Find a general expression for the phase shift produced by a scattering potential \(V(r)=A / r^{2}\), where \(A>0 .\) Is the total cross section finite? If not, does the divergence come from small or large scattering angles, and why? What modifications are necessary in the calculation if \(A<0 ?\) Are any difficulties encountered in the latter case?

In quantum mechanics, a scattering potential describes how particles interact when they approach each other. Here, the potential is given by \[V(r) = \frac{A}{r^2}\].This potential depends on the radial distance, r, and the constant, A. When A > 0, the potential is repulsive, meaning it pushes particles away from each other. This form of potential is significant in analyzing how particles scatter because it changes the way the Schrödinger equation is solved. To understand how particles are deflected or 'scattered', we need a good grasp of this potential.

The Schrödinger equation is a cornerstone of quantum mechanics. It describes how the quantum state of a system changes over time. For our problem, we use a specific form called the radial Schrödinger equation, since our potential is radially symmetric:\[\left( \frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr} - \frac{l(l+1)}{r^2} + \frac{2m}{\hbar^2} (E - V(r)) \right) R(r) = 0\]Here, R(r) is the radial part of the wave function, l is the angular momentum quantum number, m is the particle's mass, E is the energy, and \(\hbar\) is the reduced Planck's constant. This equation helps us determine the phase shift, \(\delta_l\), which is critical in understanding how particles scatter under the potential \(V(r) = \frac{A}{r^2}\). The phase shift results from changes in the wave function caused by the potential.

The total cross section, \(\sigma_t\), is a measure of the probability that a scattering event will occur. It's derived from the partial wave expansion, which considers how each quantum state, defined by the angular momentum quantum number, l, contributes to scattering. The formula for the total cross section is given by:\[\sigma_t = \frac{4\pi}{k^2} \sum_{l=0}^{\infty} (2l+1) \sin^2(\delta_l)\]Here, k is the wave number, and \(\delta_l\) is the phase shift for a given l. If \(\delta_l\) decreases sufficiently fast as l increases, the sum converges, making \(\sigma_t\) finite. For our potential \(V(r) = \frac{A}{r^2}\), substituting \( \delta_l = - \tan^{-1}\left( \frac{A}{l(l+1)} \right)\) shows that, in general, the total cross section tends to diverge for large A due to high contributions from larger values of l. This typically means the potential causes significant scattering even at larger angles, making the process more complex.

In this context, the effective potential, \(V_{\text{eff}}(r)\), combines the actual scattering potential with a centrifugal potential. The centrifugal potential term arises due to the angular momentum and is given by \(\frac{l(l+1)}{r^2}\). Thus, the effective potential for \(V(r) = \frac{A}{r^2}\) becomes:\[V_{\text{eff}}(r) = \frac{l(l+1)}{r^2} + \frac{A}{r^2}\]By combining these terms, the effective potential seen by a particle changes, affecting how it scatters. This effective potential helps determine the phase shift \(\delta_l\), using the relationship:\( \delta_l = - \tan^{-1}\left( \frac{A}{l(l+1)} \right)\). This reveals the importance of combining different components to understand how particles interact and how their paths are altered due to the scattering potential.