PHY577 Problem Set 1. Due September 26, 2005

1. Calculate the scattering solutions for a system with a separable but nonlocal s-wave potential
   

   $$H_0 = \frac{p^2_{\rm op}}{2m}$$

   $$\langle \vec r \vert V \vert \vec r' \rangle = \lambda u(r) u(r')$$ (1)

   
   
   where $\lambda$ and $u(r)$ are real. Follow these steps

   a.

   Show that
   

   $$\langle \vec k \vert V \vert \vec k'\rangle = \lambda \tilde u(k) \tilde u(k')$$ (2)

   
   

   where $\vert\vec k\rangle = \vert\Phi_{\vec k}\rangle$ are free particle states, and
   

   $$\tilde u(k) = \int d^3r e^{-i \vec k \cdot \vec r} u(r) \,.$$ (3)

   
   

   Be sure to show that $\tilde u$ does not depend on the direction of $\vec k$. You can assume for the rest of the problem that the Fourier transform is well behaved for real $\vec k$.

   b.

   Solve the Lippmann-Schwinger equation for the matrix elements of the in states
   

   $$\langle \vec k' \vert \psi_{\vec k} \rangle \equiv \tilde \psi_{\vec k}(\vec k')$$ (4)

   
   

   where the superscript $+$ is dropped here for brevity. Your result should be
   

   $$\tilde \psi_{\vec k}(\vec k') = (2\pi)^3 \delta^3(\vec k' - \vec k) + \frac{2m\lambda}{\hbar^2} \... ...pi)^3} \frac{\tilde u^2(k'')} {k^2-k''^2+i\eta} r \vert V \vert \vec r' \rangle$$ (5)

   
   

   where $\lambda$ and $u(r)$ are real. Follow these steps

   a.

   Show that
   

   $$\langle \vec k \vert V \vert \vec k'\rangle = \lambda \tilde u(k) \tilde u(k')$$ (6)

   
   

   where $\vert\vec k\rangle = \vert\Phi_{\vec k}\rangle$ are free particle states, and
   

   $$\tilde u(k) = \int d^3r e^{-i \vec k \cdot \vec r} u(r) \,.$$ (7)

   
   

   Be sure to show that $\tilde u$ does not depend on the direction of $\vec k$. You can assume for the rest of the problem that the Fourier transform is well behaved for real $\vec k$.

   b.

   Solve the Lippmann-Schwinger equation for the matrix elements of the in states
   

   $$\langle \vec k' \vert \psi_{\vec k} \rangle \equiv \tilde \psi_{\vec k}(\vec k')$$ (8)

   
   

   where the superscript $+$ is dropped here for brevity. Your result should be
   

   $$\tilde \psi_{\vec k}(\vec k') = (2\pi)^3 \delta^3(\vec k' - \vec k) + \frac{2m\lambda}{\hbar^2} \... ...{\tilde u^2(k'')} \right ]^{-1} \frac{\tilde u(k) \tilde u(k')}{k^2-k'^2+i\eta}$$

   
   

   c.

   Use the result of part b to find an expression for the scattering amplitude. Show that your result satisfies the optical theorem
   

   $$\sigma = \frac{4\pi}{k} {\rm Im} f(0)$$ (9)

   
   

   where $f(0)$ is the scattering amplitude in the forward direction.

   d.

   For the special case where $u(r)$ has a Yukawa form
   

   $$u(r) = \frac{e^{-\beta r}}{r}$$ (10)

   
   

   calculate the scattering amplitude. Your result should be
   

   $$f = \left [ -\frac{(k^2+\beta^2)^2\hbar^2}{8\pi \lambda m} - ik-\frac{\beta}{2}+\frac{k^2}{2\beta} \right ]^{-1}$$ (11)

   
   

   e.

   The effective range expansion for low energy scattering is given by expanding
   

   $$k \cot \delta_0 = -\frac{1}{a} + \frac{1}{2} r_0 k^2 + ... \,,$$ (12)

   
   

   where the phase shift is given by $k f =e^{i\delta_0}\sin\delta_0$. Find expressions for the scattering length $a$ and the effective range $r_0$ of this potential.

    $$

   Most targets are composite systems that can be excited by the projectile giving inelastic scattering. We begin with a noninteracting Hamiltonian
   

   $$H_0 = \frac{p^2}{2m} + H_T$$ (13)

   
   

   where $H_T$ describes the target. Assume that the target has just two states $\vert\rangle$ and $\vert 1\rangle$ with
   

   $$H_T \vert\rangle = \epsilon_0 \vert\rangle$$

   $$H_T \vert 1\rangle = \epsilon_1 \vert 1\rangle \,,$$ (14)

   
   

   and $\epsilon_1 > \epsilon_0$. The eigenstates of $H_0$ can then be written as
   

   $$H_0 \vert n\vec k\rangle = \left [ \frac{\hbar^2 k^2}{2m} + \epsilon_n\right ] \vert n\vec k\rangle \,.$$ (15)

   
   

   where, as usual,
   

   $$\langle n \vec r \vert m \vec k \rangle = e^{i \vec k \cdot \vec r} \delta_{nm} \,.$$ (16)

   
   

   Take the interaction between the projectile and the target to be
   

   $$\langle n \vec r \vert V \vert m \vec r' \rangle = \lambda u... ...\left [ \delta_{m1}\delta_{n0}+\delta_{m0}\delta_{n1} \right ]$$ (17)

   
   

   where $u(r)$ is real with a well behaved Fourier transform. Initially the target is in its ground state, and the projectile has incident energy $\hbar^2 k^2/2m$. We define $\kappa$ such that
   

   $$\frac{\hbar^2 \kappa^2}{2m} = \epsilon_1-\epsilon_0$$ (18)

   
   

   a.

   Solve the Lippmann-Schwinger equation for the matrix elements of the states
   

   $$\langle n \vec k'\vert\psi_{\vec k}\rangle \equiv \tilde \psi_{\vec k}(n,\vec k')$$ (19)

   
   

   where the superscript $+$ is dropped for brevity. Your result should be
   

   $$\tilde \psi_{\vec k}(0,\vec k') = (2\pi)^3 \delta^3(\vec k'-\vec k) + \frac{2m\lambda}{\hbar^2} \fr... ...a(k,\kappa)}{1-\gamma(k,0)\gamma(k,\kappa)} \frac{\tilde u(k')}{k^2-k'^2+i\eta}$$

   $$\tilde \psi_{\vec k}(1,\vec k') = \frac{2m\lambda}{\hbar^2} \frac{\tilde u(k)}{1-\gamma(k,0)\gamma(k,\kappa)} \frac{\tilde u(k')}{k^2-\kappa^2-k'^2+i\eta} \,.$$ (20)

   
   

   with
   

   $$\gamma(k,\kappa) =\frac{2m\lambda}{\hbar^2} \int \frac{d^3k'}{(2\pi)^3} \frac{\tilde u^2(k')}{k^2-\kappa^2-k'^2+i\eta} \,.$$ (21)

   
   

   b.

   Calculate the cross section, $\sigma _0$, for scattering with the target left in its ground state, and the cross section, $\sigma _1$, for scattering with the target left in its excited state. Verify that the optical theorem is satisfied, i.e.
   

   $$\frac{4\pi}{k} {\rm Im} f_{0\vec k, 0 \vec k} = -\frac{2}{v \hbar} {\rm Im} T_{0\vec k,0\vec k} = \sigma_0+\sigma_1$$ (22)

   
   

   here $v=\hbar k/m$.

   c.

   For the case where
   

   $$u(r) = \frac{e^{-\beta r}}{r}$$ (23)

   
   

   graph $\sigma_0 \beta^2$ and $\sigma_1 \beta^2$ as a function of $Em/\hbar^2 \beta^2$, where $E = \hbar^2 k^2/2m$ is the incident particle's kinetic energy. Take the values of the other parameters to be
   

   $$\kappa^2 = 2 \beta^2$$

   $$\lambda = \frac{1}{10} \frac{\hbar^2 \beta^3}{m}$$ (24)

   
   

   Use a log scale for the cross section r' &=& u(r) u(r') where $\lambda$ and $u(r)$ are real. Follow these steps

   a.

   Show that
   

   $$\langle \vec k \vert V \vert \vec k'\rangle = \lambda \tilde u(k) \tilde u(k')$$ (25)

   
   

   where $\vert\vec k\rangle = \vert\Phi_{\vec k}\rangle$ are free particle states, and
   

   $$\tilde u(k) = \int d^3r e^{-i \vec k \cdot \vec r} u(r) \,.$$ (26)

   
   

   Be sure to show that $\tilde u$ does not depend on the direction of $\vec k$. You can assume for the rest of the problem that the Fourier transform is well behaved for real $\vec k$.

   b.

   Solve the Lippmann-Schwinger equation for the matrix elements of the in states
   

   $$\langle \vec k' \vert \psi_{\vec k} \rangle \equiv \tilde \psi_{\vec k}(\vec k')$$ (27)

   
   

   where the superscript $+$ is dropped here for brevity. Your result should be
   

   $$\tilde \psi_{\vec k}(\vec k') = (2\pi)^3 \delta^3(\vec k' - \vec k) + \frac{2m\lambda}{\hbar^2} \... ...{\tilde u^2(k'')} \right ]^{-1} \frac{\tilde u(k) \tilde u(k')}{k^2-k'^2+i\eta}$$

   
   

   c.

   Use the result of part b to find an expression for the scattering amplitude. Show that your result satisfies the optical theorem
   

   $$\sigma = \frac{4\pi}{k} {\rm Im} f(0)$$ (28)

   
   

   where $f(0)$ is the scattering amplitude in the forward direction.

   d.

   For the special case where $u(r)$ has a Yukawa form
   

   $$u(r) = \frac{e^{-\beta r}}{r}$$ (29)

   
   

   calculate the scattering amplitude. Your result should be
   

   $$f = \left [ -\frac{(k^2+\beta^2)^2\hbar^2}{8\pi \lambda m} - ik-\frac{\beta}{2}+\frac{k^2}{2\beta} \right ]^{-1}$$ (30)

   
   

   e.

   The effective range expansion for low energy scattering is given by expanding
   

   $$k \cot \delta_0 = -\frac{1}{a} + \frac{1}{2} r_0 k^2 + ... \,,$$ (31)

   
   

   where the phase shift is given by $k f =e^{i\delta_0}\sin\delta_0$. Find expressions for the scattering length $a$ and the effective range $r_0$ of this potential.

    ${}$

   Most targets are composite systems that can be excited by the projectile giving inelastic scattering. We begin with a noninteracting Hamiltonian
   

   $$H_0 = \frac{p^2}{2m} + H_T$$ (32)

   
   

   where $H_T$ describes the target. Assume that the target has just two states $\vert\rangle$ and $\vert 1\rangle$ with
   

   $$H_T \vert\rangle = \epsilon_0 \vert\rangle$$

   $$H_T \vert 1\rangle = \epsilon_1 \vert 1\rangle \,,$$ (33)

   
   

   and $\epsilon_1 > \epsilon_0$. The eigenstates of $H_0$ can then be written as
   

   $$H_0 \vert n\vec k\rangle = \left [ \frac{\hbar^2 k^2}{2m} + \epsilon_n\right ] \vert n\vec k\rangle \,.$$ (34)

   
   

   where, as usual,
   

   $$\langle n \vec r \vert m \vec k \rangle = e^{i \vec k \cdot \vec r} \delta_{nm} \,.$$ (35)

   
   

   Take the interaction between the projectile and the target to be
   

   $$\langle n \vec r \vert V \vert m \vec r' \rangle = \lambda u... ...\left [ \delta_{m1}\delta_{n0}+\delta_{m0}\delta_{n1} \right ]$$ (36)

   
   

   where $u(r)$ is real with a well behaved Fourier transform. Initially the target is in its ground state, and the projectile has incident energy $\hbar^2 k^2/2m$. We define $\kappa$ such that
   

   $$\frac{\hbar^2 \kappa^2}{2m} = \epsilon_1-\epsilon_0$$ (37)

   
   

   a.

   Solve the Lippmann-Schwinger equation for the matrix elements of the states
   

   $$\langle n \vec k'\vert\psi_{\vec k}\rangle \equiv \tilde \psi_{\vec k}(n,\vec k')$$ (38)

   
   

   where the superscript $+$ is dropped for brevity. Your result should be
   

   $$\tilde \psi_{\vec k}(0,\vec k') = (2\pi)^3 \delta^3(\vec k'-\vec k) + \frac{2m\lambda}{\hbar^2} \fr... ...a(k,\kappa)}{1-\gamma(k,0)\gamma(k,\kappa)} \frac{\tilde u(k')}{k^2-k'^2+i\eta}$$

   $$\tilde \psi_{\vec k}(1,\vec k') = \frac{2m\lambda}{\hbar^2} \frac{\tilde u(k)}{1-\gamma(k,0)\gamma(k,\kappa)} \frac{\tilde u(k')}{k^2-\kappa^2-k'^2+i\eta} \,.$$ (39)

   
   

   with
   

   $$\gamma(k,\kappa) =\frac{2m\lambda}{\hbar^2} \int \frac{d^3k'}{(2\pi)^3} \frac{\tilde u^2(k')}{k^2-\kappa^2-k'^2+i\eta} \,.$$ (40)

   
   

   b.

   Calculate the cross section, $\sigma _0$, for scattering with the target left in its ground state, and the cross section, $\sigma _1$, for scattering with the target left in its excited state. Verify that the optical theorem is satisfied, i.e.
   

   $$\frac{4\pi}{k} {\rm Im} f_{0\vec k, 0 \vec k} = -\frac{2}{v \hbar} {\rm Im} T_{0\vec k,0\vec k} = \sigma_0+\sigma_1$$ (41)

   
   

   here $v=\hbar k/m$.

   c.

   For the case where
   

   $$u(r) = \frac{e^{-\beta r}}{r}$$ (42)

   
   

   graph $\sigma_0 \beta^2$ and $\sigma_1 \beta^2$ as a function of $Em/\hbar^2 \beta^2$, where $E = \hbar^2 k^2/2m$ is the incident particle's kinetic energy. Take the values of the other parameters to be
   

   $$\kappa^2 = 2 \beta^2$$

   $$\lambda = \frac{1}{10} \frac{\hbar^2 \beta^3}{m}$$ (43)

   
   

   . This is equivalent to taking natural units with $\hbar = m =\beta =1$, with numerical values, $\kappa=\sqrt{2}$, and $\lambda = 0.1$.

   My graphs are shown in figures 1 and 2.

   Figure 1: The cross sections for the parameters in the problem.

   

   Figure 2: A comparison between the total cross section calculated by adding $\sigma _0$ and $\sigma _1$, and the result from using the optical theorem. The optical theorem values were multiplied by $0.1$ to shift the graphs. Otherwise the two curves coincide.

   

    ${}$

   Baym Chapter 9, Problem 2. In this problem, the potential given should be viewed as an effective potential for the Born approximation (i.e. it is the T matrix defined in class), and the true potential when answering part b something like two short range hard sphere potentials, since a repulsive delta function potential does not scatter anything.

    ${}$

   Work through this problem, but do not turn it in.

   Calculate an analytic expression for the phase shifts $\delta_\ell(k)$ for the scattering of a particle of mass $m$ and energy $\hbar^2 k^2/2m$ for the hard sphere potential
   

   $$v(r) = \left \{ \begin{array}{ll} \infty & r < a\\ 0 & r > a \\ \end{array}\right . \,.$$ (44)

   
   

   Give a rough estimate of the maximum $\ell$ value needed in the cross section sums so that higher values of the phase shifts can be ignored. Calculate this value using the properties of the spherical Bessel functions, but also give a sentence or two semiclassical argument involving the maximum classical impact parameter and the corresponding angular momentum.

   Show using classical mechanics that the classical differential cross section takes the constant value $a^2/4$.

   Plot the differential cross section divided by the classical differential cross section as a function of the angle between the indident beam and the detector $\theta$ for energies 0.01, 1.0, and 10.0 $\hbar^2/2ma^2$. Use a log scale for the cross section axis so that you can plot the results for all three cases on the same graph.

   Plot the total cross section divided by the projected area as a function of incident energy divided by $\hbar^2/2ma^2$ for the range of energies from 0.01 to 10.0 $\hbar^2/2ma^2$. Explain why the large energy cross section, which corresponds to the classical limit, is twice the projected area. Your plot of the high energy differential cross section may help you understand this.