The Aharonov -Bohm effect

For long since the electrodynamics has developed, we are familiar with the Vector Potential. The Vector potential was for thought to be just a tool to develop the Maxwell’s equations. It was ignored in the macroscopic physical world. But there is a deep underlying physical significance of it in quantum mechanics, and also the deep connection of topology with physics (This experiment can be considered as the simplest example)

As we know any vector can be written as a curl of another vector, so the magnetic field vector B can be written as B=curl A. this A is called the vector potential.

fig: a solenoid showing Aand B

Apparently A doesn’t look very much physically significant. But it is. And it was first recognized by 2 Physicists Aharonov and Bohm. In the early 60s Aharonov and Bohm published papers and describing an effect due to a certain experimental set up of old double slit experiment. What we now know as Aharonov-Bohm effect.

(D.Bohm and Y.Aharonov, "Significance of electromagnetic potentials in the quantum theory", Phys. Ref., 115 (1959) 485 and - "Further considerations on electromagnetic potentials in the quantum theory", Phys. Rev. 123 (196) 1511.
G. M811enstedt and W.Bayh, "The Continuous variation of the phase of electron waves in field free space by means of the magnetic vector potential of a solenoid", Phys. BlUrt. 18 (1962) 299.)

In this experimental set up, think of a long solenoid is set just after the double slit. (Originally a very special kind of magnetic field is used.

[as its difficult to place a long solenoid in that tiny space] Under certain circumstances iron crystals can grow in the form of very long, microscopically thin filaments called whiskers. These whiskers can be magnetized and they then act like very tiny solenoids: Hence a purely static magnetic field concentrated in the whiskers (the field outside is extremely small and can be neglected)) So there is no magnetic and electric field present at the gap between the slits and the screen (outside the whiskers). The observed phenomenon was a shift of interference pattern just due to the placement of that solenoid. So, what could affect the charged particles on its path if no field was present?

Classically we can’t find the whisker unless we hit it by an electron, but quantum mechanically it can be detected through the electrons passing by it. This is very strange as we have set the B and E zero... Then what else could affect these particles? As they cant even go through the regions where there is non- zero (inside the solenoid) magnetic field? And there we can bring the term the vector Potential A. The only remaining term exists outside the solenoid is A. Which might affect the electrons and thus we have the shift of interference pattern. So we have no chance to think this Vector Potential Physically insignificant any more.

In the next post, we'll try to dig a little more a little calculations ...and we'll see how topology connects to physics in this very simple experiment (consider the solenoid as a hole in a flat sheet which is not simply connected) of electromagnetism at quantum level.

Ref. and for further study.
1. Lewis H. Ryder -Quantum Field Theory (1st ed) (pg-101)
2. Felsager B. - Geometry, particles and fields(2nd ed) (pg-49)
3. R.P Feynman- Lectures on Physics (Vol-II)(12th ed) (pg-895)
4. Sakurai J J- Modern Quantum Mechanics(2nd revised ed) (pg- 136)
5. M Nakahara- Geometry, Topology and Physics(1st ed) (pg- 356)
6. Peshkin M., Tonomura A.- The Aharonov-Bohm effect -(Lecture notes in Physics series of Springer-Verlag, Its a Complete resource of this topic!)
7. Wikipedia