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## Schroedinger Equation - Galilean Invariance

Hi All,

I'm new to this forum. I'm a third-year undergrad Physics Major in Australia, about to go on to Honours, very exciting project in Helium atom detection.

To the point. My 3rd year Special Rel project is an investigation of the development of relativistic QM (RQM). I have to prove all the key components of the development, for example that the TDSE is Gal but not Lorentz invariant, that the lack of invariance arises from the unequal treatment of the time/momentum operators, etc.

Anyway, what I want to ask is - everytime I go to prove Galilean invariance for the 1D TDSE with an arbitrary potential, I get an extra term appearing in the transformed equation of the form vp^, where p^ is the momentum operator and v is the relative velocity of the frame. How can this be considered "of the same form" as the original if that term is in there?

I'm not sure if I'm going about it the right way - the method I have used is to start in the S' frame and transform backwards to the S frame, ie Psi(x', t') goes to Psi(x-vt, t), etc, and using the chain rule for the partial derivatives.

I'd like to be as rigorous as possible, but I can't seem to find references where the proof is actually done and not left as a student exercise!

Thanks,

Kane

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 Quote by Kane O'Donnell Anyway, what I want to ask is - everytime I go to prove Galilean invariance for the 1D TDSE with an arbitrary potential, I get an extra term appearing in the transformed equation of the form vp^, where p^ is the momentum operator and v is the relative velocity of the frame. How can this be considered "of the same form" as the original if that term is in there?
Hi, and welcome to PF!

I've only got a second right now, but I can give you a reference: See Jackson's Classical Electrodynamics, 2ed, Chapter 11. There is a short explanation of the Galilean invariance of the Schrodinger equation there. It turns out that you also have to do a transformation on the wavefunction to recover the form of the SE under Galilean transformations. I'll post mathematical details later today.

 Recognitions: Science Advisor Thanks, our library has that reference, I'll check it out. Kane

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