http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation

 

The "Classical" HJ equation

 

[math]H + \frac{\partial S}{\partial t} = 0[/math]

 

where the action

 

[math]S = \int L dt[/math]

 

gives rise to

 

[math]\Psi = \Psi_0 e^{i \frac{S}{\hbar}}[/math]

 

so the phase of wave-functions, at some particular point, is equal to the time integral, of the action, at that point, from [math]t=-\infty[/math] to the current time ?

Yes it describes the scattered waveform in terms of the action. Notice the exponential is dimensionless as well.

Hi guys, layman here. Can you give an example (in laymen's terms) of how you'd actually apply or use the math above in a real world situation?

Hi guys, layman here. Can you give an example (in laymen's terms) of how you'd actually apply or use the math above in a real world situation?

 

The Schrodinger equation, the wave equation of all matter makes large use of the exponential in the OP. The HJ equation in particular is useful in helping to describe conserved quantities in physics.

OP? And when you say useful to describe conserved quantities...do you mean like physicists might use the HJ in accounting for data gathered from a collider, for example?