# S-Wave vs P-Wave Coupling

1. Dec 16, 2014

### Xenosum

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

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3. The attempt at a solution

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2. Dec 21, 2014