Multiplying wavefunction with complexnumber.

[olgerm]

Is it true that by multiplying wavefunction with arbitrary complexnumber, which module is 1, results another wavefunction, that has same physical meaning? aka ##\forall_\phi(\Psi\ has\ the\ same\ meaning\ as\ \Psi \cdot e^{i \cdot \phi})##
If not please give me an example of wavefunction and ##\phi## where it does not.

 

[Metmann]

##\Psi## and ##e^{i\phi}\Psi## represent the same state since ##|\Psi|^2## is the central quantity that characterizes the state.

 

[olgerm]

##\Psi## and ##e^{i\phi}\Psi## represent the same state since ##|\Psi|^2## is the central quantity that characterizes the state.

Yes, but does ##e^{i\phi}\Psi## satisfy Schrödinger equation with same potentialenergy function ##U(t;\vec X)## as ##\Psi##?

 

[Metmann]

Yes, but does ##e^{i\phi}\Psi## satisfy Schrödinger equation with same potentialenergy function ##U(t;\vec X)## as ##\Psi##?

Sure. The Schrödinger equation is ##\mathbb{C}##-linear on both sides, hence the phase drops out. In fact all quantum mechanics is ##\mathbb{C}##-linear, hence the physical space of states is the projective Hilbert space.

 

[bhobba]

Yea - all true.

Why - because states are not really members of a Hilbert space - they are really positive operators of unit trace. Such operators of the form |u><u| are called pure and can be mapped to the underlying space the operator is defined on by simply using the u - but note if you multiply |u> by a complex number of unit length to get |u'> |u><u| = |u'><u'| - that's why it is invariant to phase. In general any positive operator of unit trace can be put in the form (not necessarily uniquely BTW which has implications for the decoherence program in explaining the measurement problem I will not go into here - start another thread if interested) as U=∑pi |ui><ui| where pi are positive and sum to 1. These are called mixed states. The Born Rule then becomes the expected value of an observable O, E(O), is E(O) = trace (OU) where U is the systems state and show the pi in fact are the probability of Iui><ui| in the mixture.

In fact due to a very famous theorem by the mathematician Gleason it can be deduced from more fundamental assumptions - see post 137:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Thanks
Bill

 

[DrClaude]

Yes, but does ##e^{i\phi}\Psi## satisfy Schrödinger equation with same potentialenergy function ##U(t;\vec X)## as ##\Psi##?

Why don't you try it yourself? Write down the Schrödinger equation for ##\Psi##, make the substitution ##\Psi \rightarrow ^{i\phi}\Psi##, and see if anything changes.

 

[Gigaz]

Sure. The Schrödinger equation is ##\mathbb{C}##-linear on both sides, hence the phase drops out. In fact all quantum mechanics is ##\mathbb{C}##-linear, hence the physical space of states is the projective Hilbert space.

Yes. I think there have been some efforts in the 90s to find deviations where you have a term in the Schrödinger equation that acts on the wave function squared, but they were essentially abandoned because people could not really agree on what would happen then and how it could be seen experimentally. The linearity is really extremely fundamental to our understanding of QM.

 

[olgerm]

By degrees of freedom, any wavefunction should be unambiguously descriptive by one real-valued function , that has 4 arguments.
##f(t,\vec X)##