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This is a good read. However, there is some clarification I need.

The article has the following:

"By far the simplest

*gauge theory*is electromagnetism. And by far the simplest way to present electromagnetism as a gauge theory is through the non-relativistic Schrödinger equation of a particle moving in empty space:

iℏ∂ψ∂t=−ℏ22m∇2ψ.

Although the equation contains the wave function ψ, we know that the actual probability of finding a particle in some state is a function of |ψ|. In other words, the phase of the complex function

*ψ*can be changed without altering the outcome of physical experiments. In other words, all physical experiments will produce the same result if we perform the following substitution:

ψ→eip(x,t)ψ,

where p(x,t) is an arbitrary smooth function of space and time coordinates."

Then it goes on to derive the Schrödinger substituting in this varying function and after a lot of steps shows:

"iℏ∂ψ∂t=−ℏ22m{[∇+i∇p(x,t)]2−2mℏ∂p(x,t)∂t}ψ.

which is not the original Schrödinger equation. "

My question/line of thinking: Is the Schrodinger equation supposed to produce the same 'answer' for the wave function at every point in space? In other words two different observers at two different points in space calculating the wave function via the 'ordinary' Schrödinger should end up with the same answer? However this does not happen because each observer is allowed/has a different value for the phase of the wave function based on their location. Therefore to rectify this, the Schrodinger equation has to be modified so as to cancel out this local location-dependent phase and once that is done everyone at every point using this 'modified' Schrödinger equation will end up with the same wave function?

I would appreciate any feedback as to whether I am on the right track here. Thanks in advance.