Reformulating the Hamilton-Jacobi equation: A step-by-step guide

[QuantumClue]

The Hamilton-Jacobi equation

\frac{\partial W}{\partial t}+\frac{1}{2m}\left[\left(\frac{\partial W}{\partial x}\right)^2+\left(\frac{\partial W}{\partial y}\right)^2+\left(\frac{\partial W}{\partial z}\right)^2\right] + V(x,y,z) = 0

It is said that this can be re-formulated as |\nabla W| = \sqrt{2m(E-V)}.

This part is unclear. How do I rearrange the equation to fit that equation? I know the \nabla is the gradient expressing the three dimensional rectangular coordinates, but I am unsure as to how to rearrange the formula completely so a derivation step-by-step would be appreciated.

Thanks

 

[dextercioby]

I think one makes a separation

W(t,x,y,z) = -E t + W(x,y,z)

 

[QuantumClue]

I think one makes a separation

W(t,x,y,z,t) = -E t + W(x,y,z)

You have the time notation twice in the first parenthesis. Did you mean this?

 

[dextercioby]

Sorry, it's been corrected now. The separation is standard if the Hamiltonian is time-independent.

 

[QuantumClue]

Ok thank you.