The equally spaced energy levels of the quantum harmonic oscillator revisited: a back-to-front reconstruction of an n-body Hamiltonian

The 'back-to-front' derivation of the properties of the quantum harmonic oscillator (QHO), starting with its equally spaced energy levels, is re-examined. A new derivation that exploits the natural rotational symmetry of the QHO is proposed. The new approach allows the 'back-to-front' idea to be extended further by showing that it is possible to derive the Hamiltonian of a system of particles from the starting point that the population is represented by a natural number. This involves the symmetry properties of phasors and Schwinger's theory of angular momentum. The analysis is also extended to multi-mode bosonic systems and fermionic systems. It is suggested that these results offer an alternative way to formulate physics, based on discreteness.