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By obtaining the current of probability and multiplying by the electronic charge $e$, one can obtain the electric current density. By "replacing" the wavefunction by the field operator, one has

$$\mathbf{j}(\mathbf{r})=-i\frac{\hbar}{2m}\left[\Psi^{\dagger}(\mathbf{r})\nabla\Psi(\mathbf{r})-\left(\nabla\Psi^{\dagger}(\mathbf{r})\right)\Psi(\mathbf{r})\right].$$

How can this result be obtained by defining $$\mathbf{j}(\mathbf{r}) = e\sum_i\mathbf{v}_i\delta(\mathbf{r}-\mathbf{r}_i)$$

Thanks

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  • $\begingroup$ The j(r) composed of field operators operates on the Fock space, so you might get an expression similar to the second j(r) if applied on the right multi-particle state. It is a quantum mechanical operator, its eigenvalues will be probability currents, whereas the second j(r) consists of charge current densities. Above all, the classical expression of $j(r)$ supposes the knowledge of the position of the particles involved what is not the case of $j(r)$ composed of quantum-mechanical operators. They are very different since the first is based on QM, whereas the second on classical physics. $\endgroup$ – Frederic Thomas Sep 5 '17 at 15:37
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The two operators are not identical, however they approach to each other in the semiclassical limit. To show that, we can compute matrix elements of the two expressions in a complete set of states and show that the matrix elements approach each other. (Strictly speaking this is weak convergence).

I'll perform the computation in one dimension, the generalization to higher dimensions is straightforward.

A possible basis is the coherent state basis. The coherent states are eigenstates of the field operator $\Psi(x)$. The coherent states basis is overcomplete and especially convenient for semiclassical approximations.

In this basis each Schrodinger particle is labeled by a position and momentum and for $n$ Schroedinger particles, it has the form $| x_1, ..., x_n; p_1, ..., p_n \rangle $, such that the action of the field operator on the coherent state:

$$\Psi(x) | x_1, ..., x_n; p_1, ..., p_n \rangle = \prod_{i=1}^n \sqrt[\LARGE 4]{\frac {m \omega}{ \pi \hbar }} e^{ \large -\frac{m \omega}{2\hbar}(x-x_i)^2 + \frac{ixp_i}{\hbar}} | x_1, ..., x_n; p_1, ..., p_n \rangle $$

The factors in the right hand side consist of the coherent state wave functions that Schrodinger wrote in 1926.

The constant parameter $\omega$ is a constant having units of angular frequency.

The matrix element of the electric current ( I took the liberty of adding the particle electric charge to the expression) can be readily computed:

$$\langle x_1, ..., x_n; p_1, ..., p_n |j(x) | x_1, ..., x_n; p_1, ..., p_n \rangle = e \sum_{i=1}^n \frac{p_i}{m} \sqrt{ \frac { m \omega}{ \pi \hbar}} e^{-\frac{m \omega}{\hbar}(x-x_i)^2 }$$

The first factors are just the velocities:

$$v_i = \frac{p_i}{m}$$

The second term approaches a delta function in the semiclassical limit:

$$\lim_{\hbar \to 0} \sqrt{ \frac {m \omega }{ \pi \hbar}} e^{-\frac{m \omega}{\hbar}(x-x_i)^2 } = \delta(x-x_i)$$

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