# Questions tagged [laplace-runge-lenz-vector]

The Laplace–Runge–Lenz vector describes the shape and orientation of the orbit of one astronomical body around another. In general, the LRL vector is conserved (it's a constant of the motion) in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them (Kepler problem). Its conservation is significant in the quantization of the Hydrogen atom.

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### Textbook proof error? Runge Lenz

I was reading this proof in my textbook. They say that $$\vec{r} \cdot \dot{\vec{r}} = |\vec{r}||\dot{\vec{r}}|.$$ Doesn't that mean $\vec{r}$ is parallel to $\dot{\vec{r}}$, and if so, then the line ...
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### Poisson Bracket for the angular momentum and Runge-Lenz vector [closed]

Given the Runge-Lenz vector $$\vec{A}=\vec{p}\times\vec{L}-mk\frac{\vec{r}}{r}$$ and the angular momentum $$\vec{L}=\vec{r}\times\vec{p}$$ We can get $$\{L^i,A^j\}=\epsilon^{ijk}A^k.$$ I think ...
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### Accidental degeneracy in Hydrogen Energies [duplicate]

The energy of Hydrogen electron ground state should depend on $n$ and $\ell$, but it only depends on $n$. What is the reason behind this accidental degeneracy; I know that the reason lies in symmetry; ...
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### How I can prove the Commutation between hamiltonian and Runge-Lenz vector? [closed]

I am a undergraduate student in physics. I found this page that shows a way to prove the commutator between Runge-Lenz vector and Hamiltonian .$\left [\hat{A}_{i},\hat{H}\right]=0$ I believe he did a ...
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### A mysterious conserved quantity for a central potential

In teaching a course in classical mechanics and I have come across (from my predecessor) a to me mysterious conserved quantity. We are considering a gravitational (or electric) potential with the ...
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### Is there an analog to the Runge-Lenz vector for a 3D spherically symmetric harmonic potential?

The Runge-Lenz vector is an "extra" conserved quantity for Keplerian $\frac{1}{r}$ potentials, which is in addition to the usual energy and angular momentum conservation present in all central force ...
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### Minimal set of invariants to specify a Kepler orbit

In the Kepler problem, we know that there are various invariants, including: Energy Angular momentum vector Runge-Lenz vector All together these consist of 7 parameters. On the other hand, the ...
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### Algebraic solution of Dirac equation for Coulomb potential

The Runge-Lenz operator enables an algebraic solution of Coulomb potential energy levels without a solution of a differential equation. What is the analog for the solution of the Dirac equation in a ...
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### What symmetry operation mixes states with different $\ell$ in hydrogen atom? [duplicate]

We can mix states with different $m$ in hydrogen atom by rotating it around some axis (not coinciding with $z$). Thus rotation is the symmetry operation which mixes states with different $m$. As ...
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### Why are hydrogen energy levels degenerate in $\ell$ and $m$?

Is there a good physical picture of why the energy levels in a hydrogen atom are independent of the angular momentum quantum number $\ell$ and $m$?
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### Show that the Laplace-Runge-Lenz vector is conserved using poisson brackets

(I realise similar Phys.SE questions already exist but there is no answer with the Poisson bracket notation, I'll take this down if someone lets me know I should have commented in the existing ...
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### 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 ...
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### 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?
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### 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?
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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: $$H = (\mathbf p^2/2 \mu)-(e^2/... 3answers 3k views ### Index Manipulation and Angular Momentum Commutator Relations I have been trying for hours and cannot figure it out. I am not asking anyone to do it for me, but to understand how to proceed. We have the relations$$[L_i,p_j] ~=~ i\hbar\; \epsilon_{ijk}p_k, ...
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 ...