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I am currently working through Griffiths' textbook on quantum mechanics. The hydrogen atom was first modelled as a one body system with the proton fixed at the origin. In this case the potential was given by Coulomb's law,

[tex]V(r) = -\frac{e^2}{4\pi\epsilon_0}\frac{1}{r} \ ,[/tex]

where

This potential is easy to visualise as a "potential well" -- at least in two dimensions -- with the proton at the centre with V=-∞ and then the potential approaching 0 as r goes to ∞.

However, the hydrogen atom is then reconsidered as a multi body problem with the motion of the proton now accounted for. The positions of the particles are given by

My question is this: how can the potential

[tex]V(\vec{r}_1,\vec{r}_2) = V(\vec{r}_1-\vec{r}_2) = V(\vec{r}) = V(|\vec{r}|) = -\frac{e^2}{4\pi\epsilon_0}\frac{1}{r} [/tex]

now be visualised as a potential well. Can this only be done in some kind of 2n-dimensional "configuration space" (where n is the number of space dimensions) of tuples

[tex](\vec{r}_1,\vec{r}_2) \ ?[/tex]

Also, what is the interpretation of a general multi body potential

[tex]V = V(t,\vec{r}_1,\vec{r}_2,\ldots,\vec{r}_m) \ ?[/tex]

Is this the

[tex]V(r) = -\frac{e^2}{4\pi\epsilon_0}\frac{1}{r} \ ,[/tex]

where

*r*is the radial coordinate.This potential is easy to visualise as a "potential well" -- at least in two dimensions -- with the proton at the centre with V=-∞ and then the potential approaching 0 as r goes to ∞.

However, the hydrogen atom is then reconsidered as a multi body problem with the motion of the proton now accounted for. The positions of the particles are given by

**r**and_{1}**r**, and I understand the change into new coordinates: the separation distance_{2}**r**and the centre of mass**R**.My question is this: how can the potential

[tex]V(\vec{r}_1,\vec{r}_2) = V(\vec{r}_1-\vec{r}_2) = V(\vec{r}) = V(|\vec{r}|) = -\frac{e^2}{4\pi\epsilon_0}\frac{1}{r} [/tex]

now be visualised as a potential well. Can this only be done in some kind of 2n-dimensional "configuration space" (where n is the number of space dimensions) of tuples

[tex](\vec{r}_1,\vec{r}_2) \ ?[/tex]

Also, what is the interpretation of a general multi body potential

[tex]V = V(t,\vec{r}_1,\vec{r}_2,\ldots,\vec{r}_m) \ ?[/tex]

Is this the

*total*potential energy of the system at time*t*when particle 1 is at position**r**, particle 2 is at position_{1}**r**and so on?_{2}
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