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PhilDSP

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- TL;DR Summary
- Use of Hamilton-Jacobi formalism in QM

I'm investigating some newly conceived Hamiltonians using the approaches of de Broglie and Schrödinger as jumping off points.

Lanczos in "The Variational Principles of Mechanics" p. 278 describes and analyzes them. Neither de Broglie nor Schrödinger really completed the program of the H-J approach. De Broglie got to the point of exchanging Einstein's S = nh idea with S =Φh where Φ is the phase of the wave he described mathematically based largely on the Hamilton concept of geometric optics associated with a particle.

Schrödinger from there replaced S with his second order wave equation having the same wavelength as de Broglie. De Broglie began using Cartesian coordinates but never used the H-J formalism to cast his solution back into Cartesian coordinates and of course neither did Schrödinger.

However, J. J. Thomson, apparently retired at the time, became intrigued by his son's experimental co-discovery of de Broglie's electron waves. Rather amazingly, he found that those waves were actually predicted by the proper use of the Maxwell Equations.

My question is then, what did Bohm add to this picture? De Broglie wrote in his later years about Bohm's work and interpretations but was not convinced they continued on, or fitted his work. Neither did he believe that Schrödinger's wave idea was particularly illuminating.

Are there sources that describe the details of Bohm's H-J derivations and his thinking at the time?

Lanczos in "The Variational Principles of Mechanics" p. 278 describes and analyzes them. Neither de Broglie nor Schrödinger really completed the program of the H-J approach. De Broglie got to the point of exchanging Einstein's S = nh idea with S =Φh where Φ is the phase of the wave he described mathematically based largely on the Hamilton concept of geometric optics associated with a particle.

Schrödinger from there replaced S with his second order wave equation having the same wavelength as de Broglie. De Broglie began using Cartesian coordinates but never used the H-J formalism to cast his solution back into Cartesian coordinates and of course neither did Schrödinger.

However, J. J. Thomson, apparently retired at the time, became intrigued by his son's experimental co-discovery of de Broglie's electron waves. Rather amazingly, he found that those waves were actually predicted by the proper use of the Maxwell Equations.

My question is then, what did Bohm add to this picture? De Broglie wrote in his later years about Bohm's work and interpretations but was not convinced they continued on, or fitted his work. Neither did he believe that Schrödinger's wave idea was particularly illuminating.

Are there sources that describe the details of Bohm's H-J derivations and his thinking at the time?