quantum mechanics - The meaning of the phase in the wave function

I have just started studying QM and I got into some trouble understanding something:

Let's say there is a wave function of a particle in a 1D box ($0\leq x\leq a$):

$$\psi(x,t=0) = \frac{i}{\sqrt{5}} \sin\left(\frac{2\pi}{a}x\right) + \frac{2}{\sqrt{5}} \sin\left(\frac{5\pi}{a}x\right)$$

Then if we measure the energy, the probability of getting the energy associated with $ \sin(\frac{2\pi}{a}x) $ is $\left| \frac{i}{\sqrt{5}} \right|^2 = \frac{1}{5}$ and the probability of measuring the energy associated with $\sin\left(\frac{5\pi}{a}x\right)$ is $\left| \frac{2}{\sqrt{5}}\right|^2 = \frac{4}{5}$. So the magnitude of $ \frac{i}{\sqrt{5}} , \frac{2}{\sqrt{5}} $ determines the probability, but what is the meaning of the phase? To me, as someone who measures energy, I'll get the same thing if

$$\psi(x,t=0) = \frac{-1}{\sqrt{5}} \sin\left(\frac{2\pi}{a}x\right) + \frac{2}{\sqrt{5}} \sin\left(\frac{5\pi}{a}x\right) $$

So why does the phase matter? If it matters, how do I know to which phase the wave function collapsed after the measurement?

Answer

This is an important question. You are correct that the energy expectation values do not depend on this phase. However, consider the spatial probability density $|\psi|^{2}$. If we have an arbitrary superposition of states $\psi = c_{1} \phi_{1} + c_{2} \phi_{2}$, then this becomes

$|\psi|^{2} = |c_{1}|^{2}|\phi_{1}^{2} + |c_{2}|^{2} |\phi_{2}|^{2} + (c_{1}^{*} c_{2} \phi_{1}^{*} \phi_{2} + c.c.)$.

The first two terms do not depend on the phase, but the last term does. ($c_{1}^{*}c_{2} = |c_{1}||c_{2}|e^{i (\theta_{2} - \theta_{1})}$). Therefore, the spatial probability density can be heavily dependent on this phase. Remember, also, that the coefficients (or the wavefunctions, depending on which "picture" you are using) have a rotating phase angle if $\phi_{1,2}$ are energy eigenstates. This causes the phase difference $\theta_{2} - \theta_{1}$ to actually rotate at the energy difference, so that $|\psi|^{2}$ will exhibit oscillatory motion at the frequency $\omega = (E_{2} - E_{1})/\hbar$. This is known as Rabi oscillation, and is also related to optical transitions and many other quantum phenomena.

In summary, the phase information in a wavefunction holds information, including, but not limited to, the probability density. In a measurement of energy this is not important, but in other measurements it certainly can be.