Chapter 6, Scattering Theory Video Solutions, Modern Quantum Mechanics

Problem 1

Consider scattering in one dimension $x$ from a potential $V(x)$ localized near $x=0$. The initial state is a plane wave coming from the left, that is $\phi(x) \equiv\langle x \mid i\rangle=e^{i k x} / \sqrt{2 \pi}$.
a. Find the scattering Green's function $G\left(x, x^{\prime}\right)$, defined in one dimension analogously with (6.37), for $G_{+}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)$.
b. For the case of an attractive $\delta$-function potential $V(x)=-\gamma \hbar^{2} \delta(x) / 2 m$, with $\gamma>0$, use the Lippman-Schwinger equation to find the outgoing wave function $\psi(x) \equiv\left\langle x \mid \psi^{(+)}\right\rangle .$
c. Determine the transmission and reflection coefficients $T(k)$ and $R(k)$, defined as $\psi(x)=T(k) \phi(x)$ for $x>0 \quad$ and $\quad \psi(x)=\phi(x)+R(k) \frac{e^{-i k x}}{\sqrt{2 \pi}}$ for $x<0 .$
Show that $|T|^{2}+|R|^{2}=1$, as must be the case.
d. Confirm that you get the same result by matching right and left going waves on the left with a right going wave on the right at $x=0$.
e. We showed in Problem $2.29$ that this potential has one, and only one, bound state. Show that your results for $T(k)$ and $R(k)$ have bound-state poles at the expected positions when $k$ is treated as a complex variable.

Ameer Said

Numerade Educator

08:10

Problem 2

Prove
$$
\sigma_{\text {tot }} \simeq \frac{m^{2}}{\pi \hbar^{4}} \int d^{3} x \int d^{3} x^{\prime} V(r) V\left(r^{\prime}\right) \frac{\sin ^{2} k\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}{k^{2}\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{2}}
$$
in each of the following ways.
a. By integrating the differential cross section computed using the first-order Born approximation.
b. By applying the optical theorem to the forward-scattering amplitude in the second-order Born approximation. [Note that $f(0)$ is real if the first-order Born approximation is used.]

Carson Merrill

Numerade Educator

02:51

Problem 3

Estimate the radius of the ${ }^{40}$ Ca nucleus from the data in Figure $6.6$ and compare to that expected from the empirical value $\approx 1.4 A^{1 / 3}$ fin, where $A$ is the nuclear mass number. Check the validity of using the first-order Born approximation for these data.

Declan Nell

Numerade Educator

15:48

Problem 4

Consider a potential
$$
V=0 \text { for } r>R, \quad V=V_{0}=\text { constant } \quad \text { for } \quad r<R,
$$
where $V_{0}$ may be positive or negative. Using the method of partial waves, show that for $\left|V_{0}\right| \ll E=\hbar^{2} k^{2} / 2 m$ and $k R \ll 1$ the differential cross section is isotropic and that the total cross section is given by
$$
\sigma_{\mathrm{tot}}=\left(\frac{16 \pi}{9}\right) \frac{m^{2} V_{0}^{2} R^{6}}{\hbar^{4}}
$$
Suppose the energy is raised slightly. Show that the angular distribution can then be written as
$$
\frac{d \sigma}{d \Omega}=A+B \cos \theta \text {. }
$$
Obtain an approximate expression for $B / A$.

Mahnoor Amin

Numerade Educator

05:33

Problem 5

A spinless particle is scattered by a weak Yukawa potential
$$
V=\frac{V_{0} e^{-\mu r}}{\mu r}
$$
where $\mu>0$ but $V_{0}$ can be positive or negative. It was shown in the text that the first-order Born amplitude is given by
$$
f^{(1)}(\theta)=-\frac{2 m V_{0}}{\hbar^{2} \mu} \frac{1}{\left[2 k^{2}(1-\cos \theta)+\mu^{2}\right]} .
$$
a. Using $f^{(1)}(\theta)$ and assuming $\left|\delta_{l}\right| \ll 1$, obtain an expression for $\delta_{l}$ in terms of a Legendre function of the second kind,
$$
Q_{l}(\zeta)=\frac{1}{2} \int_{-1}^{1} \frac{P_{l}\left(\zeta^{\prime}\right)}{\zeta-\zeta^{\prime}} d \zeta^{\prime}
$$
b. Use the expansion formula
$$
\begin{aligned}
Q_{l}(\zeta)=& \frac{l !}{1 \cdot 3 \cdot 5 \cdots(2 l+1)} \\
& \times\left\{\frac{1}{\zeta l+1}+\frac{(l+1)(l+2)}{2(2 l+3)} \frac{1}{\zeta^{l+3}}\right.\\
&\left.+\frac{(l+1)(l+2)(l+3)(l+4)}{2 \cdot 4 \cdot(2 l+3)(2 l+5)} \frac{1}{\zeta l+5}+\cdots\right\} \quad(|\zeta|>1)
\end{aligned}
$$
to prove each assertion.
(i) $\delta_{l}$ is negative (positive) when the potential is repulsive (attractive).
(ii) When the de Broglie wavelength is much longer than the range of the potential, $\delta_{l}$ is proportional to $k^{2 l+1}$. Find the proportionality constant.

Mahnoor Amin

Numerade Educator

02:01

Problem 6

Check explicitly the $x-p_{x}$ uncertainty relation for the ground state of a particle confined inside a hard sphere: $V=\infty$ for $r>a, V=0$ for $r<a$. (Hint: Take advantage of spherical symmetry.)

João Gabriel Alencar Caribé

Numerade Educator

01:13

Problem 7

Consider the scattering of a particle by an impenetrable sphere
$$
V(r)=\left\{\begin{array}{cl}
0 & \text { for } r>a \\
\infty & \text { for } r<a .
\end{array}\right.
$$
a. Derive an expression for the $s$-wave $(l=0)$ phase shift. (You need not know the detailed properties of the spherical Bessel functions to be able to do this simple problem!)
b. What is the total cross section $\sigma\left[\sigma=\int(d \sigma / d \Omega) d \Omega\right]$ in the extreme low-energy limit $k \rightarrow 0$ ? Compare your answer with the geometric cross section $\pi a^{2}$. You may assume without proof:
$$
\begin{aligned}
\frac{d \sigma}{d \Omega} &=|f(\theta)|^{2} \\
f(\theta) &=\left(\frac{1}{k}\right) \sum_{l=0}^{\infty}(2 l+1) e^{i \delta_{l}} \sin \delta_{l} P_{l}(\cos \theta)
\end{aligned}
$$

Raj Bala

Numerade Educator

05:20

Problem 8

Use $\delta_{l}=\left.\Delta(b)\right|_{b=l l}$ to obtain the phase shift $\delta_{l}$ for scattering at high energies by (a) the Gaussian potential, $V=V_{0} \exp \left(-r^{2} / a^{2}\right)$, and (b) the Yukawa potential, $V=$ $V_{0} \exp (-\mu r) / \mu r$. Verify the assertion that $\delta_{l}$ goes to zero very rapidly with increasing $l$ ( $k$ fixed) for $l \gg k R$, where $R$ is the "range" of the potential. [The formula for $\Delta(b)$ is given in (6.168).] It is useful to realize that
$$
\int_{1}^{\infty} \frac{e^{-a x}}{\left(x^{2}-1\right)^{1 / 2}} d x=K_{0}(x)
$$
where $K_{0}(x)$ is the modified Bessel function of the second kind, of order zero.

Suzanne W.

Numerade Educator

03:14

Problem 9

a. Prove
$$
\frac{\hbar^{2}}{2 m}\left\langle\mathbf{x}\left|\frac{1}{E-H_{0}+i \varepsilon}\right| \mathbf{x}^{\prime}\right\rangle=-i k \sum_{l} \sum_{m} Y_{l}^{m}(\hat{\mathbf{r}}) Y_{l}^{m^{*}}\left(\hat{\mathbf{r}}^{\prime}\right) j_{l}\left(k r_{<}\right) h_{l}^{(1)}\left(k r_{>}\right)
$$
where $r_{<}\left(r_{>}\right)$stands for the smaller (larger) of $r$ and $r^{\prime} .$ Problems written for spherical waves:
$$
|E l m(+)\rangle=|E l m\rangle+\frac{1}{E-H_{0}+i \varepsilon} V|E l m(+)\rangle .
$$
Using (a), show that this equation, written in the $\mathbf{x}$-representation, leads to an equation for the radial function, $A_{l}(k ; r)$, as follows:
$$
\begin{aligned}
A_{l}(k ; r)=& j_{l}(k r)-\frac{2 m i k}{\hbar^{2}} \\
& \times \int_{0}^{\infty} j_{l}\left(k r_{<}\right) h_{l}^{(1)}\left(k r_{>}\right) V\left(r^{\prime}\right) A_{l}\left(k ; r^{\prime}\right) r^{\prime 2} d r^{\prime} .
\end{aligned}
$$
By taking $r$ very large, also obtain
$$
\begin{aligned}
f_{l}(k) &=e^{i \delta_{l}} \frac{\sin \delta_{l}}{k} \\
&=-\left(\frac{2 m}{\hbar^{2}}\right) \int_{0}^{\infty} j_{l}(k r) A_{l}(k ; r) V(r) r^{2} d r .
\end{aligned}
$$

Narayan Hari

Numerade Educator

11:38

Problem 10

Consider scattering by a repulsive $\delta$-shell potential:
$$
\left(\frac{2 m}{\hbar^{2}}\right) V(r)=\gamma \delta(r-R) \quad(\gamma>0) .
$$
a. Set up an equation that determines the $s$-wave phase shift $\delta_{0}$ as a function of $k\left(E=\hbar^{2} k^{2} / 2 m\right) .$
b. Assume now that $\gamma$ is very large,
$$
\gamma \gg \frac{1}{R}, k .
$$
Show that if $\tan k R$ is not close to zero, the $s$-wave phase shift resembles the hardsphere result discussed in the text. Show also that for tan $k R$ close to (but not exactly equal to) zero, resonance behavior is possible; that is, cot $\delta_{0}$ goes through zero from the positive side as $k$ increases. Determine approximately the positions of the resonances keeping terms of order $1 / \gamma ;$ compare them with the bound-state energies for a particle confined inside a spherical wall of the same radius,
$$
V=0, \quad r<R ; \quad V=\infty, \quad r>R .
$$
Also obtain an approximate expression for the resonance width $\Gamma$ defined by
$$
\Gamma=\frac{-2}{\left.\left[d\left(\cot \delta_{0}\right) / d E\right]\right|_{E=E_{r}}}
$$
and notice, in particular, that the resonances become extremely sharp as $\gamma$ becomes large. (Note: For a different, more sophisticated approach to this problem see Gottfried (1966), pp. 131-141, who discusses the analytic properties of the $D_{l}$-function defined by $A_{l}=j_{l} / D_{l} .$ )

Ameer Said

Numerade Educator

01:33

Problem 11

A spinless particle is scattered by a time-dependent potential
$$
\mathscr{V}(\mathbf{r}, t)=V(\mathbf{r}) \cos \omega t .
$$
Show that if the potential is treated to first order in the transition amplitude, the energy of the scattered particle is increased or decreased by $\hbar \omega$. Obtain $d \sigma / d \Omega$. Discuss qualitatively what happens if the higher-order terms are taken into account.

Ankur S

Numerade Educator

13:12

Problem 12

Show that the differential cross section for the elastic scattering of a fast positron by the ground state of the hydrogen atom is given by
$$
\frac{d \sigma}{d \Omega}=\left(\frac{4 m^{2} e^{4}}{\hbar^{4} q^{4}}\right)\left\{1-\frac{16}{\left[4+\left(q a_{0}\right)^{2}\right]^{2}}\right\}^{2}
$$

Dr. Rajveer Singh

Numerade Educator

05:03

Problem 13

Write a computer program to reproduce Figure $6.15$. Also include a plot showing the relative sizes of the energy, well depth, and effective potential, that is, the analogue of Figure $6.14$ for this problem.

Darren Wilson

Numerade Educator

05:30

Problem 14

Let the energy of a particle moving in a central field be $E\left(J_{1} J_{2} J_{3}\right)$, where $\left(J_{1}, J_{2}, J_{3}\right)$ are the three action variables. How does the functional form of $E$ specialize for the Coulomb potential? Using the recipe of the action-angle method, compare the degeneracy of the central field and the Coulomb problems and relate it to the vector $\mathbf{A}$.
If the Hamiltonian is
$$
H=\frac{p^{2}}{2 \mu}+V(r)+F\left(\mathbf{A}^{2}\right)
$$
how are these statements changed?
Describe the corresponding degeneracies of the central field and Coulomb problems in quantum theory in terms of the usual quantum numbers $(n, l, m)$ and also in terms of the quantum numbers $(k, m, n)$. Here the second set, $(k, m, n)$, labels the wave functions $\mathscr{D}_{m n}^{k}(\alpha \beta \gamma)$.

How are the wave functions $\mathscr{D}_{\operatorname{mn}}^{k}(\alpha \beta \gamma)$ related to Laguerre times spherical harmonics?

Eduard Sanchez

Numerade Educator