The time independent Schrodinger equation $$-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2}+V\psi = E\psi$$ can have many different solutions of $\psi$ for a particular value of $E$.
For example, if we found a complex solution $\psi(x)$ for a particular value of $E$, say $E_0$, we can write $\psi(x)=a(x)+ib(x)$. Then $a(x)$ and $b(x)$ will also be solutions to the T.I.S.E with $E=E_0$. Furthermore $c_1a(x)+c_2b(x)$ will also be solutions with $c_1$ and $c_2$ being arbitrary constants.
I read that one can always choose any of these solutions as the solution for the stationary state with energy $E_0$. But does that mean all these different solutions represent the same physical state of a particle?
The expectation value of any dynamical variable $Q(x,p)$ is given by $\int \psi^*Q(x,\frac{\hbar}{i}\frac{d}{dx})\psi. $ How do we know for sure that $\int a(x)^*Q(x,\frac{\hbar}{i}\frac{d}{dx})a(x) $ and $\int b(x)^*Q(x,\frac{\hbar}{i}\frac{d}{dx})b(x)$ gives the same expectation values?