# S-Wave vs P-Wave Coupling

## Homework Statement

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I'm just having trouble seeing how to determine whether or not a wave of definite angular momentum will couple to a given term. The context is in G.P. Lepage's How To Renormalize The Schrodinger Equation, in which he derives the following renormalized potential

$$V_{eff}(\mathbf{r}) = -\frac{\alpha}{r}erf(r/\sqrt{2}a) + ca^2\delta^3_a(\mathbf{r}) + d_1a^4\nabla^2\delta^3_a(\mathbf{r}) + d_2a^4\nabla \cdot \delta^3_a(\mathbf{r})\nabla.$$

The details aren't too important, but he parenthetically states at a later time that "it is obvious that the $d_2$ term couples only two P-waves in the limit $a \rightarrow 0$. When $a$ is nonzero, however, this term has a small residual coupling to S-waves." I'm not sure how he's seeing any of this. I know that P-waves have angular momentum eigenvalue l = 1, but what does this imply about what's coupled to what?

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## Answers and Replies

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?