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Schrödinger Equation in the classical limit

  1. Oct 11, 2014 #1

    Erland

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    I am currently trying to learn a little about quantum mechanics, although not on very detailed level. There is one thing I wonder:

    What happens with the Schrödinger Equation in the classical limit, i.e. when either the mass of the particle tends to infinity or when Planck's constant tends to 0?
    Somehow, this should be reduced to classical physics, similar to letting c tend to infinity in the Lorentz Transformation leads to the Galilei Transformation. But I cannot see how we get some classical equation from the Schrödinger Equation in a similar case...
     
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  3. Oct 11, 2014 #2

    mfb

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  4. Oct 11, 2014 #3

    samalkhaiat

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    Big mass does not imply classical object. So the classical limit, as you will see below, concerns only with Planck constant.
    Start from the Schrodinger equation (if you know it):
    [tex]i \hbar \frac{ \partial \Psi ( x , t )}{ \partial t } = H ( \hat{ x } , \hat{ p } ) \Psi ( x , t ) .[/tex]
    Now substitute
    [tex]\Psi ( x , t ) = R( x , t ) e^{ i S ( x , t ) / \hbar } ,[/tex]
    in the Schrodinger equation, you find
    [tex]\frac{ i \hbar }{ R } \frac{ \partial R }{ \partial t } = \frac{ \partial S }{ \partial t } + H ( \hat{ x } , \frac{ \partial S }{ \partial \hat{ x } } ) .[/tex]
    So formally, in the limit [itex]\hbar \rightarrow 0[/itex], you obtain the classical Hamilton-Jacobi equation
    [tex]\frac{ \partial S }{ \partial t } + H ( x , \frac{ \partial S }{ \partial x } ) = 0 .[/tex]
    What really is happening in the classical limit is that, there is a wave packet of width much larger than the de Broglie wave length, in the detailed Scrodinger equation you will have
    [tex] \frac{ \hbar^{ 2 } }{ 2 m } | \frac{ \nabla^{ 2 } R }{ R } | \ll | \frac{ ( \nabla S )^{ 2 } }{ 2 m } | .[/tex]
     
  5. Oct 12, 2014 #4

    Erland

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    Thanks! I guess I have to look up Hamilton-Jacobi theory.
     
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