Descriptiveness of this term

[Anton Alice]

If you want to modify the Hamiltonian by introducing the effect of an electromagnetic field, then the replacement $$\vec p \rightarrow \vec p - \frac{e}{c}\vec A$$ is applied.

Now my question is, whether there is a descriptive meaning of that extra term $- \frac{e}{c}\vec A$. As what can I think of that?
Thank you in advance

 

[say_cheese]

One way to look at it is that it is the momentum imparted through the electromagnetic force,

Momentum wikipedia -
The classical Hamiltonian ℋ for a particle in any field equals the total energy of the system – the kinetic energy T = p2/2m (where p2 = p · p, see dot product) plus the potential energy V. For a particle in an electromagnetic field, the potential energy is V = eφ, and since the kinetic energy T always corresponds to the kinetic momentum p, replacing the kinetic momentum by the above equation (p = P − qA) leads to the Hamiltonian in the table.

 

[Anton Alice]

For a particle in an electromagnetic field, the potential energy is V = eφ, and since the kinetic energy T always corresponds to the kinetic momentum p, replacing the kinetic momentum by the above equation (p = P − qA) leads to the Hamiltonian in the table.

If small type p is the kinetic momentum, what then is capital P?

 

[DrClaude]

If small type p is the kinetic momentum, what then is capital P?

It's the canonical momentum, i.e., the total momentum of the particle.