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

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- Thread starter Viraj Daniel Dsouza
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- #1

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

- #2

rubi

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A proof of what?I want the proof for a general wavefunction Ψ(x,t).

- #4

I want to prove that the time dependent Schrödinger equation is invariant under Galilean transformation.A proof of what?

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bhobba

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

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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##\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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