42
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Eigenfunctions of the Runge-Lenz vector
1. Definitions
Let's consider the nondimensionalized Hamiltonian
$$\hat H=\frac{\hat{p}^2}2-\frac1r.\tag1$$
Its standard eigenfunctions diagonalize operators $\hat H$, $\hat L_z$ and $\hat L^2$. ...
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33
votes
What exactly are the 12 conserved quantities in the Two-Body Problem?
There's not much difference in having two bodies rather than one. For two bodies, you can separate the center of mass position/velocity, and the relative position/velocity. The six center of mass ...
- 105k
17
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Is there an analog to the Runge-Lenz vector for a 3D spherically symmetric harmonic potential?
Yes there is an analog of the Laplace-Runge-Lenz vector, even more so! The N-dimensional harmonic oscillator is one of a handful superintegrable systems where you have a maximal number (2 N - 1) of ...
- 66.3k
7
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Solving Hydrogen atom with ladder operators
The Hamiltonian of the hydrogen atom, $H=\mathbf{p}^2/2m - \alpha /|\mathbf{x}|$, commutes with all components of the angular momentum operator $\mathbf{L}= \mathbf{x} \times \mathbf{p}$ and of the ...
- 7,298
6
votes
How is angular momentum defined on symplectic space?
Elementary Mechanics
Consider the elementary mechanics of a point particle moving through the space of possible positions $Q:=\mathbb R^n$. This space has a natural affine structure (see also my ...
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6
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How is angular momentum defined on symplectic space?
How is angular momentum defined on symplectic space?
The broadest definition of angular momentum is that it is the infinitesimal generator of spatial rotations. Concretely, let $\mathbf x$ be a ...
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5
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How to see the ${\rm SO}(4)$ symmetry of the classical Kepler problem?
The classical treatment is actually simpler than the quantum one, since there are no operator ordering problems. This is what Pauli relied on in his solution of the Hydrogen atom, before Schroedinger, ...
- 66.3k
4
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Show that the Laplace-Runge-Lenz vector is conserved using poisson brackets
Recall $A^2 = m^2k^2+2mEL^2$. We can first prove the magnutude is constant:
$$[A^2,H] = [m^2k^2+2mEL^2,H] = [2mEL^2,H]$$
$$ [A^2,H] = 4mEL[L,H] = 4mEL \dot{L} $$
Since $\dot{L} = 0 $, this implies $...
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4
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Is this really $SO(4)$ algebra?
Since both the $L_i$ and the $A_i$ are constants of motion, their Poisson bracket with the Hamiltonian vanishes and so it is "as good as a constant" for purposes of the algebraic structure - ...
- 129k
4
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Quantum Analog to Kepler's First Law
First things first. You know that for the Coulomb potential there are additional invariants, the Laplace-Runge-Lenz vector beyond angular momentum, nonvanishing for elliptical orbits, classically, but ...
- 66.3k
4
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Is the Laplace-Runge-Lenz vector applicable for test particle motion around black holes?
There is no direct generalization of the LRL vector for geodesic motion in black hole spacetimes, however there are some concepts in general relativity that play analogous roles:
Carter constant, a ...
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3
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Conserved Quantities in Kepler Problem?
He could (and should) count the mechanical energy as a conserved quantity. He also didn't mention that the total momentum of the 2 objects is 3 integrals of motion. However, not all quantities are ...
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3
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Why Laplace-Runge-Lenz vector in a circular motion is $0$?
We have $\vec{A}=\vec{v}\times\vec{L}-k\hat{r}$.
The magnitudes follow from the definitions: A circular motion is given exactly when the velocity is given as $v=\omega r$, therefore $\vec{v}\times\vec{...
3
votes
Textbook proof error? Runge Lenz
$\dot r$ means $$\frac{d|\mathbf r|}{dt},$$ not $$\left|\frac{d\mathbf r}{dt}\right|.$$ Your notation $|\dot{\vec{r}}|$ means the latter, not the former. The latter is the speed. The former is the ...
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3
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How is angular momentum defined on symplectic space?
In the Kepler-problem, note that only 5 of the 6 integrals of motion $(\vec{L},\vec{B})$ are algebraically independent, since $\vec{L}\cdot \vec{B}\equiv 0$. They parametrize a 5-dimensional ...
- 213k
3
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What symmetry causes the Runge-Lenz vector to be conserved?
Looking at https://arxiv.org/abs/1207.5001 one gets a very nice solution. If one is not very keen into mathematics, their basic idea is to use the infinitesimal transformation
$$\delta x^i=\epsilon L^{...
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3
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Textbook proof error? Runge Lenz
You don't have the notation quite right. The end result is that
$$\vec{r}\cdot\dot{\vec{r}} = |\vec{r}|\frac{d|\vec{r}|}{dt}.$$
In other words, $\dot{r}$ is the rate of change of the distance from the ...
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2
votes
Textbook proof error? Runge Lenz
They say that $\vec{r} \cdot \dot{\vec{r}} = |\vec{r}|\frac{d}{dt}|\vec{r}|,$ not that $\vec{r} \cdot \dot{\vec{r}} = |\vec{r}||\frac{d}{dt}\vec{r}|.$
- 213k
2
votes
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Minimal set of invariants to specify a Kepler orbit
You're correct, that there must be 5 independent constants of the motion, allowing closed 1-dimensional orbits in the 6-dimensional phase space. I'll show now that from the LRL vector, $\textbf{A}$, ...
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What does closed orbits mean in Quantum Mechanics?
If Runge-Lenz vector in CM is a constant of motion that implied the motion is closed as well as bound, as you rightly stated.
Now, in QM because we are dealing with orbitals, we have no way of knowing ...
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How to relate Laplace-Runge-Lenz vector to eccentricity?
The force is $-mkr^{-2}\hat{r}$; the special angular momentum is $h=r^2\dot{\theta}$. Since $\vec{L}=mh\hat{k}$ is conserved,$$\frac{d}{dt}\left(\vec{p}\times\vec{L}\right)=\vec{F}\times\vec{L}=m^2khr^...
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2
votes
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Symmetry and Symplectic Group of Hydrogenic Atom
Even though the intro to your question conjures up Pauli's legendary quantization of the Hydrogen atom using the rotational so(3) symmetry and the suitably normalized LRL vector which can be combined ...
- 66.3k
2
votes
Quantum Analog to Kepler's First Law
In general, you should simply stop thinking of quantum theory in terms of classical ideas.
However, this particular issue has a simple answer.
There is an analogous problem in scattering, that when a ...
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2
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Showing that the angular momentum operators and Laplace-Runge-Lenz operator together are generators of $SO(4)$
This is not the math site. Just follow the strategy of wikipedia and, noting that H commutes with L and A, it can be moved out of their commutators just like a constant number. (In its diagonal ...
- 66.3k
1
vote
Energy eigenvalue of hydrogen-like atoms using Laplace-Runge-Lenz vector
Your first task is to absorb all superfluous constants into your nondimensionalized variables, and do the same for the nice review by Valent which is required reading, if you cannot follow WP or Pauli....
- 66.3k
1
vote
What exactly are the 12 conserved quantities in the Two-Body Problem?
Before getting started, it's worth mentioning that you don't actually need twelve constants of motion, you only need six, one per degree of freedom. And you get that total energy is conserved for free,...
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1
vote
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Runge-Lenz vector when the earth is neither at the aphelion nor at the perihelion
Yes, you are right.
The Laplace-Runge-Lenz vector is constantly pointing in the same direction.
See the red vector $\mathbf{A}$ in the image below.
Figure 1: The LRL vector $\mathbf{A}$ (shown in red)...
- 41k
1
vote
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Component-free computation of Poisson bracket of Laplace-Runge-Lenz vector
This answer uses the terminology and notation of geometric algebra and geometric calculus.
A vectorial view of the Poisson bracket
The Poisson bracket of arbitrary scalar functions $G$ and $H$ of the ...
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vote
Deriving Laplace-Runge-Lenz Vector
You just got wrong the chain rule. You wrote
$$\frac{\text{d}}{\text{d}t} V(||x||) = \frac{\text{d}V(||x||)}{\text{d}t}\frac{\text{d}||x||}{\text{d}t}.$$
But it should be
$$\frac{\text{d}}{\text{d}t} ...
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Only top scored, non community-wiki answers of a minimum length are eligible