How can potentials be well-defined without violating U.P.?

[TomServo]

Tried searching for equivalent question but couldn't find it.

Presumably, a potential (like a Coulomb one) comes from another particle, which has its own momentum/position uncertainty, but in the Schroedinger equation the potential is well-defined either in terms of some coordinate system or its relation to the particle that the equation is for.

So how does this work? How do we talk about the hydrogen atom as if the nucleus were stationary without violating the uncertainty principle?

 

[Khashishi]

Treating the nucleus as stationary is just an approximation. A more accurate calculation uses the motion of a reduced mass around a stationary center of mass. This gives slightly lower energy levels.

 

[TomServo]

Treating the nucleus as stationary is just an approximation. A more accurate calculation uses the motion of a reduced mass around a stationary center of mass. This gives slightly lower energy levels.

Yeah I get that, but using a classical orbital mechanics approach still sounds like we're well-defining position and momentum.

 

[atyy]

One starts with the full multiparticle Hamiltonian, then the heavy particle is stationary only as an approximation.

https://en.wikipedia.org/wiki/Molecular_Hamiltonian

http://chemwiki.ucdavis.edu/Wikitex...r_approximation_and_the_hydrogen_molecule_ion

 

[Khashishi]

It's just an approximation. The position of the nucleus is well defined because we assume that it is stationary.

 

[Vanadium 50]

You are confusing two things. One is whether the uncertainty principle prevents you from having an arbitrarily well-defined potential. It does not. In the case of the Coulomb potential, adding mass to the central charge until you are within your tolerances will do the trick. The other is whether there is a practical limitation. Sure - for many reasons. You can't make a nucleus arbitrarily heavy. You can't make it arbitrarily small, so the point charge approximation has limits. There are environmental effects - the atom may be in a molecule, or there may be stray external electromagnetic fields, etc.

 

[DrDu]

In QM, the Coulomb potential of a nucleus is an operator, just like the nuclear position itself is represented by an operator.
As others already pointed out, it is often a good approximation to treat the nucleus classically and this is often done in praxis. However, this is only an approximation and nobody forces you to apply it.