This rotation is not in any ordinary spatial sense. Each eigenvector has an amplitude which is a complex number. This amplitude is a coefficient which multiplies one of the basis vectors. The complex coefficient has a magnitude and a direction. Therefore a state function is a linear combination of basis vectors, each one multiplied by a complex coefficient which has a magnitude and a direction in the complex plane. These coefficients can be thought of as phasors.
In the Schrödinger picture, these phasor coefficients are constantly rotating in a circle through time. The rotation operator which causes their rotation is called the propagator. The time evolution of a Schroedinger wave function can be effected mathematically by multiplying the wave function with the propagator. The propagator effects a simultaneous rotation of all the phasor coefficients of all the (infinite) basis vectors which form the state function.
Let represent an energy eigenstate at time 0. Then the rotation of the phasor coefficient of this eigenstate through time can be described by:
Thus, In the Schroedinger formulation of quantum mechanics, all unperturbed state functions are time-harmonic. State functions in the Schroedinger picture are never entirely static, they are always undulating. This is why state functions in the Schroedinger formulation are called wavefunctions. It reveals the undulatory nature of matter: the wave-particle duality. (Actually, wavefunctions also are also undulatory in space, independently of time.)
The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static. This is the Heisenberg picture.