Aharonov-Bohm Effect
Why should we use A and φ?
In classical electrodynamics we can measure electric potential φ, which is a real physical quantity. But what about the vector potential A?
According to the assumption that Magnetic Monopoles Do Not Exist, we can write the divergence of B:
and use the vector formula
to obtain
.
So in classical electrodynamics A is just a mathematical tool to simplify our problem.
What is the Aharonov-Bohm Effect[1]?
Consider a special double-slit experiment with one change: there is a small shaded region (B≠0) where magnetic flux comes out of the paper. In the classical situation, there is no magnetic field along the classical paths, so this variation can not change the result. But according to quantum mechanics, we will have an extra factor
.
What does the Aharonov-Bohm Effect mean?
In classical level, the vector potential A is just a mathematical tool. But at the quantum level, A is also a real physical quantity!
In summary, E and B can not give us all the information about the electromagnetic field; however A and φ give us more information than we need to describe the electromagnetic field.
[1] R. Shankar, Principles of Quantum Mechanics, second edition.