Symmetries and conserved quantities

[plmokn2]

I know that if a particle is in a spherically symetric potential its angular momentum will be conserved, but what about if somehow we manage to produce say an elliptically symmetric potential? Will the particle then have a momentum along the curve of the ellipse conserved?
Thanks

 

[PRB147]

It should be, because the conservation of angular momentum is from the
rotational invariance of the Hamiltonian.

 

[Avodyne]

No. Angular momentum is not "momentum along the curve of a circle", but rather $\vec x\times\vec p$. There is nothing comparable that is conserved for a generic elliptically symmetric potential.

 

[PRB147]

I think his 'momentum' here is angular momentum.
If your elliptically symmetric potential is $$V(\rho,z)=\frac{1}{\rho^2+\alpha z^2},~\alpha\neq 1$$, then angular momentum along axis
'z' is conserved.

 

[malawi_glenn]

The only thing i can remember know is that in case of an elliptic potential well, J = L + S is not a good quantum number, science the energy eigenvalues will be "mixed" with same M_J etc, c.f The Nilsson model of Atomic Nucleus.

 

[plmokn2]

Thanks for your replies, sorry for the slightly ambiguious question.

 

[blechman]

Angular momentum around the azimuthal direction will still be conserved, but the TOTAL angular momentum will not. This is basically what PRB147 said, but maybe slightly clarified.