# Why the Galileo transformation are written like this in Quantum Mechanics?

Note that first of all, in a galilean transformation, from $S$ to $S'$, a particle of constant momentum will have it's energy be different in $S'$ than $S$ because kinetic energy is proportional to the velocity squared. We know that wavefunctions oscillate like $$ e^{-i\dfrac E\hbar t} $$ so we have to take this into account when we look in our different frame.

Let's apply these results. What we'll find that it's accounting for this change in momentum gives you the term that is confusing you, and the change in energy gives you the term that you suspect would be the answer.

Consider a particle in $S$ with momentum $\vec p$ and mass $m$. It's wavefunction $\psi$ is then

$$ \psi = e^{i\left(\dfrac{\vec p \cdot \vec x}{\hbar} - \dfrac{p^2}{2m\hbar}t\right)} $$

In $S'$, with velocity $\vec v_0$, it's momentum goes to $\vec p' = \vec p - m \vec v_0$, and likewise it's energy changes. $E' = \frac{(p')^2}{2m}$, so we find that

$$ E' = E - \vec p \cdot \vec v_0 + \frac 12 m v_0^2 $$

Since the wave function must have a similar form as in $S'$,

$$ \psi' = e^{i\left(\dfrac{\vec p' \cdot \vec x}{\hbar} - \dfrac{E'}{\hbar}t\right)} $$

Plugging in these new values of energies and momentum yields this form,

$$ \psi' = e^{i\left(\dfrac{\vec p \cdot \vec x}{\hbar} - \dfrac{m\vec v_0\cdot \vec x}{\hbar}-\dfrac{Et}{\hbar}+ \dfrac{\vec p \cdot \vec v_0}{\hbar}\right)} e^{-i\dfrac {m v_0^2t}{2\hbar} } $$

we drop the very last term because it is invariant with respect to momentum. It's just a global overall phase (if there is one and only one type of mass). Rearranging leads us to find that,

$$ \psi' = e^{i\dfrac{\vec p \cdot \vec v_0 t}{\hbar}}e^{-i \dfrac{m\vec v_0\cdot \vec x}{\hbar}}\psi $$

Which is exactly the transformation you claim is the galilean transformation. Replacing $\vec p$ with an operator gives,

$$ \psi' = \hat G \psi $$

where $\hat G = e^{i\dfrac{\vec p \cdot \vec v_0 t}{\hbar}}e^{-i \dfrac{m\vec v_0\cdot \vec x}{\hbar}}$ Since the momentum states are a complete basis, this holds for any superposition of momentum states. So in general, it's true, and we've derived $\hat G$. It's because energy and momentum change in different frames.

It's straightforward to check that OP's eq. (2) indeed generates Galilean transformations. Rather it seems OP is asking

How to derive formula (2) from first principles?

*Sketched derivation of formula (2):*

Consider first the classical theory. The Hamiltonian Lagrangian for a free non-relativistic particle is $$L_H~=~\sum_{k=1}^3p_k\dot{x}^k-H, \qquad H~:=~ \frac{1}{2m}\sum_{k=1}^3p_k p_k.\tag{A}$$

Show that an infinitesimal Galilean transformation $$\delta x^k~=~t ~\delta v^k, \qquad \delta p^k~=~m ~\delta v^k, \qquad \delta t~=~0,\tag{B}$$ is a quasi-symmetry $$ \delta L_H ~=~\frac{d}{dt}\sum_{k=1}^3m x_k~\delta v^k \tag{C} $$ for the Hamiltonian Lagrangian (A). [Concerning quasi-symmetry, the reader may also enjoy reading this related Phys.SE post.]

Use Noether's theorem to find the corresponding full Noether charge $$ Q_k~=~ tp_k - m x_k.\tag{D}$$ [The first term $tp_k$ is the bare Noether charge, while the second term $m x_k$ comes from the rhs. of eq. (C).]

As a check, note that the Noether charge (D) generates the infinitesimal Galilean transformation (B), $$ \delta ~=~\sum_{k=1}^3 \{~\cdot~ , Q_k\} ~\delta v^k ,\tag{E}$$ as it should, cf. my Phys.SE answer here.

Use the correspondence principle between classical & quantum physics to deduce that $$ \delta ~=~ \sum_{k=1}^3\frac{1}{i\hbar}[~\cdot~ , \hat{Q}_k] ~\delta v^k, \tag{F}$$ where $$ \hat{Q}_k~=~ t\hat{p}_k - m \hat{x}_k.\tag{G}$$ is the Noether charge operator.

Use standard arguments to integrate the infinitesimal Galilean transformation (F) into a finite Galilean transformation in order to achieve OP's sought-for formula (2). $\Box$