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?