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Let's say, that we are in possesion of a very simple quantum system, whose state can be written as

$$ |\psi\rangle = c_0 |\psi_0 \rangle + c_1 |\psi_1\rangle.$$

Now, we can change this state in two ways:

1) By adding global phase:

$$ |\psi'\rangle = e^{i\theta}|\psi\rangle = e^{i\theta} (c_0 |\psi_0 \rangle + c_1 |\psi_1\rangle).$$

2) By adding a relative phase difference between $|\psi_0\rangle$ and $|\psi_1\rangle$:

$$ |\psi''\rangle = c_0 |\psi_0 \rangle + e^{i\phi}c_1 |\psi_1\rangle.$$

I know that in an experiment we cannot detect the global phase from case 1), but we can measure the relative phase from case 2). But I cannot figure out any tangible example of these phases. Can you give one?

(*) Additional question: I've heard something about the Berry phase. Can you explain how this one relates to the first two? If you are able to even give some "real world" example showing this relation, it would be wonderful!

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You don't even need quantum mechanics to measure the relative phase. You just need a classical optics, for example the 2-slit interference pattern.

The two-slit ($1$ and $2$) interference pattern gives you an electric field $E = E_1 + E_2$, whose intensity $I \propto |E|^2$ is shown in red below:

enter image description here

By placing a phaseplate on slit $2$, the field from that slit becomes $E_2 \mathrm{e}^{\mathrm{i}\phi}$ so that the total field is $E = E_1 + E_2\mathrm{e}^{\mathrm{i}\phi}$. In the example below, I chose $\phi = \pi$ so that all the maxima turn into minima and vice versa.

enter image description here

To jump to quantum, you can think electrons (instead of light) incident on the two slits. Instead of the electric field, you have the probability amplitude $|\psi_1\rangle$ and $|\psi_2\rangle$. Same maths.


For the Berry phase, that's the generalisation of a geometric phase. This is a phase factor that arises from the geometry/topology of the real or Hilbert space.

The easiest way to visualise it is with parallel transport (gif from here). Moving a vector along a curved surface makes it build up an angle. This angle would be related to the Berry phase. The curved surface could be the Hilbert space of a quantum system. The vector could be a spin-1/2 atom atom.

enter image description here

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  • $\begingroup$ I guess a phase plate something that just shifts the phase of an electromagnetic wave? Is there something similar in the case of electrons (as you mentioned while jumping to the quantum case)? $\endgroup$ – brzepkowski Mar 24 at 7:03
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    $\begingroup$ Yes it’s just something with a different refractive index than air so as to change the optical path length and hence the phase acquired. I can’t think of a direct analogue for electrons on top of my head, sorry. Maybe just a corner pipe where electrons bounce in and then come out with an added path length they covered? $\endgroup$ – SuperCiocia Mar 24 at 7:24
  • $\begingroup$ That's great, thank you! :) $\endgroup$ – brzepkowski Mar 24 at 8:21

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