Notes and problems for Desai Chapter VI.
Posted by peeterjoot on November 29, 2010
Chapter VI notes for .
section 6.5, interaction with orbital angular momentum
He states that we take
and that this reproduces the gauge condition , and the requirement .
These seem to imply that is constant, which also accounts for the fact that he writes .
Consider the gauge condition first, by expanding the divergence of a cross product
This gives us
With we then have
Unless is always perpendicular to we can only have a zero divergence when is constant.
Now, let’s look at . We need another auxillary identity
Here the gradients are all still acting on both and . Expanding this out by chain rule we have
With , and , we have
We note that , and
If is constant, we have
as desired. Now this would all likely be a lot more intuitive if one started with constant and derived from that what the vector potential was. That’s probably worth also thinking about.
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.