S-wave phase shift for quantum mechanical scattering

[EightBells]

Homework Statement

Consider the spherically symmetric potential energy $$\frac {2\mu V\left( r \right)} {\hbar^2} = \gamma \delta \left(r-a \right)$$ where $\gamma$ is a constant and $\delta \left( r-a \right)$ is a Dirac delta function that vanishes everywhere except on the spherical surface specified by $r=a$.
a.) Show that the S-wave phase shift $\delta_0$ for scattering from this potential satisfies the equation $$\tan \left( ka + \delta_0 \right) = \frac {\tan ka} {1+\left( \frac \gamma k \right) \tan ka}$$
b.) Evaluate the phase shift in the low-energy limit and show that the total cross section for S-wave scattering is $$ \sigma \cong 4\pi a^2 \left( \frac {\gamma a} {1+\gamma a} \right)^2$$

Relevant Equations

$$\tan \left( ka + \delta_0 \right) = \frac {\tan ka} {1+\left( \frac \gamma k \right) \tan ka}$$
$$k = \sqrt{\frac {2mE} {\hbar^2}}$$

a.) The potential is a delta function, so $V \left( r \right) = \frac {\hbar^2} {2\mu} \gamma \delta \left(r-a \right)$, therefore $V \left( r \right) = \frac {\hbar^2} {2\mu} \gamma$ at $r=a$, and $V \left( r \right) = 0$ otherwise. I've tried a few different approaches:

1.) In examples of hard sphere scattering which are easy to find online, the equation $\frac {-\hbar^2} {2m} \frac {d^2 u \left(r \right)} {dr^2} = Eu\left( r \right)$ is used for outside the hard sphere, $r \gt a$. I'm not sure how this would apply to a delta function potential however, so I got stuck using this approach.

2.)I found that for a spherically symmetric potential, the incoming and outgoing waves would accumulate the phase difference $e^{i\delta_l}$. So for $\Psi_1 = Ae^{ikr}+Be^{-ikr}$ describing the left side of the potential ($r \lt a$) and $\Psi_2 = Ce^{ikr}$, I could try to use the boundary conditions (not sure what they would be though, maybe $\Psi_1 \left(a \right) = \Psi_2 \left( a \right)$ and $\Psi_1' \left( a \right) = \Psi_2' \left( a \right)$ ?) to solve for the phase difference by looking at the ratio of coefficients, maybe $A/B$ or $A/C$?

3.) I also found an equation $a_l \left(k \right) = \frac {e^{i\delta_l}} k \sin \delta_l$ where $a_l \left(k \right)$ is a coefficient that depends on the value of the energy. I'm not sure how to determine that coefficient, so this was a dead end too.

b.) I've seen the equation $\sigma = \frac {4\pi} {k^2} \sin^2 \delta_0$, so if this is the right equation I could just plug in $\delta_0$ once I've found it in part a, and check that it matches the solution given in the problem? Does this address the low-energy limit as specified in the problem?

 

[DrClaude]

1.) In examples of hard sphere scattering which are easy to find online, the equation $\frac {-\hbar^2} {2m} \frac {d^2 u \left(r \right)} {dr^2} = Eu\left( r \right)$ is used for outside the hard sphere, $r \gt a$. I'm not sure how this would apply to a delta function potential however, so I got stuck using this approach.

This is the right approach.

What would be the solution $u(r)$ if there was no potential? What do you expect to find for $u(r \gg a)$?

 

[nrqed]

Homework Statement:: Consider the spherically symmetric potential energy $$\frac {2\mu V\left( r \right)} {\hbar^2} = \gamma \delta \left(r-a \right)$$ where $\gamma$ is a constant and $\delta \left( r-a \right)$ is a Dirac delta function that vanishes everywhere except on the spherical surface specified by $r=a$.
a.) Show that the S-wave phase shift $\delta_0$ for scattering from this potential satisfies the equation $$\tan \left( ka + \delta_0 \right) = \frac {\tan ka} {1+\left( \frac \gamma k \right) \tan ka}$$
b.) Evaluate the phase shift in the low-energy limit and show that the total cross section for S-wave scattering is $$ \sigma \cong 4\pi a^2 \left( \frac {\gamma a} {1+\gamma a} \right)^2$$
Relevant Equations:: $$\tan \left( ka + \delta_0 \right) = \frac {\tan ka} {1+\left( \frac \gamma k \right) \tan ka}$$
$$k = \sqrt{\frac {2mE} {\hbar^2}}$$

a.) The potential is a delta function, so $V \left( r \right) = \frac {\hbar^2} {2\mu} \gamma \delta \left(r-a \right)$, therefore $V \left( r \right) = \frac {\hbar^2} {2\mu} \gamma$ at $r=a$, and $V \left( r \right) = 0$ otherwise.

To add to DrClaude's useful hints, note that what you wrote in the last line is incorrect, $V$ is not $\frac {\hbar^2} {2\mu} \gamma$ at $r=a$. Have you worked with matching wavefunctions when there is a Dirac delta as boundary condition? The matching is different from usual for the derivative of the wavefunction.
Also, it looks like you think about wavefunctions on a real line (you talk about being on the `"right" or "left" of $r=a$, here we are working along the radial direction, so you have to be careful about that, in particular the condition at $r=0$.