For stationary state, the energy of the system is constant and time-independent, so for some specific form of potential, the Schrodinger can be variable separated such that the general solution will be [tex] \psi(x, t) = \phi(x) \exp(-iE t /\hbar) [/tex] The probability is given by the square of the wave function so the arbitrary phase factor [tex]\exp(iE t/\hbar)[/tex] in above solution doesn't matter. However, if we consider the wave function itself, what's the meaning of the time-dependent factor [tex]\exp(iE t/\hbar)[/tex]. If it is stationary, why wavefunction is depending on time? By the way, if the solution is non-stationary, is it no more any definite energy level? So in this case, there is no corresponding eigenvalue problem?