# Why is the generalized momentum replaced by the momentum operator but not the ordinary momentum?

I was trying to understand the derivation of the Hamiltonian for a charged particle in an electromagnetic field. https://en.wikipedia.org/wiki/Hamiltonian_mechanics#Charged_particle_in_an_electromagnetic_field

This Hamiltonian is used frequently in quantum mechanics. In quantum mechanics usually the momentum is replaced by the momentum operator. But in case of this Hamiltonian the generalized momentum is replaced by the momentum operator.

I'm very surprised about that because the generalized momentum isn't actually the momentum of the particle. When I try to calculate the expectation value of a state using this momentum operator, do I actually calculate the expectation value of the momentum of the particle then?

• Generalized momentum is conserved as a result of invariance to spatial translations. Momentum operator commutes with Hamiltonian if the latter is translationally invariant. If this is the case, the eigenvalues of the momentum operator are conserved. Thats the connection. See Weinberg's lectures on quantum mechanics – Cryo Jun 30 at 23:22
• Possible duplicates: physics.stackexchange.com/q/45796/2451 , physics.stackexchange.com/q/104178/2451 and links therein. – Qmechanic Jul 1 at 6:34