Laplace-Runge-Lenz vector problem

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In summary, the vector A is defined as the quantum version of the Laplace-Runge-Lenz vector. The system Hamiltonian is given by H and it can be shown that the commutator of Ai and Aj is equal to - 2iħ/m H εijk Lk. This calculation may seem daunting, but with a systematic approach, it can be solved effectively.
  • #1
wantommy
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The vector A is defined as the quantum version of the Laplace-Runge-Lenz vector,
YLRSKU2.jpg


where
hPMi7Th.jpg


The system Hamiltonian is given by
kP7BCI0.jpg


Show that
4ntaIGk.jpg




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this problem confuse me for at least 2 years...
i can't derive it :cry:

Can any expert help me?
hope someone can write down specific derivation

Thanks a lot!
 
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  • #2
Mathematician's reply: Let X be any vector XxX = 0 (vector).
 
  • #3
wantommy said:
this problem confuse me for at least 2 years...
i can't derive it :cry:
wantommy, This is an excellent exercise. Even if you can find a complete solution on the web, I encourage you to work it out for yourself. I know of no shortcut, but with a systematic approach it shouldn't take you two years!

Hints: First of all, ditch the cross product notation and write out the components. This problem is about commutators, and the cross product obscures the factor ordering. For example, what you want to show is really

[Ai, Aj] = - 2iħ/m H εijk Lk

Second, build the result up a piece at a time. Start with the commutators that are obvious:

[xi, xj] = 0
[pi, pj] = 0
[pi, xj] = iħ δij

and most importantly,

[Li, Vj] = - i εijk Vk

where V is any vector operator.

Collect intermediate results like [pi, Lj], [xi, Lj] and [Ai, Lj].

Whoops, were you paying attention? :smile: You don't even have to work these out! They're all instances of [Li, Vj].

Finally, make use of well-known identities like

εijk εklm = δil δjm - δim δjl and

[A, BC] = [A,B] C + B [A,C]

Learning how to do a calculation systematically (and learning how to get it right!) is an important part of a physics education.
 

Related to Laplace-Runge-Lenz vector problem

1. What is the Laplace-Runge-Lenz vector problem?

The Laplace-Runge-Lenz vector problem is a mathematical concept in classical mechanics that describes the motion of a particle in a central force field. It is named after Pierre-Simon Laplace, Carl Runge, and Wilhelm Lenz, who all contributed to its development.

2. What is the significance of the Laplace-Runge-Lenz vector in physics?

The Laplace-Runge-Lenz vector is significant because it is a conserved quantity in a central force field, meaning that its magnitude and direction remain constant throughout the motion of the particle. This vector also plays a crucial role in determining the shape and orientation of the trajectory of the particle.

3. How is the Laplace-Runge-Lenz vector calculated?

The Laplace-Runge-Lenz vector is calculated using the position and velocity of the particle, as well as the mass and gravitational constant of the central body. It can be expressed as a vector equation or in terms of its components.

4. What are some real-life applications of the Laplace-Runge-Lenz vector problem?

The Laplace-Runge-Lenz vector problem has applications in various fields, including celestial mechanics, astrodynamics, and spacecraft trajectory planning. It is also used in the study of planetary orbits and the motion of comets and other celestial bodies.

5. What are some limitations of the Laplace-Runge-Lenz vector problem?

The Laplace-Runge-Lenz vector problem is a simplified model that assumes a spherical central body and a circular orbit. In reality, many celestial bodies and their orbits are not perfectly spherical or circular, so this model may not accurately predict their motion. Additionally, it does not account for other forces such as atmospheric drag or the gravitational influence of other bodies.

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