how to estimate the phase parameter of a complex function?

Quantum Mechanics: Why is it impossible to measure the phase of a wave function, but it is possible to measure a difference in phase (eg. Berry's phase, or an Aharonov-Bohm effect)?

• For example, in the Aharonov-Bohm effect we can experimentally measure a phase shift from the incoming beam of particles relative to the outgoing. However, when performing measurements in quantum mechanics, we can only measure the amplitude 'psi^2' of a wave function, and so we cannot measure the phase (b/c it is usually in the form of an imaginary exponential term). What is the physical explanation that allows a phase difference to be measured?

• Answer:

  Consider the classic two slit experiment with electrons (which I won't describe as I assume you know it). You end up with the an interference pattern on your screen. Let's say the wavefunction for a beam through through the left slit is L(x) and the one for the right slit is R(x). The interference pattern is given by |L(x)+R(x)|^2. Now introduce a magnetic field between the slits. Both of the wavefunctions will acquire some phase, let's say α and β. Then the resulting interference pattern will be |exp(iα)L(x)+exp(iβ)R(x)|^2. In general this is different from |L(x)+R(x)|^2 and is observable in the lab. If the whole setup had acquired one phase factor it would make no visible difference. But we can see the difference because different parts of the total wave acquired different phases and so the absolute value of their sum does change.