Answer to Question #124879 in Optics for kashaf

Question

Calculate phase shifts for quantum scattering, and then in terms of these phases give an account on scattering amplitude. As an application consider a sphere of radius r, assuming a certain potential V such that it is infinite inside the sphere and minimum outside the sphere,calculate phase shifts for this sphere.

Answer 1

As per the question,

Radius of the sphere$=r$

Potential of the sphere$=V$

Phase shit- Angle $\delta_l$ is known as the phase shift.

The partial wave equation of the plane wave $e^{ikt}=\Sigma_{l=0}^\infty\sqrt{4\pi(l+1)}i^lj_l(K_r)Y_{lo}(\theta)$

We know that asymptotically spherical Bessel function, $j_l{(kr)}$ has the form,

$j_l{(kr)} \ \ \overrightarrow{r\rightarrow\infty} \ \ \frac{\sin(kr-\frac{l\pi}{2})}{kr}$

Now, substituting the value and solving it further,

$e^{ikt}=\Sigma_{l=0}^\infty\sqrt{4\pi(l+1)}i^lj_l(K_r)Y_{lo}(\theta)$

$\Rightarrow e^{ikt}=\Sigma_{l=0}^\infty\sqrt{4\pi(l+1)}\{\frac{e^{ikr}}{2ikr}-\frac{i^le^{-i(kr-\frac{l\pi}{2}))}}{2ikr} \}Y_{lo}(\theta)$