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Is there a known good way to visualize a quantum state, composed of the sum of eigenstates, with a phase rotating on each state. I am looking for a way to keep up with the state and the phase.

In a two state system, the position of the bloch vector on the bloch sphere represents the state, and the phase is represented by the angle in the equatorial plane.

But where do you represent the kinetic energy phase and a Berry's phase?

I can sorta imagine how to do the kinetic energy could be represented, as an overall rotation... so perhaps it's the Bloch Sphere axes rotating? But I am stuck on the Berry phase. Does anyone know of a good representation of the coefficients and phase of a quantum state, generally.

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The Berry phase is equal to the spin times the solid angle $\Omega$ enclosed by the trajectory on the Bloch sphere as shown in the figureenter image description here

It is easy to prove that using the Stokes theorem.

The dynamical phase does not have a representation on the Bloch sphere, since it is not a geometric object. It's value depends on the Hamiltonian.

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