Mathematical misconception in scattering: switching from cartesian to spherical

[M. next]

If we were to consider a nucleon-nucleon interaction:
We know that the incident wave (plane wave) is ψ= Ae$^{ikz}$, propagating in z direction

But for some mathematical facilities, we tend to use spherical coordinates, the wave becomes = $\frac{A}{2ik}$[e$^{ikr}$/r - e$^{-ikr}$/r]

How come?
Where did the '2ik' in the denominator come from?

 

[andrien]

where you have seen that.
writing e^(ikr) - e^(-ikr)=2i sin(kr),we have second one as
sin(kr)/kr which is different from e^(ikz)=e^{(ikr cosθ)}

 

[M. next]

What? Sorry, but I didn't get where you're pointing to.
What was written is as follows:

The incident plane wave traveling in z direction:
ψ=Ae$^{ikz}$

They then mentioned that it was mathematically easier to work with spherical waves e$^{ikr}$/r and e$^{-ikr}$/r.

Lastly, for l=0,
ψ=$\frac{A}{2ik}$( e$^{ikr}$/r - e$^{-ikr}$/r)

That was what's written in some nuclear course.

 

[andrien]

e^(ikz)=Ʃ_li^l(2l+1)j_l(kr)p_l(cosθ)
=Ʃ_li^l(2l+1)(i/2k)[e^{-i(kr-l∏/2)}/r-e^i(kr-l∏/2)/r] p_l(cosθ)
j_l(kr) is spherical bessel function and P_l you know.this form of spherical bessel can be gotten from spherical hankel functions of first and second kind(their addition).for l=o this reduces to the required form.