Hamilton-Jacobi Equation related to Schrodinger?

[Casco]

Hamilton-Jacobi Equation related to Schrödinger??

Where it comes from the Schrödinger equation? Is it related to Hamilton-Jacobi equation? And
any good text to consult??

 

[dextercioby]

You mean the JWKB approximation ? It's typically discussed in almost all serious books on quantum mechanics.

 

[Demystifier]

Where it comes from the Schrödinger equation? Is it related to Hamilton-Jacobi equation? And
any good text to consult??

See e.g. Sec. 2 of
http://xxx.lanl.gov/abs/quant-ph/0505143 [Found.Phys.Lett. 19 (2006) 553-566]

 

[jfy4]

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

These can be shown to be the Hamilton-Jacobi equation and the continuity equation respectively.

 

[Bill_K]

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 Schrödinger Equation is second-order, and has ∂²S/∂x² in place of (∂S/∂x)². Books like Goldstein explain the relationship -

Hamilton-Jacobi Eq : Schrödinger 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."