# Conjugate momentum in the hamiltonian

1. May 25, 2010

### moobox

Hey,

I just have a quick question that I haven't quite been able to find a definitive answer to, regarding conjugate momenta in the Hamiltonian.

Ok, so it regards the following term for the hamiltonian in a magnetic field:

$$H=\frac{1}{2m}(p-qA)^2$$

I'd like to ask whether $$p$$ is the conjugate momentum or if $$p_c=p-qA$$ is the conjugate momentum. As a guess, I would say that $$p_c=p-qA$$ is the conjugate momentum, as it seems to me that the hamiltonian should take into account the magnetic field. Would this then mean that the hamiltonian could be written as $$H=\frac{1}{2m}(p_c)^2$$

Also, very important, does $$-i\hbar\nabla$$ represent the canonical momentum operator or the classical/mechanical momentum operatpor?

Im sure the answers are around somewhere on the internet, but it strikes me that there are some conflicting statements and a tendency to just go "oh yeah, now we swap the canonical momentum, $$p$$ for mechanical momentum $$p$$" and the like, so it would be nice to get a definitive answer.

2. May 25, 2010

### genneth

p is the conjugate/canonical momentum. p-eA is the mechanical momentum.

3. May 25, 2010

### moobox

Ah, brilliant! Thanks for your quick response!

So that would make $$i\hbar\nabla$$ the conjugate/canonical momentum operator?