Checkpoint: Scattering Calculations

This checkpoint pulls the scattering half of the module together. You will derive the Born amplitude for a finite-range spherical square well, evaluate it, and check the low-energy limit against the s-wave intuition from the phase-shift lesson.

The setup

Consider a uniform spherical-well potential

V(r) = \begin{cases} V_0, & r < a, \\ 0, & r > a. \end{cases}

The first Born amplitude for a central potential is

f^{(1)}(\theta) = -\frac{2m}{\hbar^2 q}\int_0^\infty r\,V(r)\,\sin(qr)\,dr, \qquad q = 2k\sin(\theta/2).

For the square well the integral runs only to $a$ . Using $\int_0^a r\sin(qr)\,dr = \big[\sin(qa) - qa\cos(qa)\big]/q^2$ , the amplitude is the exact closed form

\boxed{\;f^{(1)}(\theta) = -\frac{2mV_0}{\hbar^2}\,\frac{\sin(qa) - qa\cos(qa)}{q^3}\;}

The low-energy check

As $q\to 0$ , expand $\sin(qa) - qa\cos(qa) = \tfrac{1}{3}(qa)^3 - \tfrac{1}{30}(qa)^5 + \cdots$ , so

f^{(1)} \;\xrightarrow{\,q\to 0\,}\; -\frac{2mV_0}{\hbar^2}\,\frac{a^3}{3},

an isotropic (angle-independent) amplitude — exactly as expected when only the s-wave survives at low energy. The forward and backward amplitudes coincide in this limit, and the total cross-section is simply $\sigma = 4\pi |f^{(1)}|^2$ .

Your task

Work in units where $2m/\hbar^2 = 1$ . Take $V_0 = 1$ , $a = 1$ , and evaluate the Born amplitude at a finite momentum transfer $q = 1$ (so $qa = 1$ ). Compute

f^{(1)} = -V_0\,\frac{\sin(qa) - qa\cos(qa)}{q^3},

and return its value. (You can sanity-check the sign and magnitude against the $q\to 0$ value $-V_0 a^3/3 \approx -0.333$ ; at $qa=1$ the amplitude is a bit smaller in magnitude.)

Run your code to see the quantum state.