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Time-Dependant Schro Composite System

  1. Sep 14, 2009 #1
    1. The problem statement, all variables and given/known data
    Given:
    [tex]a\partial\psi/\partial t = \widehat{H}\psi[/tex]

    Consider a is an unspecified constant. Show this equation has the following property. Let [tex]\widehat{H}[/tex] be the Hamiltonian of the system composed of two independant parts:
    [tex]\widehat{H}(x_{1},x_{2}) = \widehat{H_{1}}(x_{1}) + \widehat{H_{2}}(x_{2})[/tex]

    and the stationary states of the composite system are:
    [tex]\psi(x_{1},x_{2}) = \psi_{1}(x_{1},t)\psi_{2}(x_{2},t)[/tex]

    Show that this product form is a solution to the preceding equation for the given Hamiltonian.


    2. The attempt at a solution
    I have a feeling this is the wrong starting point but I do this (I assume I can use separation of variables)

    [tex]\psi(x_{1},x_{2}) = \psi_{1}(x_{1},t)\psi_{2}(x_{2},t) = \vartheta_{1}(x_{1})T_{1}(t)\vartheta_{2}(x_{2})T_{2}(t)[/tex]


    After plugging that into the first equation and some algebra I seperate the variables and then set them equal to some constant E. I then have:

    [tex]\partial/\partial t (T_{1}T_{2}) = (1/a)ET_{1}T_{2}[/tex]

    and

    [tex]\widehat{H}_{1}\vartheta_{1} + \widehat{H}_{2}\vartheta_{2} = E\vartheta_{1}\vartheta_{2}[/tex]

    and from here, assuming I did everything correctly, I don't know how to continue.
     
  2. jcsd
  3. Sep 15, 2009 #2

    kuruman

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    ** Edit ** Removed incorrect answer.
     
    Last edited: Sep 15, 2009
  4. Sep 15, 2009 #3

    kuruman

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    I interpret

    [tex]
    \widehat{H}(x_{1},x_{2}) = \widehat{H_{1}}(x_{1}) + \widehat{H_{2}}(x_{2})
    [/tex]

    to mean that

    [tex]
    \widehat{H}(\vartheta_{1}\vartheta_{2}) = \vartheta_{2}\widehat{H_{1}}(\vartheta_{1}) + \vartheta_{1}\widehat{H_{2}}(\vartheta_{2})
    [/tex]

    Your interpretation is dimensionally incorrect, if nothing else. This should make a difference.
     
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