Homework 2

1. Show that
   
   $$\int\limits_{-\infty}^\infty\psi^*(x)x\psi(x)dx =\int\limits_... ...phi^*(p)\left(i\hbar{\partial\over\partial p}\right)\phi(p)dp.$$
   
   
   Remember that the wave functions go to zero at infinity.

2. Directly calculate the the RMS uncertainty in $x$ for the state $\psi(x)=\left({a\over\pi}\right)^{1\over 4} e^{-ax^2\over 2}$ by computing
   
   $$\Delta x=\sqrt{\langle\psi\vert(x-\langle x\rangle)^2\vert\psi\rangle} .$$
   
   

3. Calculate $\langle p^n\rangle$ for the state in the previous problem. Use this to calculate $\Delta p$ in a similar way to the $\Delta x$ calculation.

4. Calculate the commutator $[p^2,x^2]$.

5. Consider the functions of one angle $\psi(\theta)$ with $-\pi\leq\theta\leq\pi$ and $\psi(-\pi)=\psi(\pi)$. Show that the angular momentum operator $L={\hbar\over i}{d\over d\theta}$ has real expectation values.

6. A particle is in the first excited state of a box of length $L$. What is that state? Now one wall of the box is suddenly moved outward so that the new box has length $D$. What is the probability for the particle to be in the ground state of the new box? What is the probability for the particle to be in the first excited state of the new box? You may find it useful to know that
   
   $$\int\sin(Ax)\sin(Bx)dx={\sin\left((A-B)x\right)\over 2(A-B)}-{\sin\left((A+B)x\right)\over 2(A+B)}.$$
   
   

7. A particle is initially in the $n^{th}$ eigenstate of a box of length L. Suddenly the walls of the box are completely removed. Calculate the probability to find that the particle has momentum between $p$ and $p+dp$. Is energy conserved?

8. A particle is in a box with solid walls at $x=\pm {a\over 2}$. The state at $t=0$ is constant $\psi(x,0)=\sqrt{2\over a}$ for $-{a\over 2}<x<0$ and the $\psi(x,0)=0$ everywhere else. Write this state as a sum of energy eigenstates of the particle in a box. Write $\psi(x,t)$ in terms of the energy eigenstates. Write the state at $t=0$ as $\phi(p)$. Would it be correct (and why) to use $\phi(p)$ to compute $\psi(x,t)$?

9. The wave function for a particle is initially $\psi(x)=Ae^{ikx}+Be^{-ikx}$. What is the probability flux $j(x)$?

10. Prove that the parity operator defined by $P\psi(x)=\psi(-x)$ is a hermitian operator and find its possible eigenvalues.