Exercises

1. Monochromatic light with a wavelength of $6000 \AA$ passes through a fast shutter that opens for $10^{-9}$ sec. What is the subsequent spread in wavelengths of the no longer monochromatic light?

2. Calculate $\langle x\rangle$, $\langle x^2\rangle$, and $\sigma_x$, as well as $\langle p\rangle$, $\langle p^2\rangle$, and $\sigma_p$, for the normalized wavefunction
   
   $$\psi(x) = \sqrt{\frac{2 a^3}{\pi}} \frac{1}{x^2+a^2}.$$
   
   
   Use these to find $\sigma_x \sigma_p$. Note that $\int_{-\infty}^{\infty} dx/(x^2+a^2) = \pi/a$.

3. Classically, if a particle is not observed then the probability of finding it in a one-dimensional box of length $L$, which extends from $x=0$ to $x=L$, is a constant $1/L$ per unit length. Show that the classical expectation value of $x$ is $L/2$, the expectation value of $x^2$ is $L^2/3$, and the standard deviation of $x$ is $L/\sqrt{12}$.

4. Demonstrate that if a particle in a one-dimensional stationary state is bound then the expectation value of its momentum must be zero.

5. Suppose that $V(x)$ is complex. Obtain an expression for $\partial P(x,t)/\partial t$ and $d/dt \int P(x,t) dx$ from Schrödinger's equation. What does this tell us about a complex $V(x)$?

6. $\psi_1(x)$ and $\psi_2(x)$ are normalized eigenfunctions corresponding to the same eigenvalue. If
   
   $$\int_{-\infty}^\infty \psi_1^\ast \psi_2 dx = c,$$
   
   
   where $c$ is real, find normalized linear combinations of $\psi_1$ and $\psi_2$ which are orthogonal to (a) $\psi_1$, (b) $\psi_1+\psi_2$.

7. Demonstrate that $p=-{\rm i} \hbar \partial/\partial x$ is an Hermitian operator. Find the Hermitian conjugate of $a = x + {\rm i} p$.

8. An operator $A$, corresponding to a physical quantity $\alpha$, has two normalized eigenfunctions $\psi_1(x)$ and $\psi_2(x)$, with eigenvalues $a_1$ and $a_2$. An operator $B$, corresponding to another physical quantity $\beta$, has normalized eigenfunctions $\phi_1(x)$ and $\phi_2(x)$, with eigenvalues $b_1$ and $b_2$. The eigenfunctions are related via
   

   $$\psi_1 = (2 \phi_1+3 \phi_2) \left/ \sqrt{13},\right.$$

   $$\psi_2 = (3 \phi_1-2 \phi_2) \left/ \sqrt{13}.\right.$$

   
   
   $\alpha$ is measured and the value $a_1$ is obtained. If $\beta$ is then measured and then $\alpha$ again, show that the probability of obtaining $a_1$ a second time is $97/169$.

9. Demonstrate that an operator which commutes with the Hamiltonian, and contains no explicit time dependence, has an expectation value which is constant in time.

10. For a certain system, the operator corresponding to the physical quantity $A$ does not commute with the Hamiltonian. It has eigenvalues $a_1$ and $a_2$, corresponding to properly normalized eigenfunctions
    

    $$\phi_1 = (u_1+u_2)\left/\sqrt{2},\right.$$

    $$\phi_2 = (u_1-u_2)\left/\sqrt{2},\right.$$

    
    
    where $u_1$ and $u_2$ are properly normalized eigenfunctions of the Hamiltonian with eigenvalues $E_1$ and $E_2$. If the system is in the state $\psi=\phi_1$ at time $t=0$, show that the expectation value of $A$ at time $t$ is
    
    $$\langle A\rangle = \left(\frac{a_1+a_2}{2}\right) + \left(\f... ...a_1-a_2}{2}\right)\cos\left(\frac{[E_1-E_2] t}{\hbar}\right).$$