4.13: Exercises

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

2. Calculate \(\begin{equation}\langle x\rangle,\left\langle x^{2}\right\rangle, \text { and } \sigma_{x}, \text { as well as }\langle p\rangle,\left\langle p^{2}\right\rangle, \text { and } \sigma_{p}\end{equation}\), for the normalized wavefunction 

\begin{equation}\psi(x)=\sqrt{\frac{2 a^{3}}{\pi}} \frac{1}{x^{2}+a^{2}}\end{equation}

Use these to find \(\begin{equation}\sigma_{x} \sigma_{p}\end{equation}\). Note that \(\begin{equation}\int_{-\infty}^{\infty} d x /\left(x^{2}+a^{2}\right)=\pi / a\end{equation}\). 

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

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 \(\begin{equation}V(x)\end{equation}\) is complex. Obtain an expression for \(\begin{equation}\partial P(x, t) / \partial t\end{equation}\) and \(\begin{equation}d / d t \int P(x, t) d x\end{equation}\) from Schrödinger's equation. What does this tell us about a complex \(\begin{equation}V(x)\end{equation}\)?

6. \(\begin{equation}\psi_{1}(x) \text { and } \psi_{2}(x)\end{equation}\) are normalized eigenfunctions corresponding to the same eigenvalue. If 

\begin{equation}\int_{-\infty}^{\infty} \psi_{1}^{*} \psi_{2} d x=c\end{equation}

where \(\begin{equation}c\end{equation}\) is real, find normalized linear combinations of  \(\begin{equation}\psi_{1} \text { and } \psi_{2}\end{equation}\) which are orthogonal to (a) \(\begin{equation}\psi_{1},(\mathrm{b}) \psi_{1}+\psi_{2}\end{equation}\)

7. \(\begin{equation}\begin{aligned}
&\text { Demonstrate that } p=-\mathrm{i} \hbar \partial / \partial x \text { is an Hermitian operator. Find the Hermitian conjugate of }\\
&a=x+i p
\end{aligned}\end{equation}\)

8. An operator , corresponding to a physical quantity \(\begin{equation}\alpha\end{equation}\), has two normalized eigenfunctions \(\begin{equation}\psi_{1}(x) \text { and } \psi_{2}(x)\end{equation}\), with eigenvalues \(\begin{equation}a_{1} \text { and } a_{2}\end{equation}\).  An operator , corresponding to another physical quantity \(\begin{equation}\beta\end{equation}\), has normalized eigenfunctions \(\begin{equation}\phi_{1}(x) \text { and } \phi_{2}(x)\end{equation}\), with eigenvalues \(\begin{equation}b_{1}\end{equation}\) and \(\begin{equation}b_{2}\end{equation}\). The eigenfunctions are related via

\begin{equation}\begin{array}{l}
\psi_{1}=\left(2 \phi_{1}+3 \phi_{2}\right) / \sqrt{13} \\
\psi_{2}=\left(3 \phi_{1}-2 \phi_{2}\right) / \sqrt{13}
\end{array}\end{equation}

\(\begin{equation}\alpha\end{equation}\) is measured and the value \(\begin{equation}a_{1}\end{equation}\) is obtained. If \(\begin{equation}\beta\end{equation}\) is then measured and then \(\begin{equation}\alpha\end{equation}\) again, show that the probability of obtaining \(\begin{equation}a_{1} \text { a second time is } 97 / 169 \text { . }\end{equation}\).

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  does not commute with the Hamiltonian. It has eigenvalues \(\begin{equation}a_{1} \text { and } a_{2}\end{equation}\), corresponding to properly normalized eigenfunctions

\begin{equation}\begin{aligned}
\phi_{1} &=\left(u_{1}+u_{2}\right) / \sqrt{2} \\
\phi_{2} &=\left(u_{1}-u_{2}\right) / \sqrt{2}
\end{aligned}\end{equation}

where \(\begin{equation}u_{1} \text { and } u_{2}\end{equation}\) are properly normalized eigenfunctions of the Hamiltonian with eigenvalues \(\begin{equation}E_{1} \text { and } E_{2}\end{equation}\). If the system is in the state \(\begin{equation}\psi=\phi_{1} \text { at time } t=0\end{equation}\), , show that the expectation value of  at time  is

\begin{equation}\langle A\rangle=\left(\frac{a_{1}+a_{2}}{2}\right)+\left(\frac{a_{1}-a_{2}}{2}\right) \cos \left(\frac{\left[E_{1}-E_{2}\right] t}{\hbar}\right)\end{equation}