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Laplace-Runge-Lenz vector problem

  1. Mar 18, 2014 #1
    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



    ------------------------------------------------------------------

    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!!!!
     
  2. jcsd
  3. Mar 18, 2014 #2

    mathman

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    Gold Member

    Mathematician's reply: Let X be any vector XxX = 0 (vector).
     
  4. Mar 18, 2014 #3

    Bill_K

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    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.
     
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