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(As opposed to a Lorentz transformation which leaves Maxwell's equations unchanged.) I've actually seen the equations obtained before. They were formulated by some 19th century physicist, maybe Hertz or Heaviside, and they involve adding velocity-dependent terms to the Ampere-Maxwell law and Faraday's law.
Newton's second law $\vec F=m\vec a$ will only be invariant under Galilean transformation provided the force between the interacting objects depends on the separation vector between the interacting objects, $(\vec{r}_1-\vec{r}_2)$ and is directed along $(\vec{r}_1-\vec{r}_2)$.
This 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.
Galilean transformation of the wave equation. Ask Question Asked 12 years, 6 months ago. Modified 7 years ...
That way you can remember that the Galilean transformation is more of a crude approximation of the motion of particles, while Lorentz transformation are more exact. Galilean Transformation. This is what most people's intuitive understanding of a particle in motion would be. Say we are inside a train that is moving.
Transformation law of momentum under Galilean transformation 4 Is Newton's law really invariant under Galilean transformation (for velocity-dependent Lorentz force)?
Galilean transformation does not affect the momentum of an object itself. However, it can change the way momentum is measured or observed in different reference frames. In other words, the momentum of an object may appear different depending on the reference frame in which it is measured, but the actual momentum of the object remains the same.
The Galilean transformation is not proven from experiment. The Galilean transformation should be used only when two frames of reference are moving at a constant speed (possibly zero) relative to each other. But even with that limitation there is experimental evidence that it is incorrect and that the Lorentz transformation should be preferred.
I then introduce the full "Galilean field transformtions" and ask where does the formula (III) come from and if this new transformation resolves the paradox. $\endgroup$ – math_lover Commented Jul 10, 2017 at 1:44
I'm reading Weinberg's Lectures on Quantum Mechanics and in chapter 3 he discusses invariance under Galilean transformations in the general context of non-relativistic quantum mechanics. Being a sy...