Good evening.

@stargazersmith Sure!

@aducharme I had no idea!

So the vector potential is more than just a mathematical convenience. It's a lovely little result.

If that Physics Today article is paywalled, you can access a copy that one of the authors hosts on his university's Digital Commons:

https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1118&context=physicsfacpub

Even though the magnetic field is zero outside the solenoid, the vector potential is not. And when a charged particle like an electron passes through this region, it "feels" the vector potential and acquires a shift in its quantum mechanical phase.

The shift in the electron’s phase affects the way it interferes with itself in a double-slit experiment. You run the experiment with and without the solenoid turned on, and you measure a change in the interference pattern.

https://pubs.aip.org/physicstoday/article/62/9/38/895362/The-Aharonov-Bohm-effects-Variations-on-a-subtle

Aharonov and Bohm pointed out that this vector potential has real physical consequences; it isn’t just a mathematical convenience.

Imagine coiling a wire very tightly around a long cylinder. This is called a (cylindrical) “solenoid.” If you send a current through the wire you get an approximately constant magnetic field inside the cylinder, but essentially no magnetic field outside the cylinder.

The electric and magnetic fields can be recovered from the way these potentials change from point-to-point in space, and over time.

One of the potentials has a handy interpretation in terms of the work done on a charged particle as it is pushed around by an electric field.

But the other potential, which we call the “vector potential,” has no such interpretation. It initially seems like a mathematical convenience that one introduces to simplify magnetic field calculations.

While at Bristol, Bohm and Yakir Aharonov worked out what is usually called the "Aharonov-Bohm Effect." (They were not aware that Ehrenberg and Siday had suggested the same phenomenon a few years earlier.)

The Aharonov-Bohm effect demonstrates of the reality of the vector potential in electrodynamics.

Rather than working with the electric and magnetic fields that exert forces on charged particles, one can formulate electrodynamics in terms of what we call "potentials."

Bohm was still formally at Princeton when he submitted the two papers on his formulation of quantum mechanics, but he had relocated to Brazil by the time they were published. He would remain there for a few more years before moving to Israel, and eventually to Bristol in the UK.

He would spend four years in Bristol before being appointed professor at Birkbeck College, University of London in 1961. Bohm remained there until he retired in the late 80s.

Bohm became a professor at Princeton after the war, but eventually the House Un-American Activities Committee came for him. Princeton caved and suspended him; he was eventually arrested for refusing to testify.

Even though Bohm was acquitted, Princeton no longer wanted him on the faculty. Einstein unsuccessfully lobbied to keep him at the Institute for Advanced Study as a personal research assistant, and other jobs fell through. Bohm eventually secured a position in São Paolo.

Since his work was now classified, and he did not have security clearance, Bohm was not allowed to write his own thesis or present a defense!

Oppenheimer (his advisor) had to go to the University on Bohm's behalf, and make special arrangements to grant his PhD.

Bohm belonged to a number of organizations supporting communism when he was a graduate student. As a result, the military wouldn't grant him security clearance to work on the Manhattan Project.

This was probably fine with Bohm, who opposed US involvement in WWII. But it put him on an unusual trajectory for completing his doctorate. Bohm had performed scattering calculations that were needed by folks working on the Manhattan Project. His work was immediately classified.

Several years later, John Stewart Bell revisited the question of hidden variable theories. He was not satisfied with attempts to explain measurement in the Copenhagen Interpretation, and was encouraged by how the issue was addressed in Bohm’s theory.

He proved a result now referred to as Bell’s Theorem, which essentially rules out the possibility of a local hidden variable theory reproducing the predictions of quantum mechanics. Bohm’s theory evades Bell's argument because it is non-local.

When one first learns QM, a very natural question is “particle or wave?” It’s natural because our macroscopic intuition, unattuned to the microscopic world, thinks of this as a one-or-the-other proposition.

Bohm’s approach provides a simple answer: there’s a particle AND a wave.

But Bohm’s theory is also non-local, exhibiting immediate action-at-a-distance. In that sense it was no less spooky to many physicists than the interpretation he sought to replace. So it didn’t win many adherents.

Unlike the Copenhagen Interpretation, Bohm’s formulation is completely deterministic and asserts that systems have a real and definite configuration that exists even when we haven't performed a measurement to observe it.

The wave function is still there, shepherding particles along. Each particle would travel through one or the other opening in a double-slit experiment, while the wave function would pass through both and interfere, driving particles towards some spots and away from others.

(2) ”A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables. II"
David Bohm, Phys. Rev. 85, 180
https://journals.aps.org/pr/abstract/10.1103/PhysRev.85.180

This formulation, often referred to as De Brogle-Bohm Theory, was published in 1952. It appeared in a pair of papers, both in the January 15th issue of Physical Review.

(1) "A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables. I"
David Bohm, Phys. Rev. 85, 166
https://journals.aps.org/pr/abstract/10.1103/PhysRev.85.166

Bohm devised his own formulation of QM that he felt evaded von Neumann's objections, and might avoid some troubling ideas associated with the Copenhagen Interpretation.

His approach was similar to the “pilot wave” idea first presented by de Broglie at the 1927 Solvay Conference.

That same year, a crotchety old faculty member named Einstein encouraged the young Bohm to study von Neumann’s work scrutinizing “hidden variable theories.”

These are theories that assert additional deterministic variables whose state and dynamics, were they known to us, would resolve the indeterminacy of quantum mechanics.

Bohm’s quantum mechanics textbook was published in 1951. It was very successful, and is still available from Dover as a reprint. It's a great book.

I almost said “inexpensive reprint," but then I checked the Dover website. It's $40 now! A few years ago it was $25, and when I bought it, the book was $14. Dover-flation!

https://store.doverpublications.com/0486659690.html

Anyway, here's my copy: