# Electron in a field: Canonical momentum versus kinematical momentum

1. May 30, 2013

### IFNT

Can anyone explain to me the definition of canonical and kinetic momentum?

The Schrödinger equation for an electron in a laser field can be written in the length gauge or the velocity gauge.
In the litterature it is often said that in the length gauge the kinematical momentum is equal to the canonical momentum k=p, while in the velocity gauge the kinematical momentum is the canonical momentum added by the vector potential k=p+A.

Can anyone elaborate on this or has any ref I can look at?

2. Jun 2, 2013

### daveyrocket

The canonical momentum is defined as the derivative of the Lagrangian with respect to velocity. In a system where the potential doesn't have a velocity dependence (ie. one that doesn't have a magnetic field), this will just be the kinematic momentum, p = mv.

For a system with a charged particle in a magnetic field, the potential will have a term with the dot product of the velocity with the vector potential. So when you take a derivative with respect to velocity, in addition to the mv that you get from the kinetic energy, you also get a contribution from the vector potential.

See here for example:
http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/ParticleMagneticField.htm