# Symmetries and conserved quantities

1. Dec 14, 2007

### 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 partical then have a momentum along the curve of the ellipse conserved?
Thanks

2. Dec 14, 2007

### PRB147

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

3. Dec 14, 2007

### 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.

4. Dec 14, 2007

### 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.

Last edited: Dec 14, 2007
5. Dec 15, 2007

### 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, scince the energy eigenvalues will be "mixed" with same M_J etc, c.f The Nilsson model of Atomic Nucleus.

6. Dec 15, 2007

### plmokn2

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

7. Dec 20, 2007

### 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.