The dynamical behaviour of simple harmonic motion can be found in numerous natural phenomena. Within the quantum realm of atomic, molecular and optical systems, two main features are associated with harmonic oscillations: a finite ground-state energy and equally spaced quantum energy levels. Here it is shown that there is in fact a one-to-one mapping between the provision of equally spaced quantum energy levels and simple harmonic motion. The analysis establishes that the Hamiltonian of any system featuring quantized energy levels in an evenly spaced, infinite set must have a quadratic dependence on a pair of canonically conjugate variables. Moreover, specific physical inferences can be drawn. For example, exploiting this "back-to-front" derivation, and based on the known existence of photons, it can be proved that an electromagnetic energy density is quadratic in both the electric and magnetic fields.