Galilean covariance of the Schrödinger equation without choosing a representation The most general form of Schrödinger equation is
$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$
where $\psi(t)$ is an element of a Hilbert space $\mathcal H$ (not necessarily $L^2$), and $H$ is a self-adjoint operator.
Any kind of proof of the Galilean covariance of the Schrödinger equation I have seen so far, ψ
is in the coordinate representation and the Hamiltonian operator is taken in its explicit form.  Is there a way to deal with Galilean covariance of the Schrödinger equation in the general setting, without picking a specific representation?
 A: Hint: The links provided above address/settle your question, but, as a complementary setup, you might consider restricting to one dimension, starting from the general solution to the general TDSE, namely
$$ |\Psi(t)\rangle = e^{-it\hat H/\hbar}|\Psi(0) \rangle\tag 2,$$
and never look back!
Just dot on the left with $\langle x|$, a bra,  $\langle x|\Psi(t)\rangle=\psi(x,t)$ ;   and monitor the braiding of exponential operators under the action of a Galilean transformation,
$$
\hat G=  \exp\left ({\frac{iv}{\hbar}(m\hat x-t\hat p)} \right )= e^{-imv^2t/2\hbar}e^{imv\hat x/\hbar}e^{-itv\hat p /\hbar }   = e^{imv^2t/2\hbar}e^{-itv\hat p /\hbar } e^{imv\hat x/\hbar} \\ \implies \qquad  \hat G^\dagger \hat x \hat G =\hat x-vt, \qquad\qquad\\
 \hat G^\dagger \hat p \hat G =\hat p-mv,\\  
\hat G^\dagger {\hat{ p} ^2 \over 2m}  \hat G = {(\hat{ p}-mv )^2 \over 2m} ,\\
\hat G^\dagger V(\hat  x) \hat G = V(\hat x -vt).
$$
To get to the expressions you might have,  defining $|\Psi'\rangle= \hat G^\dagger |\Psi\rangle$, you   have, from above,
$$
\psi'(x,t)=\langle x|\Psi'(t)\rangle=\langle x|  \hat G^\dagger   |\Psi(t)\rangle\\
 = e^{-imvx/\hbar+imv^2t/2\hbar}\langle x+vt|\Psi(t)\rangle= e^{-imvx/\hbar+imv^2t/2\hbar} \psi(x+tv,t).
$$
$\hat G^\dagger  $ acted on the left, the bra. I have used $e^{tv\partial_x} \langle x| = \langle x+vt|$, of course.  No fuss, no muss. It’s not the TDSE you care about, it’s its solutions!
