Chapter 3, The Schrödinger Equation, the Wave Function, and Operator Algebra Video Solutions, Quantum mechanics

Problem 1

If the state $\psi(\mathbf{r})$ is a superposition,
$$
\psi(\mathbf{r})=c_1 \psi_1(\mathbf{r})+c_2 \psi_2(\mathbf{r})
$$
where $\psi_1(\mathbf{r})$ and $\psi_2(\mathbf{r})$ are related to one another by time reversal, show that the probability current density can be expressed without an interference term involving $\psi_1$ and $\psi_2$.

Sheh Lit Chang

University of Washington

Problem 2

For a free particle in one dimension, calculate the variance at time $t,(\Delta x)_t^2 \equiv$ $\left\langle\left(x-\langle x\rangle_t\right)^2\right\rangle_t=\left\langle x^2\right\rangle_t-\langle x\rangle_t^2$ without explicit use of the wave function by applying (3.44) repeatedly. Show that
$$
(\Delta x)_t^2=(\Delta x)_0^2+\frac{2}{m}\left[\frac{1}{2}\left\langle x p_x+p_x x\right\rangle_0-\langle x\rangle_0\left\langle p_x\right\rangle\right] t+\frac{\left(\Delta p_x\right)^2}{m^2} t^2
$$
and
$$
\left(\Delta p_x\right)_t^2=\left(\Delta p_x\right)_0^2=\left(\Delta p_x\right)^2
$$

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10:42

Problem 3

Consider a linear harmonic oscillator with Hamiltonian
$$
H=T+V=\frac{p_x^2}{2 m}+\frac{1}{2} m \omega^2 x^2
$$
(a) Derive the equation of motion for the expectation value $\langle x\rangle_t$ and show that it oscillates, similarly to the classical oscillator, as
$$
\langle x\rangle_t=\langle x\rangle_0 \cos \omega t+\frac{\left\langle p_x\right\rangle_0}{m \omega} \sin \omega t
$$
(b) Derive a second-order differential equation of motion for the expectation value $\langle T-V\rangle_{\text {, }}$, by repeated application of (3.44) and use of the virial theorem. Integrate this equation and, remembering conservation of energy, calculate $\left\langle x^2\right\rangle_t$.
(c) Show that
$$
\begin{aligned}
(\Delta x)_t^2=\left\langle x^2\right\rangle_t-\langle x\rangle_t^2= & (\Delta x)_0^2 \cos ^2 \omega t+\frac{\left(\Delta p_x\right)_0^2}{m^2 \omega^2} \sin ^2 \omega t \\
& +\left[\frac{1}{2}\left\langle x p_x+p_x x\right\rangle_0-\langle x\rangle_0\left\langle p_x\right\rangle_0\right] \frac{\sin 2 \omega t}{m \omega}
\end{aligned}
$$

Verify that this reduces to the result of Problem 1 in the limit $\omega \rightarrow 0$.
(d) Work out the corresponding formula for the variance $\left(\Delta p_x\right)_t^2$.

Mirza Aslam Beig

Numerade Educator

16:39

Problem 4

Prove that the probability density and the probability current density at position $\mathbf{r}_0$ can be expressed in terms of the operators $\mathbf{r}$ and $\mathbf{p}$ as expectation values of the operators
$$
\rho\left(\mathbf{r}_0\right) \rightarrow \delta\left(\mathbf{r}-\mathbf{r}_0\right) \quad \mathbf{j}\left(\mathbf{r}_0\right) \rightarrow \frac{1}{2 m}\left[\mathbf{p} \delta\left(\mathbf{r}-\mathbf{r}_0\right)+\delta\left(\mathbf{r}-\mathbf{r}_0\right) \mathbf{p}\right]
$$

Derive expressions for these densities in the momentum representation.

Sheh Lit Chang

University of Washington

07:40

Problem 5

For a system described by the wave function $\psi\left(\mathbf{r}^{\prime}\right)$, the Wigner distribution function is defined as
$$
W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right)=\frac{1}{(2 \pi \hbar)^3} \int e^{-i \mathbf{p}^{\prime} \cdot \mathbf{r}^{\prime \prime} / \hbar} \psi^*\left(\mathbf{r}^{\prime}-\frac{\mathbf{r}^{\prime \prime}}{2}\right) \psi\left(\mathbf{r}^{\prime}+\frac{\mathbf{r}^{\prime \prime}}{2}\right) d^3 r^{\prime \prime}
$$
(In formulas involving the Wigner distribution, it is essential to make a notational distinction between the unprimed operators, $\mathbf{r}$ and $\mathbf{p}$, and the real number variables, which carry primes.)
(a) Show that $W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right)$ is a real-valued function, defined over the sixdimensional "phase space"' $\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right)$.? $^?$
(b) Prove that
$$
\int W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right) d^3 p^{\prime}=\left|\psi\left(\mathbf{r}^{\prime}\right)\right|^2
$$
and that the expectation value of a function of the operator $\mathbf{r}$ in a normalized state is
$$
\langle f(\mathbf{r})\rangle=\iint f\left(\mathbf{r}^{\prime}\right) W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right) d^3 r^{\prime} d^2 p^{\prime}
$$
(c) Show that the Wigner distribution is normalized as
$$
\int W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right) d^3 r^{\prime} d^3 p^{\prime}=1
$$
(d) Show that the probability density $\rho\left(\mathbf{r}_0\right)$ at position $\mathbf{r}_0$ is obtained from the Wigner distribution with ${ }^8$
$$
\rho\left(\mathbf{r}_0\right) \rightarrow f(\mathbf{r})=\delta\left(\mathbf{r}-\mathbf{r}_0\right)
$$

Guilherme Barros

Numerade Educator

06:30

Problem 6

(a) Show that if $\phi\left(\mathbf{p}^{\prime}\right)$ is the momentum wave function representing the state, the Wigner distribution is
$$
W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right)=\frac{1}{(2 \pi \hbar)^3} \int \phi^*\left(\mathbf{p}^{\prime}-\mathbf{p}^{\prime \prime}\right) \phi\left(\mathbf{p}^{\prime}+\mathbf{p}^{\prime \prime}\right) e^{2 i \mathbf{p}^{\prime \prime} \cdot \mathbf{r}^{\prime} / \hbar} d^3 p^{\prime \prime}
$$
(b) Verify that
$$
\int W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right) d^\beta r^{\prime}=\left|\phi\left(\mathbf{p}^{\prime}\right)\right|^2
$$
and that the expectation value of a function of the operator $\mathbf{p}$ is
$$
\langle g(\mathbf{p})\rangle=\iint g\left(\mathbf{p}^{\prime}\right) W\left(\mathbf{r}^{\prime}, \mathbf{p}^{\prime}\right) d^3 r^{\prime} d^3 p^{\prime}
$$

Eduard Sanchez

Numerade Educator