In introductory quantum mechanics I have always heard the mantra
The phase of a wave function doesn't have physical meaning. So the states $| \psi \rangle$ and $\lambda|\psi \rangle$ with $|\lambda| = 1$ are physically equivalent and indiscernible.
In fact by this motivation it is said that the state space of a physical system shouldn't be a Hilbert space, but rather a projective Hilbert space, where vectors which only differ up to a multiplicative constant of magnitude 1 are identified.
But I also heard that one of the defining "feature" of quantum mechanics is the superposition principle: We can combine states $| \psi_1 \rangle, |\psi_2 \rangle$ to a new state $| \psi_1 \rangle + | \psi_2 \rangle$. This should for example explain the constructive / destructive interference we see in the double slit.
But if two states with the same phase are physically equivalent, so should the states $| \psi \rangle, -|\psi \rangle$. But their sum is zero. I have seen experiments which exploit this and measure the relative phase difference between two different states. But if relative phase difference is measurable, then surely the phase of a wave function does have physical meaning? This should mean that we can identify the phases of all states of a quantum system up to a $U(1)$ transformation by gauging some state to have phase $1$. Is this correct? How can this be solidified with the above mantra?
I have asked a second question here ("The superposition principle in quantum mechanics") regarding the superposition principle which is closely related to this question.
Best Answer
When people say that the phase doesn't matter, they mean the overall, "global" phase. In other words, the state $|0 \rangle$ is equivalent to $e^{i \theta} |0 \rangle$, the state $|1\rangle$ is equivalent to $e^{i \theta'} |1 \rangle$, and the state $|0\rangle + |1 \rangle$ is equivalent to $e^{i \theta''} (|0 \rangle + |1 \rangle)$.
Note that "equivalence" is not preserved under addition, since $e^{i \theta} |0 \rangle + e^{i \theta'} |1 \rangle$ is not equivalent to $|0 \rangle + |1 \rangle$, because there can be a relative phase $e^{i (\theta - \theta')}$. If we wanted to describe this very simple fact with unnecessarily big words, we could say something like "the complex projective Hilbert space of rays, the set of equivalence classes of nonzero vectors in the Hilbert space under multiplication by complex phase, cannot be endowed with the structure of a vector space".
Because the equivalence doesn't play nicely with addition, it's best to just ignore the global phase ambiguity whenever you're doing real calculations. Finally, when you're done with the entire calculation, and arrive at a state, you are free to multiply that final result by an overall phase.