Orbital angular momentum and powers of k

[Einj]

Hi guys,
I have a question which is probably stupid. I am studying angular momentum and I have found almost everywhere that, if we have orbital angular momentum higher than $\ell=0$ then it brings powers of the momentum of the system like $k^\ell$. What is the reason of that? Can someone suggest a book where I can find it or something?

Thank you

 

[diegzumillo]

I may be able to help but I need to know where this is coming from. You say this appears in a few books, can you name one so I can check what it says?

 

[Einj]

You can find it, for example, in Weinberg - Lectures on Quantum Mechanics, in the chapter on Shallow Bound States, when he talks about the request for the state to be in S-wave in order to avoid suppression by factors $k^\ell$.

 

[andrien]

It comes out by solving the radial part of wave eqn which contains the factor of l(l+1),there we use
ρ=kr as the variable and in some simplification with potential,the solutions are of the form (ρ^l)+(1/ρ^l+1),second part is rejected for finiteness near the origin which does give k^l type term.All quantum mechanics books containing scattering theory deals with it.

 

[Einj]

Ok, thank you very much. I'll definitely take a look at that!