Phy5645/Problem3

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_l (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_l (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)\!}

Also, with regards to the strength of the delta function held proportional to the discontinuity of the derivative of the wave function at 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 Aj'_l(ka) + Bn'_l(ka) - Cj'_l(ka) = -\lambda A j_l(ka)\!}

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}}

Therefore by algebraic manipulation:

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)}{j_l(ka)n'_l(ka) - n_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)}{n_0(ka)j_1(ka) - n_1(ka)j_0(ka) + \lambda n_0(ka)j_0(ka)}}

which simplifies to:

$tan(\delta _{0})={\frac {\lambda sin^{2}(ka)}{1-\lambda sin(ka)cos(ka)}}$

From here, recall that the scattering amplitude $f_{k}(\theta )={\frac {1}{k}}\sum _{l=0}^{\infty }(2l+1)e^{i\delta _{l}(k)}sin(\delta _{l}(k))P_{l}(cos(\theta ))$

For $l=0\!$ and in conjunction with the derived result for $\delta _{l}\!$ above:

$f_{k}(\theta )={\frac {e^{i\delta _{0}}}{k}}sin(\delta _{0})$