Worked Problem for Scattering on a Delta-Shell Potential

Consider an attractive delta-shell potential (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda > 0} ) of the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(\textbf{r})=-\frac{\hbar^2 \lambda}{2m} \delta(r-a)}

1) Derive the equation for the phase shift caused by this potential for arbitrary angular momentum.

2) Obtain the expression for the s-wave phase shift.

3) Obtain the scattering amplitude for the s-wave.

Solutions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -\frac{\hbar^2}{2m}\frac{d^2}{dr^2} + \frac{\hbar^2 l(l+1)}{2mr^2} + V(r)\right]u_l (r)=Eu_l (r)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(\textbf{r})=f_l (r)Y_m^l(\theta, \phi)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_l (r)=\frac{u_l (r)}{r}}

In region one, r < a, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{1l} (r)=C j_l(kr)\!} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=\sqrt{\frac{2mE}{\hbar^2}}}

In region two, r > a, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{2l} (r)=A j_l(kr) + B n_l(kr)\!}

Invoking continuity of the wave function on either side of the boundary:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C j_l(ka) = A j_l(ka) + B n_l(ka)\!}

The first derivative is discontinuous due to the behavior of the delta function, so we must find the second condition needed a slightly different way. The delta function is most easily evaluated with an integral, so we consider the integral of the Schrodinger equation from a+Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} to a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \epsilon} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int^{a+\epsilon}_{a-\epsilon}\left[\frac{d^2}{dr^2} + \frac{l(l+1)}{r^2} - \lambda\delta(r - a) - k^2\right] u_l(r)dr = 0}

Now, we take the limit as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon \rightarrow 0} , and note that only the following two terms remain (the other integrals have the same value on either side of a):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int^{a+\epsilon}_{a-\epsilon}\frac{d^2}{dr^2}u_l(r)dr = \int^{a+\epsilon}_{a-\epsilon}\lambda\delta(r-a)dr }

Which now becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dr}u_l(r)|_{a+\epsilon} - \frac{d}{dr}u_l(r)|_{a-\epsilon} = \lambda u_l(a) = f'_{1l}(a) - f'_{2l}(a) }

Which, combined with the above boundary condition for continuity gives that:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Akj'_l(ka) + Bkn'_l(ka) - Ckj'_l(ka) = -\lambda C j_l(ka)\!}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Akj'_l(ka) + Bkn'_l(ka) - Ckj'_l(ka) = -\lambda Aj_l(ka) + Bn_l(ka)\!}

As usual, let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle tan(\delta_l)=\frac{-B}{A}}

By solving the two equations obtained from the boundary conditions for the ratio Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{-B}{A}} , we find:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle tan(\delta_l)=\frac{\lambda j^2_l(ka)}{kj_l(ka)n'_l(ka) - kn_l(ka)j'_l(ka) + \lambda n_l(ka)j_l(ka)}}

For s-waves, set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l=0\!}

Therefore:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle tan(\delta_0)=\frac{\lambda j^2_0(ka)}{kn_0(ka)j_1(ka) - kn_1(ka)j_0(ka) + \lambda n_0(ka)j_0(ka)}}

which simplifies to:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle tan(\delta_0)=\frac{\lambda sin^2(ka)}{k - \lambda sin(ka)cos(ka)}}

From here, recall that the scattering amplitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_k (\theta)=\frac{1}{k}\sum_{l=0}^\infty (2l+1)e^{i\delta_l(k)}sin(\delta_l (k)) P_l(cos(\theta))}

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l=0\!} and in conjunction with the derived result for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_l\!} above:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_k (\theta)=\frac{e^{i\delta_0}}{k}sin(\delta_0)}