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Hamilton-Jacobi Equation related to Schrodinger?

  1. Oct 14, 2011 #1
    Hamilton-Jacobi Equation related to Schrodinger??

    Where it comes from the Schrodinger equation? Is it related to Hamilton-Jacobi equation? And
    any good text to consult??
     
  2. jcsd
  3. Oct 14, 2011 #2

    dextercioby

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    Re: Hamilton-Jacobi Equation related to Schrodinger??

    You mean the JWKB approximation ? It's typically discussed in almost all serious books on quantum mechanics.
     
  4. Oct 15, 2011 #3

    Demystifier

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    Re: Hamilton-Jacobi Equation related to Schrodinger??

    See e.g. Sec. 2 of
    http://xxx.lanl.gov/abs/quant-ph/0505143 [Found.Phys.Lett. 19 (2006) 553-566]
     
  5. Oct 16, 2011 #4
    Re: Hamilton-Jacobi Equation related to Schrodinger??

    If one considers the wave function in terms of it's amplitude and phase
    [tex]
    \Psi(\vec{x},t)=A(\vec{x},t)e^{i S(\vec{x},t)/\hbar}
    [/tex]
    and substitutes this into the Schrodinger equation one gets two equations
    [tex]
    -\frac{\hbar^2}{2M}\nabla^2 A+\frac{1}{2M}A(\vec{\nabla}S)^2+WA=-A\frac{\partial S}{\partial t}
    [/tex]
    [tex]
    -\frac{1}{2M}[A\nabla^2 S+2(\vec{\nabla} A)\cdot (\vec{\nabla}S)]=\frac{\partial A}{\partial t}
    [/tex]

    These can be shown to be the Hamilton-Jacobi equation and the continuity equation respectively.
     
  6. Oct 16, 2011 #5

    Bill_K

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    Re: Hamilton-Jacobi Equation related to Schrodinger??

    They're not quite the same, are they. For a simple Hamiltonian H = (1/2m) p2 + V(x), the Hamilton-Jacobi Equation is a first-order equation while the Schrodinger Equation is second-order, and has ∂2S/∂x2 in place of (∂S/∂x)2. Books like Goldstein explain the relationship -

    Hamilton-Jacobi Eq : Schrodinger Eq :: Eikonal Eq : Wave Eq

    In words, "The Hamilton-Jacobi Equation tells us that classical mechanics corresponds to the geometrical optics limit of a wave motion."
     
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