Worked Problem for Scattering on a Delta-Shell Potential
The effective one-dimensional Schrödinger equation is
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).}
In region one, given by 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 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 u_{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, given by 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 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 u_{2l} (r)=A j_l(kr) + B n_l(kr).\!}
Continuity of the wave function on either side of the boundary requires 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 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 in a manner similar to how we found it for the one-dimensional delta function potential. We first integrate the effective Schrödinger equation from 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 a+\epsilon\!} 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 a-\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}
We now 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 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 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 \int^{a+\epsilon}_{a-\epsilon}\frac{d^2u_l(r)}{dr^2}\,dr = \int^{a+\epsilon}_{a-\epsilon}\lambda\delta(r-a)\,dr }
This 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 \left. \frac{du_l(r)}{dr}\right |_{a+\epsilon} - \left. \frac{du_l(r)}{dr}\right |_{a-\epsilon} = u'_{2l}(a) - u'_{1l}(a) = \lambda u_l(a).}
This, combined with the above continuity condition, gives us
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).\!}
We now define the phase shift,
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 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 \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 the 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 s\!} wave case, 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.\!} In this case,
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)},}
or
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 is given by
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}).}
Keeping only the 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\!} term and in conjunction with the 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\!} derived above, we find 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 f_k (\theta)=\frac{e^{i\delta_0}}{k}\sin{\delta_0}.}