Coordinate transformation properties of the wave-function in the Heisenberg picture?

Question

Are the coordinate transformation properties of the wave-function in the Heisenberg picture simply:

$$ psi(x_1,x_2,x_3) to psi( x’_1,x’_2,x’_3)$$

where the coordinates $(x_1,x_2,x_3)$ and $( x’_1,x’_2,x’_3)$ are related to each other by coordinate transformations.(I’m searching of a proof of the same).

Or is there some phase factors as well? If not what is the correct transformation properties?

For Example

Consider a wave function $Psi(r,t)$ and suppose that the potential
energy is constant. Now let’s switch$^dagger$ to a reference frame,
which moves with respect to original one with velocity $At$ with $A$
constant:

$$Psi'(r,t)=Psileft(r-frac{At^2}2,tright)expleft[frac
{im}hbarleft(Atr-frac {A^2t^3}6right)right]*$$

$Psi'(r,t)$ is the wavefunction in accelerated (with acceleration
$A$) frame.

If we assume Schrödinger’s equation for free particle

$$ihbar partial_tPsi(r,t)=-frac{hbar^2}{2m}partial_{rr}Psi(r,t),$$

we can get the effective potential energy for the $Psi'(r,t)$ wave
function:

$$U_text{eff}(r)=frac{ihbar
partial_tPsi'(r,t)+frac{hbar^2}{2m}partial_{rr}Psi'(r,t)}{Psi'(r,t)}=-mAr.$$

But this is nothing than potential energy in uniform gravitational
field:

$$U_text{grav}(r)=mgh,$$

where we use $g=-A$ is free fall acceleration and $h=r$ is height.

_{$^dagger$ The switch is similar to the one described in e.g.
Landau, Lifshitz "Quantum mechanics. Non-relativistic theory" — in a
problem after $S$17, but taking time-dependent velocity into account
(i.e. not forgetting to integrate $frac12mV^2$ with respect to time
instead of just multiplying by $t$).}

*The pre-factor is simply the unitary operator in that reference frame.