# Trying to understand how two particles interact

box
I'm wondering how the wave functions of two particles would evolve through time if they cross paths. Say you have an electron whose wave functon comes close to another charged particle, like an alpha particle. Now I believe if the alpha particle is well localised you can treat it has a point charge and use the time independant schrodinger equation to determine how the electrons wave function changes through time. But what if the alpha particle is not well localised and its wavefunction is spread out? How will the electrons wave function evolve? Is there anyway to give me an idea without going into too much detail?

btw I am guessing since the alpha particle is so heavy it won't be affected much by an electron moving at a moderate speed.

DaTario
Supose you have an atom trapped by an harmonic potential. Then you send light on this atom. Light is an electromagnetic wave. The quantum spatially extended atom will experience the light's electromagnetic forces in a extended way as well. In other words: the atom will not in general experience the electromagnetic field in just one point of space at a given instant. The ratio between the wave vector k and the width of the fundamental state of the atom inside this hamonic potential [itex], \Delta x [\itex], will give us a notion of how point-like is the interaction between the atom and the light. This parameter is the so called Lamb-Dicke parameter.
This question is somewhat related to the dipole approximation, but I believe strongly it is not the same notion.

Although the systems are not the same, I hope this serves as a begining.

Note: As long as you can treat the alpha particle as the source of a potential, it seems that you can profit from Schroedinger's equation for the electron in order to obtain the time evolution of its wave function.

Best Regards,

DaTario

Last edited:
In fact, the cross section of two charged particles interacting via photon exchange (basic qED) is identical to the Rutherford cross section, which, of course, is calculated using classical physics. One of the few cases in which classical-quantum (no spin, or average over spin, no internal structure )

Localized or spread out, all can be handled. This is done in any text on basic scattering theory, classical or quantum.

regards,
Reilly Atkinson

Antiphon
box said:
I'm wondering how the wave functions of two particles would evolve through time if they cross paths. Say you have an electron whose wave functon comes close to another charged particle, like an alpha particle. Now I believe if the alpha particle is well localised you can treat it has a point charge and use the time independant schrodinger equation to determine how the electrons wave function changes through time. But what if the alpha particle is not well localised and its wavefunction is spread out? How will the electrons wave function evolve? Is there anyway to give me an idea without going into too much detail?

btw I am guessing since the alpha particle is so heavy it won't be affected much by an electron moving at a moderate speed.

The wavefunction is a function of the coordinates of both particles
and must be interpreted as in the following language: the amplitude
squared of the wave function is the probability that particle A is at x1
AND particle B is at x2. It is a joint probabliity distribution.

Of course this language is overly simplistic. One particle could be in a state
of definite momentum with equal amplitude everywhere in space. This is
why the Psi function is considered to have only a probabilistic interpretation
and not a directly physical one.