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Goldstein schodinger's equation Lagragian problem.

  1. Oct 12, 2008 #1
    Problem 3 in the continuous systems and fields chapter of (the first edition, 1956 printing) of Goldstein's classical mechanics has the following Lagrangian:

    [tex]
    L = \frac{h^2}{8 \pi^2 m} \nabla \psi \cdot \nabla \psi^{*}
    + V \psi \psi^{*}
    + \frac{h}{2\pi i}
    ( \psi^{*} \dot{\psi}
    - \psi \dot{\psi}^{*} )
    [/tex]

    The problem is to treat [itex]\psi[/itex] and its conjugate as independent field variables and show that this generates Schodinger's equation and its conjugate.

    Doing the problem I find I need [itex]h/4\pi i[/itex] in this last term to make it work out. Could somebody with a newer edition of this text see if this is a corrected typo?
     
  2. jcsd
  3. Oct 13, 2008 #2

    malawi_glenn

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    Still the same eq for the Lagrangian density.

    I have third edition.
     
  4. Oct 13, 2008 #3

    Ben Niehoff

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  5. Oct 13, 2008 #4
    Does anybody see where I went wrong:

    [tex]
    L
    = \frac{h^2}{8 \pi^2 m} \nabla \psi \cdot \nabla \psi^{*} + V \psi \psi^{*} + \frac{h}{2 \pi i} ( \psi^{*} \partial_t \psi - \psi \partial_t \psi^{*} )
    [/tex]
    [tex]
    = \frac{h^2}{8 \pi^2 m} \partial_k \psi \partial_k \psi^{*} + V \psi \psi^{*} +
    \frac{h}{2 \pi i} ( \psi^{*} \partial_t \psi - \psi \partial_t \psi^{*} )
    [/tex]

    We have
    [tex]
    \frac{\partial L}{\partial \psi^{*} } = V\psi + \frac{h}{2 \pi i} \partial_t \psi
    [/tex]

    and canonical momenta
    [tex]
    \frac{\partial L}{\partial{(\partial_k \psi^{*})}} = \frac{h^2}{8 \pi^2 m} \partial_{k} \psi
    [/tex]
    [tex]
    \frac{\partial L}{\partial{(\partial_t \psi^{*})}} = -\frac{h}{2 \pi i} {\psi}
    [/tex]

    [tex]
    \frac{\partial L}{\partial \psi^{*}} = \sum_k \partial_k \frac{\partial L}{\partial{(\partial_k \psi^{*})}} + \partial_t \frac{\partial L}{\partial{(\partial_t \psi^{*})}}
    [/tex]
    [tex]
    V\psi + \frac{h}{2 \pi i} \partial_t \psi = \frac{h^2}{8 \pi^2 m} \sum_k \partial_{kk} \psi -\frac{h}{2 \pi i} \frac{\partial \psi}{\partial{t}}
    [/tex]

    which is off by a factor of two in the time term
    [tex]
    -\frac{h^2}{8 \pi^2 m} \nabla^2 \psi + V\psi = \frac{h i}{\pi} \frac{\partial \psi}{\partial{t}}
    [/tex]
     
  6. Oct 13, 2008 #5

    samalkhaiat

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  7. Oct 13, 2008 #6
    Thanks Sam.

    Malawi,

    What page and chapter is this problem in, in the third edition?

    EDIT: fyi. I reported the problem and got the following response:

    > I looked in the 1st, 4th and 6th printings of the 3rd edition. The problem appears on page 599 (#4). The correction you sent was made before the 1st printing since the 1/4 appears on page 599 of all printings.
     
    Last edited: Oct 14, 2008
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