Visualising Potential Wells in Two Body Systems

[Xander314]

I am currently working through Griffiths' textbook on quantum mechanics. The hydrogen atom was first modeled 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 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 r₁ and r₂, and I understand the change into new coordinates: the separation distance 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 r₁, particle 2 is at position r₂ and so on?

 

[The_Duck]

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 r₁ and r₂, and I understand the change into new coordinates: the separation distance 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]

Well, in the two-body case since the potential only depends on the separation vector ##\vec{r}##, you can visualize it in the 3D space that ##\vec{r}## lives in. In general though, you are right: when you have ##n## particles the potential is a function of ##3n## spatial coordinates, and so becomes much harder to visualize. Note that the wave function also depends on ##3n## spatial coordinates, so when you have many particles you can no longer visualize the wave function as a wave propagating in real space. Instead it is a wave in configuration space.

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 r₁, particle 2 is at position r₂ and so on?

Yes.

 

[Xander314]

Thanks for the help!