Quantum Phase interpretation I've been learning (on my own) how to solve the Time Independent Schrödinger Equation, with the solution being:
$$\psi(x,t)=\psi(x)e^{-i(Et/\hbar)}$$
And after reading about the phase factor, I'm still unsure about what the physical interpretation of the phase $(Et/\hbar)$ is.
Any help with understanding the meaning of the phase would be greatly appreciated.
 A: While the other answers are technically correct, I am not sure they are of much use to a novice - which is what implied in the question. I therefore would try to add a couple of rather pedestrian points.
The equation in the OP is generally not correct. If $\psi_n(x)$ is an eigenfunction of the time-independent Schrödinger equation, $$H\psi_n(x)=E_n\psi_n(x),$$ then the time-dependent solution corresponding to this eigenfunction is
$$
\psi_n(x,t) = e^{-\frac{iE_nt}{\hbar}}\psi_n(x),
$$
whereas the full solution of the time-dependent Schrödinger equation is
$$
\Psi(x,t) = \sum_nc_ne^{-\frac{iE_nt}{\hbar}}\psi_n(x).
$$
We can readily see from the solution above that the phase becomes important when we are dealing with more than one state. Let us consider, for example, an electron in a potential, which is in a superposition of its ground and excited states, $n=0,1$. Its wave function is then written as
$$
\Psi(x,t) = c_0e^{-\frac{iE_0t}{\hbar}}\psi_0(x) + c_1e^{-\frac{iE_1t}{\hbar}}\psi_1(x),
$$
whereas the expectation value of its position operator is
$$
\langle x(t)\rangle = \int dx \Psi^*(x,t)x\Psi (x,t) = x_{00} + x_{11} + 2\Re\left[x_{01}e^{-\frac{i(E_0-E_1)t}{\hbar}}\right],
$$
here
$$
x_{nm} = \int\psi_n(x)^*x\psi_n(x).$$
In other words, the position operator oscillates with frequency $\omega_{01}=\frac{E_1-E_0}{\hbar}$. This will readily become relevant when it interacts with time-dependent electric field, where the system will absorb or emit EM radiation at this frequency.
A: I would say the phase per se does not have a meaningful physical interpretation, as evidenced by gauge invariance (one can change the phase and the electromagnetic 4-potential simultaneously and obtain a physically equivalent situation). Note, for example, that the value of the energy in the phase factor strongly depends on whether you include the rest mass in it or not.
A: In quantum mechanics, given a Hilbert space $\mathcal{H}$ that represents an algebra of (un)-bounded operators $\hat{A}_i$, the set of states of the theory is the projective Hilbert space $P(\mathcal{H})$, namely the set of equivalence classes $[\psi]$ with respect to the equivalence relation $\psi\sim \phi$ whenever $\psi = e^{i\theta}\phi$. For finite dimensional Hilbert spaces (i. e. spin only systems) one can see that $P(\mathcal{H}_n) = \mathbb{C}P^{n-1}$, namely projectivity reduces spheres to circles, circles to lines and so on and so forth.
As such, any two elements of the Hilbert space that differ by a global phase factor do in fact represent the same physical state, as they belong to the same equivalence class, by definition. In particular, since observables are defined as modulus square of scalar products, given any two states $\phi, \psi$ then
$$
|\langle\phi|\psi\rangle|^2
$$
does not see any global phase present in either vectors.
That global phases set state definitions in the same equivalence class does not however mean that the elements $\psi$ and $e^{i\theta}\psi$ are interchangeable; in fact when taking linear combinations with other vectors of the space the two formulae
$$
|a \rangle + |b \rangle
$$
$$
|a \rangle + e^{i\theta}|b \rangle
$$
generate different distinct elements, because the phase factor that was global initially is no more so in the linear combination. This has as a result the effect that when taking again scalar products the phase factor $\theta$ does in fact play a role and appears eventually as a modulation coefficient $\cos(\theta)$ in the overall probability distribution. Standard example is a spin system in a Hamiltonian that generates spin-spin interactions (magnetic field) with corresponding transition probabilities that depend strongly on the initial phases.
