In this paper we experimentally demonstrate an oscillating energy shift of center-of-mass quantum-confined exciton levels in a semiconductor quantum well after excitation into a superposition of two quantum confined exciton states of different parity. Oscillations are observed at frequencies corresponding to the quantum beats between these states. We show that the observed effect is a manifestation of the exciton density oscillations in the real space similar to the dynamics of an electron in a Hertzian dipole. The effect is caused by the exciton-exciton exchange interaction and appears only if the excitons are in a superposition quantum state. This effect exists until the Raman coherence in the interacting excitons has decayed. This effect may be harnessed for quantum technologies requiring the quantum coherence of states.