# Help understanding the origin of scattering phase shift

I'm having a conceptual problem. The time-independent Schrodinger equation for a continuum state $$\psi_{k, l}(r)$$ is given by

$$0 = \left( - \frac{\nabla^2}{2} + v(r) - \frac{k^2}{2} \right) \psi_{k,l}(r) ,$$

where $$\nabla^2$$ is the Laplacian, $$v(r)$$ some potential, $$k$$ the electron momentum. Expressing this as a matrix, I find real-valued eigenstates. However, textbooks and papers say the solution is

$$\psi_{k,l}(r) \propto R_{k,l}(r) Y_{l,m}(\theta,\phi) e^{i \sigma_{k}},$$

where $$\sigma_k$$ is known as the scattering phase shift. I understand how to interpret the phase shift - it is the phase difference in the asymptotic region between the eigenstate and a plane wave of the same momentum. Where, however, does this comparison come in when solving the Schrodinger equation?

• How were you arriving at real eigenstates? If nothing else in the limit that $v=0$ the eigenstates are plane waves and clearly not real in general. Aug 29, 2021 at 13:54