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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Energy eigenvalue of hydrogen-like atoms using Laplace-Runge-Lenz vector
I have a basic question about a few calculations involving the quantum mechanical Laplace-Runge-Lenz vector.
In classical mechanics there is the Laplace-Runge-Lenz vector, which for a hydrogen-like ...
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What exactly are the 12 conserved quantities in the Two-Body Problem?
The Two-Body problem consists of 6 2nd-order differential equations
\begin{equation}
\ddot{\mathbf{r}}_1 = \frac{1}{m_1}\ \mathbf{F_g} \\
\ddot{\mathbf{r}}_2 = -\ \frac{1}{m_2}\ \mathbf{F_g}
\end{...
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Runge-Lenz vector when the earth is neither at the aphelion nor at the perihelion
Assuming the earth is either at the perihelion or at the aphelion, it is easy to see the Runge-Lenz (RL) vector is directed along the line joining the perihelion and aphelion. Since the RL vector is a ...
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Conserved Quantities in Kepler Problem?
In our classical mechanics class, professor said that Kepler's problem is a kind of Integrable System such that the number of conserved quantities would be equal to the number of degrees of freedom. ...
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Why Kepler problem is equivalent to a free particle on 4 dimensional sphere?
In trying to understand Laplace-Runge-Lenz vector, I read in Wikipedia
that the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four dimensional hypersphere....
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Quantum Analog to Kepler's First Law
According to Kepler's First Law, the orbit of a planet is an ellipse round the sun with the sun at one focus. There's an inherent asymmetry in this. Instead of the sun being in the dead center, its ...
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Component-free computation of Poisson bracket of Laplace-Runge-Lenz vector
How can the Poisson bracket $\{A,H\}$ be computed directly without components, where $H$ is the Hamiltonian for the inverse square force, $$H=\frac{p^2}{2m} - \frac{k}{|r|}\ ,$$
and $A$ is the ...
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Is the Laplace-Runge-Lenz vector applicable for test particle motion around black holes?
In classical mechanics , the Laplace-Runge-Lenz (LRL) vector is a characteristic feature of the Kepler problem. This enables a very simple discussion of the properties of the orbit for the problem. It ...
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Solving Hydrogen atom with ladder operators
Start with a system I fairly understood, the harmonic oscillator.
Here all possible states fulfilling the eigenvalue-equation $H |n\rangle = E_n |n\rangle$ are given by
$$|n\rangle = \dfrac{1}{\sqrt{n!...
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Relation between $SO(4)$ symmetry and conservation of LRL (Lagrangian-Runge-Lentz) vector [duplicate]
In this answer, conservation of LRL vector (classical and quantum)
$$
\vec A=\frac1{2m}(\vec p\times \vec L-\vec L\times \vec p)-\frac{q^2}{r}\hat r,
$$
was related to $SO(4)$ symmetry, but no further ...
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Symmetry and Symplectic Group of Hydrogenic Atom
New version of the question:
A simmetry needs to be canonical, following the first answer of this post which states:
the symmetry requirement is not necessary in the definition of canonical ...
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Why Laplace-Runge-Lenz vector in a circular motion is $0$?
This vector is the sum of two vectors, and I understand why their direction is opposite, but I don't understand why their magnitude is the same.
I know that the direction of this vector is always the ...
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Hydrogen atom as a free particle moving on a three dimensional sphere?
I have heard on various occasions that the Hamiltonian/Lagrangian of the Hydrogen atom or that of a particle moving in $1/r$-potential can be transformed into that of a free particle moving on the ...
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Is this really $SO(4)$ algebra?
The commutation relations involving the components of Runge-Lenz vector of the Hydrogen atom problem, ${\vec A}$ and the angular momentum ${\vec L}$ are given by
$$
[L_i,L_j]=i\hslash\varepsilon_{ijk}...
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How to relate Laplace-Runge-Lenz vector to eccentricity?
So the eccentricity can be written in this form
$$\mathbf{e}=\frac{\mathbf{A}}{mk}
=\frac{1}{mk}(\mathbf{p}\times\mathbf{L})-\hat{\mathbf{r}}$$
but I cannot find a proof or figure it out on my own.
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