Chapter 13, Scattering theory Video Solutions, The Physics of Quantum Mechanics

Problem 1

Show that the operators $\widetilde{\Omega}_{\pm}$defined by equation (13.23) obey
$$
H \widetilde{\Omega}_{\pm}=\widetilde{\Omega}_{\pm}\left(H_{\mathrm{K}} \pm \mathrm{i} \epsilon\right) \mp \mathrm{i} \epsilon
$$

Allison Stroman

Numerade Educator

01:38

Problem 2

Obtain the first- and second-order contributions to the S-matrix from the Feynman rules given in $\S 13.3$.

Ankit Gupta

Numerade Educator

04:57

Problem 3

Derive the Lippmann-Schwinger equation
$$
|\pm\rangle=|E\rangle+\frac{1}{E-H_{\mathrm{K}} \pm \mathrm{i} \epsilon} V|\pm\rangle
$$
where $|\pm\rangle$ are in and out states of energy $E$ and $|E\rangle$ is a free-particle state of the same energy. In the case that the potential $V=V_{0}|\chi\rangle\langle\chi|$ for some state $|\chi\rangle$ and constant $V_{0}$, solve the Lippmann-Schwinger equation to find $\langle\chi \mid \pm\rangle$.

Laszlo Zalavari

Numerade Educator

01:40

Problem 4

A certain central potential $V(r)$ falls as $r^{-n}$ at large distances. Show that the Born approximation to the total cross-section is finite if $n>2$. Is this a problem with the Born approximation?

Michael Twiton

Numerade Educator

04:57

Problem 5

Compute the differential cross-section in the Born approximation for the potential $V(\mathbf{r})=V_{0} \exp \left(-r^{2} / 2 r_{0}^{2}\right)$. For what energies is the Born approximation justified?

Hafiz Shahzaib

Numerade Educator

13:41

Problem 6

When an electron scatters off an atom, the atom may be excited (or even ionised). Consider an electron scattering off a hydrogen atom. The Hamiltonian may be written as $H=H_{0}+H_{1}$ where
$$
H_{0}=\frac{\hat{\mathbf{p}}_{1}^{2}}{2 M}-\frac{e^{2}}{4 \pi \epsilon_{0} r_{1}}+\frac{\hat{\mathbf{p}}_{2}^{2}}{2 m}
$$
is the Hamiltonian of the hydrogen atom (of mass $M$ ) whose electron is described by coordinate $\mathbf{r}_{1}$, together with the kinetic Hamiltonian of the scattering electron (of mass $m$ ), while
$$
H_{1}=\frac{e^{2}}{4 \pi \epsilon_{0}}\left(\frac{1}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|}-\frac{1}{r_{2}}\right)
$$
is the interaction of the scattering electron with the atom.
By using $H_{0}$ in the evolution operators, show that in the Born approximation the amplitude for a collision to scatter the electron from momentum $\mathbf{p}_{2}$ to $\mathbf{p}_{2}^{\prime}$ whilst exciting the atom from the state $|n, l, m\rangle$ to the state $\left|n^{\prime}, l^{\prime}, m^{\prime}\right\rangle$ is
$$
\begin{aligned}
&f\left(\mathbf{p}_{2} ; n, l, m \rightarrow \mathbf{p}_{2}^{\prime} ; n^{\prime}, l^{\prime}, m^{\prime}\right) \\
&=-\frac{4 \pi^{2} \hbar m}{(2 \pi \hbar)^{3}} \int \mathrm{d}^{3} \mathbf{r}_{1} \mathrm{~d}^{3} \mathbf{r}_{2} \mathrm{e}^{-\mathrm{i} \boldsymbol{q}_{2} \cdot \mathbf{r}_{2}}\left\langle n^{\prime}, l^{\prime}, m^{\prime} \mid \mathbf{r}_{1}\right\rangle\left\langle\mathbf{r}_{1} \mid n, l, m\right\rangle H_{1}\left(\mathbf{r}_{1}, \mathbf{r}_{2}\right)
\end{aligned}
$$
where $\mathbf{q}_{2}$ is the momentum transferred to the scattering electron. (Neglect the possibility that the two electrons exchange places, and the recoil of the hydrogen atom.)

Compute the differential cross-section for the $|1,0,0\rangle \rightarrow|2,0,0\rangle$ transition and show that when the energy of the scattering particle is very high it varies as $\operatorname{cosec}^{12}(\theta / 2)$. Hint: perform the $\mathrm{d}^{3} \mathbf{r}_{2}$ integral by including a factor $\mathrm{e}^{-\alpha r_{2}}$ and then let $\alpha \rightarrow 0$.

Robert Schnibbe

Numerade Educator

01:57

Problem 7

Use the optical theorem to show that the first Born approximation is not a valid description of forward scattering.

Keshav Singh

Numerade Educator

11:38

Problem 8

A particle scatters off a hard sphere, described by the potential
$$
V(\mathbf{r})= \begin{cases}\infty & \text { for }|\mathbf{r}| \leq a \\ 0 & \text { otherwise. }\end{cases}
$$
By considering the form of the radial wavefunction $u(r)$ in the region $r>$ $a$, show that the phase shifts are given by $\tan \delta_{l}=j_{l}(k a) / n_{l}(k a)$, where $k=\sqrt{2 m E} / \hbar$ and $j_{l}(k r)$ and $n_{l}(k r)$ are the two independent solutions (spherical Bessel functions) of the second-order radial equation
$$
\frac{1}{r^{2}} \frac{\mathrm{d}}{\mathrm{d} r}\left(r^{2} \frac{\mathrm{d}}{\mathrm{d} r} u(r)\right)=\left(\frac{l(l+1)}{r^{2}}-\frac{2 m E}{\hbar^{2}}\right) u(r)
$$
with $j_{l}(k r)$ regular at the origin.
In the limit $k r \rightarrow 0$, show that these functions behave as
$$
j_{l}(k r) \sim(k r)^{l} \quad n_{l}(k r) \sim 1 /(k r)^{l+1}
$$
In fact, the limiting values are $j_{l}(k r) \rightarrow(k r)^{l} /(2 l+1) ! !$ and $n_{l}(k r) \rightarrow$ $-(2 l-1) ! ! /(k r)^{l+1}$ where $(2 l+1) ! ! \equiv(2 l+1)(2 l-1)(2 l-3) \cdots 3 \cdot 1$ Use this to show that in the low-energy limit, the scattering is spherically symmetric and the total cross-section is four times the classical value.

Ameer Said

Numerade Educator

01:13

Problem 9

Show that in the Born approximation the phase shifts $\delta_{l}(E)$ for scattering off a spherical potential $V(r)$ are given by
$$
\delta_{l}(E) \simeq-\frac{2 m k}{\hbar^{2}} \int_{0}^{\infty} \mathrm{d} r r^{2} V(r)\left(j_{l}(k r)\right)^{2}
$$
When is the Born approximation valid? Hint: with $q=2 k \sin (\theta / 2)$ we have
$$
\frac{\sin (q r)}{q r}=\sum_{l=0}^{\infty}(2 l+1) j_{l}(k r)^{2} \mathrm{P}_{l}(\cos \theta)
$$

Raj Bala

Numerade Educator

05:56

Problem 10

Two $\alpha$ particles collide. Show that when the $\alpha$ particles initially have equal and opposite momenta, the differential cross-section is
$$
\frac{\mathrm{d} \sigma}{\mathrm{d} \Omega}=|f(\theta)+f(\theta-\pi)|^{2}
$$
Using the formula for $f(\theta)$ in terms of partial waves, show that the differential cross-section at $\theta=\pi / 2$ is twice what would be expected had the $\alpha$ particles been distinguishable.
A moving electron crashes into an electron that is initially at rest. Assuming both electrons are in the same spin state, show that the differential cross-section falls to zero at $\theta=\pi / 4$.

Suzanne W.

Numerade Educator