Electron in a Bubble

[Erasmus00]

I'm having a bit of trouble with a question. My brain doesn't seem to be working.

 

An electron is contained within a bubble of radius R, What is the energy of the ground state, and how does it depend on R?

 

I was thinking of using the uncertainty principle to get the momentum, hence the energy. Any better way?

-Will

[Qfwfq]

An electron is contained within a bubble of radius R

I suppose you mean a potential well, which may be finite or infinite.

 

Any better way?

The standard way is to make the boundary conditions (simple in the more ideal case of infinite well, from R outward the amplitude is just plain zero). In both cases, with radial symmetry, the eigenstates are series of Bessel and Neuman functions but I can't remember details from years ago, exercises in basic theoretical physics course, but that's the most exact path.

 

Complicated, you might say. Perhaps the ground state energy can be estimated well enough, as you suggest. Heisenberg gives you the delta p and the particle whacking around inside the well can be modelled as a free particle between whacks. It's momentum p is therefore only changing in direction so delta p would be twice the modulus, apply E^2 - p^2 = m^2 and you should have a quite good estimate.

[Erasmus00]

I suppose you mean a potential well, which may be finite or infinite.

 

The standard way is to make the boundary conditions (simple in the more ideal case of infinite well, from R outward the amplitude is just plain zero). In both cases, with radial symmetry, the eigenstates are series of Bessel and Neuman functions but I can't remember details from years ago, exercises in basic theoretical physics course, but that's the most exact path.

 

Complicated, you might say. Perhaps the ground state energy can be estimated well enough, as you suggest. Heisenberg gives you the delta p and the particle whacking around inside the well can be modelled as a free particle between whacks. It's momentum p is therefore only changing in direction so delta p would be twice the modulus, apply E^2 - p^2 = m^2 and you should have a quite good estimate.

 

Yea, the problem was for the girl I'm tutoring. It's a freshman year intro to modern physics type thing, so I imagine it must be the uncertainty estimate. I doubt they want you to solve explicitly and deal with the Bessel functions. Although, I suppose for the ground state you could set l=0, and then you've got a simple infinite potential well.

-Will