I thought I saw ..., well, not a pussy cat, but a comment in Dirac$^1$, going on about the "Normalization" of a scattering wave function being insensitive to the form of such a wavefunction close to the scattering centre ( hence origin of coordinates ).
How well, can this idea be justified?
If the idea is justifiable, it could be of use in setting up an expression I would like to use, in working out the 'Delta Function Normalization' of the 'Irregular Coulomb Wave Function', see at
Reference
- P.A.M. Dirac, The Principles Of Quantum Mechanics 4th Ed., Clarendon Press, Oxford, 1958
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Related Questions
Is this the way to delta-function normalise a continuum wave function?
How do you work out, the 'Delta Function Normalization', of a 'Regular Coulomb Wave Function'?
and
The idea can be justified from a mathematical point of view. This justification uses the idea that, adding or subtracting a finite quantity, to or from an infinite one, cannot really make a difference to the infinity.
Let $r_a$ be the radial coordinate beyond which a real valued continuum wave function adopts its asymptotic form, and let $\Psi_{k,L}(r)$ be the radial part of such a wave function, in some central potential, so that $\Psi_{k,L,a}(r)$, is it's asymptotic form for any $r>r_a$.
For $\mathbf{k=k^\prime}$ , so that the first integral below is infinite
We have
\begin{align*} \int _0^\infty \Psi_{k,L}(r) \Psi_{{k^\prime} ,L}(r) dr &= \int _0^{r_a} \Psi_{k,L}(r) \Psi_{{k^\prime },L}(r) dr + \int _{r_a}^\infty \Psi_{k,L,a}(r) \Psi_{{k^\prime} ,L,a}(r) dr \\ &=\int _{r_a}^\infty \Psi_{k,L,a}(r) \Psi_{{k^\prime} ,L,a}(r) dr \\ &=\int ^{r_a}_0 \Psi_{k,L,a}(r) \Psi_{{k^\prime} ,L,a}(r) dr + \int _{r_a}^\infty \Psi_{k,L,a}(r) \Psi_{{k^\prime} ,L,a}(r) dr\\ &=\int _0^\infty \Psi_{k,L,a}(r) \Psi_{{k^\prime} ,L,a}(r) dr \end{align*}
The above "equations", should be approximately OK for $k$ approximately equal to $ k^\prime$ as well. So, how $ \int _0^\infty \Psi_{k,L}(r) \Psi_{{k^\prime} ,L}( r) dr $ behaves, as a multiple of a delta function $\delta(k-k^\prime)$ is determined by the asymptotic forms of $\Psi_{k,L}$ and $\Psi_{k^\prime ,L}$.
For some number $N_L$ we will have
\begin{equation*} \int ^\infty_0 \Psi_{k,L,a}(r) \Psi_{{k^\prime} ,L,a}(r) dr = N_L \delta(k-k^\prime) \end{equation*}
There is no mention of the form of $\Psi_{k,L}$ or $\Psi_{k^\prime ,L}$ close to the scattering centre in the above.
Hence, the normalisation of a continuum/scattering wave function is insensitive to the functions form near to the scattering centre.