How Do S-Waves and P-Waves Couple in Quantum Mechanics?

[Xenosum]

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?

Homework Equations

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The Attempt at a Solution

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[eljose79]

Thank you for your question. In order to determine whether or not a wave of definite angular momentum will couple to a given term, you will need to consider the angular momentum quantum number, l, and the total angular momentum quantum number, j, of the term in question. In this case, the term in question is the d_2a^4\nabla \cdot \delta^3_a(\mathbf{r})\nabla term.

In the limit of a \rightarrow 0 , the term in question has a small residual coupling to S-waves. This means that it will primarily couple to waves with l=0 and j=0, which are S-waves. However, when a is nonzero, there may be a small coupling to P-waves, which have l=1 and j=1 or j=0. This is because P-waves have a nonzero angular momentum and can therefore couple to terms with a nonzero gradient.

I hope this helps to clarify the situation for you. If you have any further questions, please don't hesitate to ask.