I can't find any good resources online that explain how we can derive the effect of a uniform electric field on a free electron using the quantum mechanics Hamiltonian formalism.

In our condensed matter notes, it explains that the effect of an electric field on the occupied electron states in an unfilled band is to 'tilt' it towards electron k states in the positive x-direction but I don't understand how this can be proved. Firslty it seems like the electric field would change the the band (since it would affect the energies) rather than just tilt it, also using this classical argument I would expect constant acceleration to higher and higher k states in the positive x-direction.

Therefore I wanted to go back to basics to understand exactly how electric field causing an 'current' (or acceleration of a charged particle) is treated in QM.

How does this work in QM?