- #1

- 3

- 0

I am reading up on scattering theory and I'm having difficulty rederiving some of the results.

In 'Inverse problems in Quantum Scattering theory' (2nd Ed.) by Chadan and Sabatier they state that the solution for the s-wave regular solution, which is defined by the boundary condition [itex]{\phi(k,0) = 0}[/itex] and [itex]{\phi'(k,0) = 1}[/itex], is

[itex]\phi(k,r) = \frac{sin(kr)}{r} + \int^r_0 \frac{sink(r-r')}{k} V(r')\phi(k,r')dr'[/itex].

It is clear to me that this is a solution of the reduced radial Schrodinger equation for l=0, and I also follow the argument in the book that establishes convergence. But they state that this solution follows from 'the variation of the constants of Lagrange'. As far as I know this is simply the variation of parameters used to solve nonhomogeneous linear differential equations. I have tried to solve the radial equation (for l=0) using this approach but it doesn't work. 'Scattering theory of waves and particles' by Newton gives the same result using the more sophisticated approach of Green's functions, with which I am not very comfortable yet.

I could sketch my (incorrect) attempt using variation of parameters but I think a few comments should make it clear why it fails.

Generally, for a 2nd order DE of the type

[itex]\phi'' + p(r)\phi' + q(r)\phi = f(r)[/itex]

we look for solutions of the form [itex]\phi_p = u_1(r)\phi_1(r) + u_2(r)\phi_2[/itex]

where [itex]\phi_1[/itex] and [itex]\phi_2[/itex] are solutions of the homogeneous equation

[itex]\phi'' + p(r)\phi' + q(r)\phi = 0[/itex].

The problem is... the reduced Schrodinger is a homogeneous equation. So obviously the method can't work. I managed to get something like the correct solution by pretending that [itex]f(r) = V(r)\phi(r)[/itex] (in which case the homogeneous equation is the harmonic oscillator) but the procedure is mathematically flawed. As a reminder, the reduced radial Schrodinger equation for s-wave is

[itex]\phi'' + k^2\phi=V(r)\phi[/itex].

The only restriction on the potential is that [itex]\int_b^∞ |V(r)|rdr<\infty[/itex] where [itex]b≥0[/itex]

I would appreciate suggestions as to the proper method to approach this problem. Any ideas? (This is NOT a homework problem)