- #1
xahdoom
- 3
- 0
This isn't a homework problem - I can't understand a particular statement in my professor's notes. As such, I hope it's in the correct forum.
The Hamiltonian for a charged particle in a potential field A is
[itex]\hat{H}[/itex] = (1/2m) ( -i [itex]\hbar[/itex] [itex]\nabla[/itex] - q A)[itex]^{2}[/itex]
The square bracket can be expanded.
In my professor's notes, this expands to [itex]\hat{H}[/itex] = (1/2m) ( -[itex]\hbar[/itex][itex]^{2}[/itex][itex]\nabla[/itex][itex]^{2}[/itex] + q[itex]^{2}[/itex]A[itex]^{2}[/itex] + 2 q i [itex]\hbar[/itex] A[itex]\bullet \nabla[/itex] + q i [itex]\hbar[/itex] ( [itex]\nabla \bullet [/itex] A )
When I attempt the expansion myself, I don't get the factor of 2 present in the 3rd term of the expansion. I know that it must be there - subsequent proofs using the Landau gauge don't work without it - but I don't understand where it came from.
Any help in understanding the reasoning behind this would be greatly appreciated.
Homework Statement
The Hamiltonian for a charged particle in a potential field A is
[itex]\hat{H}[/itex] = (1/2m) ( -i [itex]\hbar[/itex] [itex]\nabla[/itex] - q A)[itex]^{2}[/itex]
The square bracket can be expanded.
Homework Equations
In my professor's notes, this expands to [itex]\hat{H}[/itex] = (1/2m) ( -[itex]\hbar[/itex][itex]^{2}[/itex][itex]\nabla[/itex][itex]^{2}[/itex] + q[itex]^{2}[/itex]A[itex]^{2}[/itex] + 2 q i [itex]\hbar[/itex] A[itex]\bullet \nabla[/itex] + q i [itex]\hbar[/itex] ( [itex]\nabla \bullet [/itex] A )
The Attempt at a Solution
When I attempt the expansion myself, I don't get the factor of 2 present in the 3rd term of the expansion. I know that it must be there - subsequent proofs using the Landau gauge don't work without it - but I don't understand where it came from.
Any help in understanding the reasoning behind this would be greatly appreciated.