Prove that TDSE is invariant under Galilean Transformation.

[Viraj Daniel Dsouza]

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

 

[rubi]

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.

 

[PeterDonis]

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

A proof of what?

 

[Viraj Daniel Dsouza]

A proof of what?

I want to prove that the time dependent Schrödinger equation is invariant under Galilean transformation.

 

[bhobba]

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 Schrödinger 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

 

[hilbert2]

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 $t$.