Electric current in quantum mechanics?

Quick question:

I want to find an expression for the (electric) current density of an electron, in quantum mechanics. Either a single electron or a general charge distribution $\rho$.

Classically j=$\rho$ v.

What should I use here?

Maybe the electric charge multiplied by the probability current?

Thanks.

Answer

Yes, $\vec \jmath(x,y,z)$ should be defined as $e$ times the Schrodinger probability current.

$$\begin{equation*} \vec \jmath = \frac{e\hbar}{2mi}\left(\Psi^* \frac{\partial \Psi }{\partial x}- \left(\frac{\partial \Psi^* }{\partial x}\right)\Psi \right) , \quad e\lt 0. \end{equation*}$$ That's possible to explicitly see in the formalism of quantum field theory. The definition $\vec v(x,y,z)/\rho$ would be no good because "the velocity of the electron at a particular point $(x,y,z)$" isn't too well-defined due to the uncertainty principle (if the position is given, the velocity is not).

One may be puzzled because the expression for $\vec\jmath$ above isn't an operator – it is quadratic in the wave function. But in quantum field theory, it is an operator – an observable – because it is a function of the field operators $\Psi$.

If we consider non-relativistic quantum mechanics with fixed coordinates of particles and we still want to define $\vec\jmath(x,y,z)$ as a linear operator, an observable, we must appreciate that this operator is only nonzero is the particle is located in the infinitesimal vicinity of the point $(x,y,z)$. So we have

$$\rho (x_0,y_0,z_0) = e \delta^{(3)}(\hat{\vec r}-\vec r_0)$$ and $$\vec\jmath (x_0,y_0,z_0) = \frac e2\{\delta^{(3)}(\hat{\vec r}-\vec r_0),\frac{\hat{\vec p}}{m} \}$$ I had to write one-half of the commutator with the velocity operator because functions of positions and velocities don't commute but we still need a Hermitian operator.

If there are $N$ charged particles, the operators $\hat{\vec r}$ and $\hat{\vec p}$ acquire an extra index from $1$ to $N$ and $\rho(x_0,y_0,z_0)$ and $\vec\jmath(x_0,y_0,z_0)$ must be written as a sum of the expressions over this index.

One may verify that e.g. for wave packets, the integrals over $\vec r_0$ (some regions) give us what we would expect.