Sample Test Problems

1. A 1D harmonic oscillator is in a linear combination of the energy eigenstates
   
   $$\psi=\sqrt{2\over 3}u_0-i\sqrt{1\over 3}u_1$$
   
   
   Find the expected value of $p$.
   Answer
   
   $$\begin{eqnarray*} \langle\psi\vert p\vert\psi\rangle &=&-i\sqrt{m\hbar\omega\ov... ...r 9}\sqrt{m\hbar\omega\over 2} ={2\over 3}\sqrt{m\hbar\omega}\\ \end{eqnarray*}$$
   
   
   
   

2. Assuming $u_n$ represents the $n^{th}$ 1D harmonic oscillator energy eigenstate, calculate $\langle u_n\vert p\vert u_m\rangle$.
   Answer
   
   $$\begin{eqnarray*} p&=&\sqrt{m\hbar\omega\over 2}{A-A^\dagger\over i} \\ \langle... ...a\over 2}(\sqrt{m}\delta_{n(m-1)}-\sqrt{m+1}\delta_{n(m+1)}) \\ \end{eqnarray*}$$
   
   
   
   

3. Evaluate the ``uncertainty'' in $x$ for the 1D HO ground state $\sqrt{\langle u_0\vert(x-\bar{x})^2\vert u_0\rangle}$ where $\bar{x}=\langle u_0\vert x\vert u_0\rangle$. Similarly, evaluate the uncertainty in $p$ for the ground state. What is the product $\Delta p\Delta x$?
   Answer
   Its easy to see that $\bar{x}=0$ either from the integral or using operators. I'll use operators to compute the rest.
   $$\begin{eqnarray*} x&=&\sqrt{\hbar\over 2m\omega}(A+A^\dagger) \\ \langle u_0\ve... ...t{m\hbar\omega\over 2} \\ \Delta p\Delta x&=&{\hbar\over 2} \\ \end{eqnarray*}$$
   
   
   
   

4. Use the commutator relation between $A$ and $A^\dagger$ to derive $[H,A]$. Now show that $A$ is the lowering operator for the harmonic oscillator energy.
5. At $t=0$, a one dimensional harmonic oscillator is in the state $\psi(t=0)=\sqrt{3\over 4}u_0+i\sqrt{1\over 4}u_1$. Calculate the expected value of $p$ as a function of time.
6. At $t=0$, a harmonic oscillator is in a linear combination of the $n=1$ and $n=2$ states.
   
   $$\psi=\sqrt{2\over 3}u_1-\sqrt{1\over 3}u_2$$
   
   
   Find $\langle x \rangle$ and $\langle x^2 \rangle$ as a function of time.
7. A 1D harmonic oscillator is in a linear combination of the energy eigenstates
   
   $$\psi=\sqrt{2\over 3}u_0+\sqrt{1\over 3}u_2 .$$
   
   
   Find $\langle x^2 \rangle$.