- #1

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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- #3

PeterDonis

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

A proof of what?

- #4

A proof of what?

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

- #5

bhobba

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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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##\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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