- #1
spaghetti3451
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The following extract is taken from Appendix A of the following paper: http://arxiv.org/abs/0810.0713.Any solution of the Schrödinger 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?
##-\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?