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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Conserved Quantities in the Two-Body Problem
I'm having trouble matching the number of conserved quantities in the two-body problem with the number of differential equations.
We know that, considering positions and velocities in the two-body ...
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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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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 but I cannot find a proof or figure it out on my own.
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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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How is angular momentum defined on symplectic space?
In classical mechanics on a flat 3D vector space (which we will refer to as configuration space), we can define position and momentum vectors $x$ and $p$. We can then define angular momentum as $L = r ...
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Physical interpretation of the symmetry for the Runge-Lenz vector
In the post What symmetry causes the Runge-Lenz vector to be conserved?, and based on the results of https://arxiv.org/abs/1207.5001, it was it was discussed that the Runge-Lenz vector is the ...
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What does closed orbits mean in Quantum Mechanics?
In classical mechanics, The inverse potential
$$\phi=-\frac{C}{r}$$
leads to closed orbits. The planetary motion is one example. The the conservation of the
$$\mathbf{n}=\frac{\mathbf{p}\times\mathbf{...
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How to see the ${\rm SO}(4)$ symmetry of the classical Kepler problem?
It is well-known that the hidden symmetry of the $1/r$-problem $$H=\frac{{\bf p}^2}{2m}-\frac{k}{r}\tag{1}$$ is ${\rm SO(4)}$ in the sense that the components of angular momentum $L_i$ and those of ...
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Deriving Laplace-Runge-Lenz Vector
consider a particle in a central potential, i.e. the potential $V$ only depends on the distance $ r = \| \vec{x} \|$ to the origin. The equation of motion thus reads
$$ m\ddot{\vec{x}}=-\frac{\partial}...
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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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Connection between Kepler Problem and Harmonic Oscillator
Background. Take the Kepler Lagrangian as
$L^K = \frac{1}{2}\dot{q}_i\dot{q}_i + \frac{k}{q}$,
and the Lagrangian for the isotropic harmonic oscillator as
$L^H = \frac{1}{2}\dot{q}_i\dot{q}_i - \...
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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 ...
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Degeneracy of Hamiltonian for Hydrogen Atom (Coulomb Problem)
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 ...
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$SO(4,2)$ symmetry of the hydrogen atom
The hydrogen atom with Hamiltonian obviously has $SO(3)$ symmetry since it just depends on the radius.
$$ H = \frac{\mathbf{p}^2}{2m} - \frac{k}{r}$$
This is generated by angular momentum $\mathbf{L}...
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How I can prove the Commutation between hamiltonian and Runge-Lenz vector? [closed]
I am a undergraduate student in physics. I found this page that shows a way to prove the commutator between Runge-Lenz vector and Hamiltonian
.$\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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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:
$$\hat{A}=\frac{i}{\hbar}\left[\hat{p},\,\frac{1}{2}\hat{L}^{2}-k\left|\hat{r}\right|\right]=\frac{1}{2}\left(\hat{p}\...
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Kepler problem: flows generated by constants of motion
This is part of an admission exam problem, found at
http://www.sissa.it/mp/admission/tests//2008_common.pdf
Consider the Hamiltonian of Kepler problem
$$H(\boldsymbol{r},\boldsymbol{p})= \frac{|\...
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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:
$$
H = (\mathbf p^2/2 \mu)-(e^2/...
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Index Manipulation and Angular Momentum Commutator Relations
I have been trying for hours and cannot figure it out. I am not asking anyone to do it for me, but to understand how to proceed.
We have the relations
$$[L_i,p_j] ~=~ i\hbar\; \epsilon_{ijk}p_k,$$
...
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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 ...
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