Solution of Schrodinger equation in axially symmetric case

[spaghetti3451]

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.

Firstly, 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?

Secondly, 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 $r \rightarrow \infty$?

Thirdly, what does it mean to normalise $R_{kl}(r)$ as above?

 

[DrClaude]

Firstly, 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?

I guess this is a typo, as otherwise there is a problem with dimensions.

Secondly, 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 $r \rightarrow \infty$?

Thirdly, what does it mean to normalise $R_{kl}(r)$ as above?

As they say, this is a representation of a spherical wave, phase-shifted due to the interaction at r = 0. It is standard in scattering theory to normalize wave functions that way, in order to obtain stationary states, at the expense of wave function being no longer square integrable. You should find more details in a good book on quantum scattering. Sakurai's Modern Quantum Mechanics can be a good starting point.