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HOMEWORK 1 TEXTBOOKS: Quantum Mechanics, Vol. 2, by Claude Cohen-Tannoudji, et al., Wiley-VCH. Quantum Mechanics with Basic Field Theory, by Bipin R. Desai, Cambridge. Quantum Mechanics, by Eugen Merzbacher, Hamilton. Modern Quantum Mechanics, by J. J. Sakurai, Addison-Wesley. due date: Tue., February 1, 2011 Problem 1.1 Show that, when a particle of mass $m_1$ collides elastically with a particle of mass $m_2$ that is initially at rest, all the recoil (mass $m_2$) particles are scattered in the forward hemisphere ($\theta \lt \frac{\pi}{2}$). Problem 1.2 Consider the potential \[ V(r) = \left\{ \begin{array}{ccc} V_0 & , & r \lt r_0 \\ 0 & , & r \gt r_0 \end{array} \right. \] Using the method of partial waves, show that for \[ |V_0| \ll E \ \ , \ \ \ \ kr_0 \ll 1 \] the differential cross section is isotropic and the total cross section is given by \[ \sigma = \frac{16\pi \mu^2 V_0^2 r_0^6}{9\hbar^4} \] Problem 1.3 Compute and make a polar plot of the differential scattering cross section for a perfectly rigid sphere of radius $r_0$ when \[ kr_0 = \frac{1}{2} \] using the first three partial waves ($\ell = 0,1,2$). Compute the total cross section using these three terms. Problem 1.4 Consider scattering by a repulsive $\delta$-shell potential \[ V(r) = \frac{\gamma \hbar^2}{2\mu} \ \delta (r-r_0) \] where $\gamma \gt 0$. Set up an equation that determines the $s$-wave phase shift $\delta_0$ as a function of $k$. Show that for large $\gamma$ your result resembles the corresponding result for a hard sphere as long as $\tan (kr_0)$ is not close to zero. What happens when $\tan (kr_0)$ is close to zero? Problem 1.5 Using the Born approximation for a Yukawa potential \[ V(r) = V_0\ \frac{e^{-\alpha r}}{r} \] calculate the first three phase shifts $\delta_\ell$ ($\ell = 0,1,2$) assuming they are small ($|\delta_\ell| \ll 1$). When the de Broglie wavelength is much longer than the range of the potential, show that $\delta_\ell$ is proportional to $k^{2\ell +1}$ and find the proportionality constant in each of the three cases $\ell = 0,1,2$. Problem 1.6 Calculate the elastic scattering and absorption cross sections for a potential of the form \[ V(r) = \left\{ \begin{array}{ccc} - V_0 (1+i\xi) & , & r \lt r_0 \\ 0 & , & r \gt r_0 \end{array} \right. \] where $V_0, \xi \gt 0$, in the low energy limit. You may assume that $\xi \ll 1$ and there is neither a resonance nor a Ramsauer-Townsend effect. Problem 1.7 Using the optical theorem or otherwise, show that the differential elastic cross section in the forward direction obeys the inequality \[ \left. \frac{d\sigma}{d\Omega} \right|_{\theta =0} \ge \left( \frac{k\sigma}{4\pi} \right)^2 \] where $\sigma$ is the total cross section.

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