QM and relative phases I recently started formally learning about QM. I have studied thus far that any global phase difference is irrelevant when taking energy expectation values. However, that is not evidently the case for relative phases.
Say there is a position expectation value $\langle x\rangle$ for a given wavefunction that is a mixture of two different states. Will $\langle x\rangle$ change depending on the phase difference? Will the energy expectation value not change for relative phase, or is it a feature that only works for global phases?
 A: Let's take an example.
Suppose the wave function can be expressed as:
$$\Psi=c_1\psi_1+c_2\psi_2$$
If we add a global phase difference, say $e^{i\varphi}$:
$$\Psi\to\Psi'=e^{i\varphi}(c_1\psi_1+c_2\psi_2)$$
then, the expectation value of $O$:
$$\langle O\rangle _{\Psi'}=\langle \Psi'|O|\Psi'\rangle =e^{-i\varphi+i\varphi}\langle \Psi|O|\Psi\rangle =\langle \Psi|O|\Psi\rangle =\langle O\rangle _{\Psi}$$
We can see that there is no change here.
If we add a relative phase difference between $\psi_2$ and $\psi_1$:
$$\Psi\to\Psi'=c_1\psi_1+e^{i\phi}c_2\psi_2$$
then:
$$\langle O\rangle _{\Psi'}=\langle \Psi'|O|\Psi'\rangle =\langle c_1\psi_1+e^{i\phi}c_2\psi_2|O|c_1\psi_1+e^{i\phi}c_2\psi_2\rangle \\=|c_1|^2\langle \psi_1|O|\psi_1\rangle +|c_2|^2\langle \psi_2|O|\psi_2\rangle +e^{i\varphi}c_1^*c_2\langle \psi_1|O|\psi_2\rangle +e^{-i\varphi}c_2^*c_1\langle \psi_2|O|\psi_1\rangle $$
$$\langle O\rangle _\Psi=|c_1|^2\langle \psi_1|O|\psi_1\rangle +|c_2|^2\langle \psi_2|O|\psi_2\rangle $$
So, in general, $\langle O\rangle _{\Psi'}\ne \langle O\rangle _{\Psi}$ for the case of relative phase.
