# 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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### Question about Sakurai's $SO(4)$ symmetry section

In Sakurai's Quantum mechanics book, he says the hydrogen atom has $SO(4)$ symmetry by explicitly exhibiting operators $I_i,K_i$ that satisfy the commutation relation of the Lie algebra $so(4)$. ...
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### Physical implication of conservation of Laplace-Runge-Lenz vector

I'm having trouble wrapping my mind around the Laplace-Runge-Lenz vector. Conservation of momentum can be visualized as an object moving in a straight line with constant speed. One can even visualize ...
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### Showing The Laplace–Runge–Lenz vector (per unit mass) is constant

Given an inverse square law $\ddot{\vec{r}}=-\frac{\mu}{r^2}\hat{r}$, I define the Angular momentum per unit mass as $\vec{H}=\vec{r}\times\dot{\vec{r}}$. Showing it's constant is strightfoward. Then ...
It can be proven that the commutation relations $$\Big[ N,H\Big]=0$$ $$\Big[ L,H\Big]=0$$ $$\Big[ N,L^2\Big] \neq 0$$ Where $N$ is the Runge-Lenz vector. I'm wondering if it is possible to write ...