Prove that TDSE is invariant under Galilean Transformation.

  • #1
I want the proof for a general wavefunction Ψ(x,t).
 

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  • #2
rubi
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The Schrödinger equation isn't invariant under Galilean transformations. It's invariant under a central extension of the universal cover of the Galilean group. Specifically, this means that you cannot just apply a Galilean transformation to the wave-function, but you need to add a (mass and time dependent) phase factor in order to transform a solution into another solution. In contrast to the Lorentz group, the phase factor cannot be made to vanish in case of the Galilean group.
 
  • #4
A proof of what?
I want to prove that the time dependent Schrödinger equation is invariant under Galilean transformation.
 
  • #5
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I want to prove that the time dependent Schrödinger equation is invariant under Galilean transformation.
See chapter 3 Ballentine - QM - A Modern Development - he does the converse - proves the Schrodinger equation from the Galilean principle of relativity. Specifically he assumes the probabilities of an observation are invariant.

BTW I think Rubi is right although I haven't gone deep into it.

Added Later:
Had a quick scan through Ballentine and Rubi is correct. Here is a paper I found on it - note the phase change mentioned by Rubi.
http://www.ijqf.org/wps/wp-content/uploads/2015/07/IJQF-2486.pdf

It says it is invariant - but with that phase change things are a bit murkier. Interesting question - before going into it I would have said yes it is - however its a bit more subtle.

Thanks
Bill
 
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  • #6
hilbert2
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Could this be somehow deduced from the fact that the canonical commutation relation ##[\hat{x},\hat{p}]=i\hbar## doesn't change in the transformation

##\hat{x} \rightarrow \hat{x}+vt##,
##\hat{p} \rightarrow \hat{p} + mv##

with ##m## and ##v## real numbers and ##t## the time?

On the other hand, the commutator at any time ##t## will also remain the same if the ##vt## is replaced by ##\beta t^2## or something else that is nonlinear in ##t##.
 

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