[Q-e-developers] Divergence in PAW UPF

[Lorenzo Paulatto]

On Monday, 16 May 2016 11:34:24 CEST Anton Kozhevnikov wrote:
> Dear Lorenzo!
> Can you please clarify this statement: "scattering states are not
> normalizable: they always diverge". Are the scattering states differ from
> the unbound states (i.e. plane waves), which are also not normalizable, but
> at least they don't diverge? The scattering states in the UPF files have a
> value of ~10^8 at the end of the radial grid. What's the physics behind the
> states with psi(r)->infinity for r->infinity?

I'm relatively sure you can get diverging states in any quantum mechanical 
system if the external potential is integrable (or some similar condition 
which I cannot recall), see for example the finite potential well <https://
en.wikipedia.org/wiki/Finite_potential_well>.

To be honest, I haven't spent much time about the physical meaning of these 
scattering states, so what follows is probably wrong. I guess, it physically 
means that a particle can escape the potential well, and, since the rest of 
the universe is much larger that the specific point where the potential is 
sizeable, the probability of having the particle anywhere else is diverging.

hth

-- 
Dr. Lorenzo Paulatto
IdR @ IMPMC -- CNRS & Université Paris 6
+33 (0)1 44 275 084 / skype: paulatz
http://www.impmc.upmc.fr/~paulatto/
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