A De Broglie to Schrödinger to Thomson to Bohm?

[PhilDSP]

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?

 

[Demystifier]

Summary:: Use of Hamilton-Jacobi formalism in QM

My question is then, what did Bohm add to this picture?

The most important contribution of Bohm was understanding what happens during the measurement and showing that all measurable predictions agree with those of standard QM.

 

[PhilDSP]

Okay, thanks. That is a very worthy contribution - more or less what I understand from what I've read so far.

On the other hand, Bohm's take is philosophically a bit apologetic possibly. He is saying, in effect, that the formulations that de Broglie and Schrödinger created, with extension, stand up to the formulations of more popular QM interpretations, isn't he? Whereas the aspirations of de Broglie, Schrödinger and Thomson were to significantly transcend and extend the current understanding of any formulation of QM.

To add some additional philosophic perspective, de Broglie arguably discovered one of the few major physical aspects associated with QM. Three other major physical discoveries I can think of are superconductivity, anti-particles and non-local phenomena.

Being generally more interested in mathematical demonstrations, I would like to know more specifics of what Bohm's mathematical discoveries were and what more recent investigations have highlighted. Perhaps the idea that Lorentz invariance is "emergent" is one such finding and that that is even isotropic to de Broglie's and Thomson's mathematical findings? After all, the world of sub-atomic particles seems to be governed by SU2 and SU3 topologies (which do not essentially lead to the expression of Lorentz invariance).

 

[PeterDonis]

Bohm's take is philosophically a bit apologetic possibly

Please bear in mind that "philosophy" discussions fall into the category of interpretations, and belong in the QM foundations and interpretations forum. A separate thread should be started for such discussions.

The mathematical question you are asking is a valid topic for this forum. Please keep this thread focused on that.