The Lippmann-Schwinger formalism can also be applied to a one-dimensional transmission-reflection problem with a finite-range potential, $V(x) \neq 0$ for $0<$ $|x|<a$ only.

(a) Suppose we have an incident wave coming from the left: $\langle x | \phi\rangle=e^{i k x} / \sqrt{2 \pi}$ How must we handle the singular $1 /\left(E-H_{0}\right)$ operator if we are to have a transmitted wave only for $x>a$ and a reflected wave and the original wave for $x<-a ?$ Is the $E \rightarrow E+i \varepsilon$ prescription still correct? Obtain an expression for the appropriate Green's function and write an integral equation for $\left\langle x | \psi^{(+)}\right\rangle$.

(b) Consider the special case of an attractive $\delta$ -function potential $$V=-\left(\frac{\gamma \hbar^{2}}{2 m}\right) \delta(x) \quad(\gamma>0)$$ Solve the integral equation to obtain the transmission and reflection amplitudes. Check your results with Gottfried $1966,$ p. 52.

(c) The one-dimensional $\delta$ -function potential with $\gamma>0$ admits one (and only one) bound state for any value of $\gamma$. Show that the transmission and reflection amplitudes you computed have bound-state poles at the expected positions when $k$ is regarded as a complex variable.

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Prove $$\sigma_{\mathrm{tot}} \simeq \frac{m^{2}}{\pi \hbar^{4}} \int d^{3} x \int d^{3} x^{\prime} V(r) V\left(r^{\prime}\right) \frac{\sin ^{2} k\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}{k^{2}\left|\mathbf{x}-\mathbf{x}^{\prime}\right|^{2}}$$

(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. $[\text { Note that } f(0)$ is real if the first-order Born approximation is used.]

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

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Consider a potential $V=0 \quad$ for $r>R, \quad V=V_{0}=$ constant $\quad$ for $r<R$ 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 $$\frac{d \sigma}{d \Omega}=A+B \cos \theta$$ Obtain an approximate expression for $B / A$.

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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.

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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.)

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Consider the scattering of a particle by an impenetrable sphere $$V(r)=\left\{\begin{array}{ll}

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 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{array}{l}

\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{array}$$.

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Use $\delta_{l}=\left.\Delta(b)\right|_{b=l / k}$ 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 $l(k \text { fixed ) for } l \gg k R, \text { where } R$ is the "range" of the potential. [The formula for $\Delta(b) \text { is given in }(6.5 .14)]$.

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(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 $r^{\prime}$.

(b) For spherically symmetrical potentials, the Lippmann-Schwinger equation can be 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 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}$$

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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 hard-sphere 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. $131-41,$ who discusses the analytic properties of the $D_{l}$ -function defined by $A_{l}=j_{l} / D_{l} .$ )

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A spinless particle is scattered by a time-dependent potential $$\mathcal{V}(\mathbf{r}, t)=V(\mathbf{r}) \cos \omega t$$ 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.

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

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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 problem to that of the Coulomb problem, 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 $\mathcal{D}_{m n}^{k}(\alpha \beta \gamma)$. How are the wave functions $\mathcal{D}_{m n}^{k}(\alpha \beta \gamma)$ related to Laguerre times spherical harmonics?

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