The following extract is taken from Appendix A of the following paper: http://arxiv.org/abs/0810.0713.

Any solution of the Schrodinger equation with rotational invariance around the $z$ axis can be expanded as $$\psi_{k}=\Sigma_{l}A_{l}P_{l}(\cos \theta)R_{kl}(r),$$ where $R_{kl}(r)$ are the continuum radial functions associated with angular momentum $l$ satisfying $$-\frac{1}{2M}\frac{1}{r^{2}}\frac{d}{dr}\big(r^{2}\frac{d}{dr}R_{kl}\big)+\big(\frac{l(l+1)}{r^{2}}+V(r)\big)R_{kl} = \frac{k^{2}}{2M}R_{kl}$$

The $R_{kl}(r)$ are real, and at infinity look like a spherical plane wave which we can choose to normalise as $$R_{kl}(r) \rightarrow \frac{1}{r}\sin(kr-\frac{1}{2}l\pi+\delta_{l}(r))$$ where $\delta_{l}(r) << kr$ as $r \rightarrow \infty$.

The phase shift $\delta_{l}$ is determined by the requirement that $R_{kl}(r)$ is regular as $r \rightarrow 0$. Indeed, if the potential $V(r)$ does not blow up faster than $\frac{1}{r}$ near $r \rightarrow 0$, then we can ignore it relative to the kinetic terms, and we have that $R_{kl} \sim r^{l}$ as $r \rightarrow 0$; all but the $l=0$ terms vanish at the origin.

I have a couple of questions regarding the extract.

  1. There is a factor of $2M$ in front of $r^{2}$ in the expression $\frac{l(l+1)}{r^{2}}$ in quantum mechanics textbooks. Why is this factor missing here? Is this a typo?

  2. Why is $R_{kl}(r)$ at infinity normalised as $$R_{kl}(r) \rightarrow \frac{1}{r}\sin(kr-\frac{1}{2}l\pi+\delta_{l}(r))$$ where $\delta_{l}(r) << kr$ as $r \rightarrow \infty$?

  3. What does it mean to normalise $R_{kl}(r)$ as above?


Firstly this equations applicable for full rotational invariance where potential dependence of $r^2$ only.

Factor $2M$ can be written. It is a typo.

The solution of free Schrodinger equation in 3d is $ R_{kl}^0=\sqrt{\frac{\pi}{2 k r}}J_{l+1/2}(kr) $ where $J$ is a Bessel function also for $x\gg1$ asymptotic of Bessel function has the following form $J_n(x)\sim\sqrt{\frac{2}{\pi x}}\cos(x-\frac{\pi n}{2}-\frac{\pi}{4})$. For the big distance where $\frac{U(r)}{E}\ll1$ one can use free Schrodinger equation. All influense of the potential in additional phase.

In fact this normalization is the normalization for the Delta function for the free particles.


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