangle\over dt}&=&{1\over i\hbar}\langle[A,H]\rang... ...},V(x)]\rangle \\ &=&-\left\langle{dV\over dx}\right\rangle \\ \end{eqnarray*}$$
   
   
   
   

3. Compute the commutators $[A^\dagger,A^n]$ and $[A,e^{iHt}]$ for the 1D harmonic oscillator.
   Answer
   
   $$\begin{eqnarray*}[A^\dagger,A^n]&=&n[A^\dagger,A]A^{n-1}=-nA^{n-1} \\ {[A,e^{iH... ...n=0}^\infty {(it)^{n}H^{n}\over (n)!}=it\hbar\omega Ae^{iHt} \\ \end{eqnarray*}$$
   
   
   
   

4. * Assume that the states $\vert u_i>$ are the eigenstates of the Hamiltonian with eigenvalues $E_i$, ( $H\vert u_i>=E_i\vert u_i>$).

   a)

   Prove that $<u_i\vert[H,A]\vert u_i>=0$ for an arbitrary linear operator $A$.

   b)

   For a particle of mass $m$ moving in 1-dimension, the Hamiltonian is given by $H={p^2\over 2m}+V(x)$. Compute the commutator [H,X] where $X$ is the position operator.

   c)

   Compute $<u_i\vert P\vert u_i>$ the mean momentum in the state $\vert u_i>$.

5. * At $t=0$, a particle of mass $m$ is in the Harmonic Oscillator state $\psi(t=0)={1\over\sqrt{2}}(u_0+u_1)$. Use the Heisenberg picture to find the expected value of $x$ as a function of time.