Scattering

Abstract

Scattering experiments are discussed and the concept of the Scattering Cross Section is introduced. The Scattering Amplitude is introduced in terms of the asymptotic form of the wave function and its relation to the scattering cross section is derived. The integral scattering equation is introduced and the Born Approximation is discussed. Scattering in central potentials is considered and the partial wave analysis of the scattering amplitude is studied. Low-energy resonances in finite range central potentials are analyzed. Scattering in the Coulomb potential is considered and the Coulomb differential cross section is derived. The Formal Theory of Scattering is developed and the Lippmann-Schwinger equation is derived. The concept of the Scattering Matrix is introduced.

Problems and Exercises

20.1

Calculate the differential cross section for the central potentials $V(r)=V_0e^{-r/a}$ and $V(r)=V_0e^{-r^2/a^2}$ in the Born approximation.

20.2

Consider the scattering of a particle of mass m by a repulsive potential $V(r)=\Theta (a-r)V_0$.

(a) Calculate the phase shift at high energies $(\hbar k)^2>>2mV_0$.

(b) In the opposite limit $(ka)<<1$, show that the differential cross section has the form $\sigma (\theta )=A+B\cos \theta$ with $B<<A$.

(c) Suppose that $\xi \equiv \hbar ^2/mV_0a^2<<1$. What is the meaning of that in the case $ka<<1$? Calculate A and B in the limit $\xi \rightarrow 0,\,(ka)\rightarrow 0$.

(d) Calculate the phase shift for $\ell =0$ exactly.

20.3

Consider the scattering of particles from a periodic potential $V(\mathbf{{ r}})=V(\mathbf{{ r}}+\mathbf{{ a}})$. Write down the expression for the scattering amplitude in the Born approximation and show that it is nonzero for special values of the momentum transfer $\mathbf{{ q}}$.

20.4

Consider a particle of mass m and energy $E>0$ moving in the short-range central potential V(r). Show that the radial wave function satisfies the integral equation

$$R_{E,\ell }(r)\,=\,j_{\ell }(kr)\,-\frac{m}{2\hbar ^2\pi }\int _0^{\infty }dr'\,{r'}^2\,\mathcal{{G}}_{k,\ell }(r,r')\,V(r')\,R_{E,\ell }(r')\,,$$

where the radial Green’s function is defined through the equation

$$\left( \,\frac{1}{r^2}\frac{d}{dr}r^2\frac{d}{dr}\,-\frac{\ell (\ell +1)}{r^2}\,+\,k^2\,\right) \mathcal{{G}}_{k,\ell }(r,r')\,=\,-\frac{4\pi }{r^2}\,\delta (r-r')\,.$$

Prove that the solution to this equation is

$$\mathcal{{G}}_{k,\ell }(r,r')\,=\,\left\{ \begin{array}{cc} 4i\pi k\,j_{\ell }(kr)\,h_{\ell }(kr')\,&{}\,(r'>r)\\ \,&{}\,\\ 4i\pi k\,j_{\ell }(kr')\,h_{\ell }(kr)\,&{}\,(r>r') \end{array}\right. $$

where $h_{\ell }(x)=j_{\ell }(x)+in_{\ell }(x)$.

20.5

Consider the system of two electrons interacting through their mutual Coulomb repulsion $e^2/|\mathbf{{ r}}_1-\mathbf{{ r}}_2|$. Classify all the possible cases of total spin of the system and calculate the differential cross section for each case.

20.6

Two spin-0 bosons of masses $m_1$ and $m_2$ interact through the potential

$$V(\mathbf{{ r}}_1,\,\mathbf{{r}}_2)\,=\,\left\{ \begin{array}{cc} +\infty \,&{}\,|\mathbf{{ r}}_1-\mathbf{{ r}}_2|<a\\ \,&{}\,\\ 0\,&{}\,|\mathbf{{ r}}_1-\mathbf{{ r}}_2|>a \end{array}\right. $$

(a) Calculate the total scattering cross section at low energies.

(b) Do the same in the case that the two bosons are identical ($m_1=m_2$).

20.7

The potential

$$V\,=\,V_{ab}|a\rangle \langle b|+V_{ba}|b\rangle \langle a|\,.$$

Consider the Schwinger–Lippmann equation $|\psi \rangle =|\psi _0\rangle +G(E)V|\psi \rangle$ and determine the state $|\psi \rangle$.

20.8

Consider the potential

$$V(r)\,=\,-\frac{g^2}{(r^2+a^2)^2}\,.$$

(a) Calculate the scattering amplitude $f_k(\theta )$ in the Born approximation.

(b) Using (a) calculate the total cross section.

20.9

Consider a double slit cut in a very thin material on the (x, y)-plane and a beam of particles incident along the $\hat{z}$-axis. The situation can be modeled with a potential

$$V(x,y,z)\,=\,\left\{ \begin{array}{cc} V_0\delta (z)\,&{}\,{\textit{for}}\,\,\,\left\{ \begin{array}{cc} x\in [-b-a, -b+a]\,&{}\,-c<y<c\\ \,&{}\,\\ x\in [b-a,b+a]\,&{}\,-c<y<c \end{array}\right. \\ \,&{}\,\\ 0\,&{}\,{\textit{otherwise}} \end{array}\right. $$

where the centers of the slits are at the points $x=\pm b$ and their widths $2a<<b$. Calculate the differential scattering cross section in the Born approximation for incidence along the z-axis.

20.10

Consider scattering in a hard sphere ($V(r)=+\infty$ for $r\le a$) in the case $\ell =0$. Investigate the existence of resonances, defined by $\delta _0(E)=\pi /2$.