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11
votes
2answers
221 views

Eigenfunctions of the Runge-Lenz vector

The hamiltonian for the hydrogen atom, $$ H = \frac{\mathbf{p}^2}{2m} - \frac{k}{r} $$ is spherically symmetric and it therefore commutes with the angular momentum $\mathbf{L}$; this causes all its ...
6
votes
2answers
277 views

How can one see that the Hydrogen atom has $SO(4)$ symmetry?

For solving hydrogen atom energy level by $SO(4)$ symmetry, where does the symmetry come from? How can one see it directly from the Hamiltonian?
4
votes
2answers
161 views

Runge-Lenz vector and Keplerian Orbits

Is the loss of closed Keplerian orbits in relativistic mechanics directly tied to the absence of the Runge-Lenz vector?
2
votes
1answer
834 views

A Short way to show Conservation of Quantum Laplace–Runge–Lenz Vector?

I had been asked to prove the conservation of Quantum Laplace–Runge–Lenz Vector: ...
2
votes
1answer
122 views

Kepler problem: flows generated by constants of motion

This is part of an admission exam problem, found at http://www.sissa.it/mp/admission/tests//2008_common.pdf Consider the Hamiltonian of Kepler problem $$H(\boldsymbol{r},\boldsymbol{p})= ...
1
vote
0answers
431 views

Square of Laplace–Runge–Lenz vector in Hydrogen atom [closed]

I have a problem. I've tried this question, but I don't get the correct expression. Can someone give me some ideas? Thanks! Consider the Hydrogen Atom Hamiltonian: $$ H = (\mathbf p^2/2 ...
4
votes
3answers
846 views

Index Manipulation and Angular Momentum Commutator Relations

I have been trying for hours and cannot figure it out. I am not asking anyone to do it for me, but to understand how to proceed. We have the relations $$[L_i,p_j] ~=~ i\hbar\; \epsilon_{ijk}p_k,$$ ...
21
votes
5answers
2k views

What symmetry causes the Runge-Lenz vector to be conserved?

Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the ...