3.11: Exercises

1. Monochromatic light with a wavelength of $6000 \unicode{x212b}$ 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 $\psi_2$ that 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 $$\begin{aligned} \psi_1 &= (2\,\phi_1+3\,\phi_2) \left/ \sqrt{13},\right.\nonumber\\[0.5ex] \psi_2 &= (3\,\phi_1-2\,\phi_2) \left/ \sqrt{13}.\right.\nonumber\end{aligned}$$ $\alpha$ 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 that commutes with the Hamiltonian, and contains no explicit time dependence, has an expectation value that 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 $$\begin{aligned} \phi_1 &= (u_1+u_2)\left/\sqrt{2},\right.\nonumber\\[0.5ex] \phi_2 &= (u_1-u_2)\left/\sqrt{2},\right.\nonumber\end{aligned}$$ 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 $$\langle A\rangle = \left(\frac{a_1+a_2}{2}\right) + \left(\frac{a_1-a_2}{2}\right)\cos\left(\frac{[E_1-E_2]\,t}{\hbar}\right).$$

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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