Probability Current and Current Operators in Quantum Mechanics

In the real space representation of QM, the momentum operator is p̂ = −ih̄~ ∇, and the vector potential depends only on the space coordinates and time. The main topic of these notes is to find and describe the operations and operators that would be used to calculate the electric currents associated with the motion of these particles, assumed to be electrons. That is a useful thing to have, because it will allow for determining their induced electric and magnetic dipole moments, which exhibit themselves in the optical properties of materials. The main question to be addressed here is: What is the operator for the electric current density? Or, given a particular state of a quantum system, how does one find the distribution of electric current density? By the form of this question, mainly we are concerned with states described in real space, so the interest is primarily in their real space representations, or their real space wave functions. Charge current is associated with the quantum motion of the charges. But motion in quantum mechanics is probabilistic, hence, the motion one talks about is how the probability for finding the particle moves aroud with time. So the main idea is that one needs to find a ”probability current” that relates to how the probability for locating the electron might be changing with time, when a wave function satisfies a Schrodinger equation based on the above Hamiltonian. In a semi-classical sense, we need to find the effective velocity operator v̂ or current density operator ĵ for one quantum particle. The electric charge density ρe for an individual electron needs 1The charge will be written as e, although for real electrons that is a negative number, e = −4.8 × 10−10 esu, in CGS units, or e = −1.602× 10−19 coulombs, in SI units. These notes use CGS.