How to write velocity operator of a given Hamiltonian in quantum mechanics?

Question

Let us have a 2D lattice model with three sites in one unit-cell (basically 2D Kagome lattice). In Fourier space, the Hamiltonian is written as
$$
H = sum_{k_x,k_y}
begin{bmatrix}
a^dagger & b^dagger & c^dagger
end{bmatrix}
begin{bmatrix}
h_{11}&h_{12}&h_{13}
h_{21}&h_{22}&h_{23}
h_{31}&h_{32}&h_{33}
end{bmatrix}
begin{bmatrix}
abc
end{bmatrix}equivPsi^dagger h({k_x,k_y}) Psi
$$
here $a,b,c$ are operators of particles on 3 sites of the unit-cell.
Question:
How to write the velocity operator for $a,b,c$ particles in x-direction and y-direction separately.
My attempt:
Using Ehrenfest theorem for particles $a$, we have:
$$partial_t a^dagger a = frac{1}{ihbar}[a^dagger a,H]$$
(to not make it a shopping-question:)
I think the Ehrenfest theorem is the right way to write velocity operators. But I don't see how exactly one can separate velocity of particles in x- and y-directions or $k_x,k_y$ directions using this theorem. How does quantum mechanics define velocity in different directions?

Roger Vadim · Answer

In the first quantization representation the velocity operator is obtained using
$$hat{dot{x}} = frac{1}{ihbar}left[hat{x},hat{H}right]_-,$$
which defines the operator of the derivative or the derivative of the operator, depending on whether you work in Schrödinger or Heisenberg picture.
In second quantization one switches to the operators using the usual prescription  - calculating the matrix element between the field operators $psi^dagger(x),psi(x)$.
For trancated Hamiltonians, like it seems to be the case in the question, one is usually interested in the current operator, which is obtained using the continuity equation, as the derivative of the charge operator in the region of interest, e.g.
$$partial_t a^dagger a = frac{1}{ihbar}left[a^dagger a,hat{H}right]_-.$$
A caveat is that one may have some challenges when switching the gauge, e.g., when trying to apply the Kubo formula (doable, but not straightforward).

How to write velocity operator of a given Hamiltonian in quantum mechanics?

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