A new look at the quantum mechanics of the harmonic oscillator

In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\varphi \in \mathbb{R} mod 2 \pi$\end{document} and I > 0 which played a prominent role in the “old” Bohr‐Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ, I )‐model for the HO: the transformation $q = \sqrt{2I}\cos \varphi, p = -\sqrt{2I}\sin \varphi$ is only locally symplectic and singular for ( q , p ) = (0,0). Globally the phase space {( q , p )} has the topological structure of the plane ℝ 2 , whereas the phase space {(φ, I )} corresponds globally to the punctured plane ℝ 2 ‐(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h 0 = I , h 1 = I cos φ and h 2 = ‐ I sin φ as the basic coordinates on {(φ, I )}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ, I )} compared to the usual one for {( q , p )}: In the quantum mechanics for the (φ, I )‐model of the HO the three h j correspond to the self‐adjoint generators K j , j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy $E_{k,n=0} = \hbar \omega k$ of the (φ, I )‐Hamiltonian $H\smash[t]{(\vec{K})}= \hbar \omega K_0$ . For an m ‐fold covering the lowest possible value for k is k = 1/ m , which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the K j , but keep their usual properties. The richer structure of the K j quantum model of the HO is “erased” when passing to the simpler ( Q , P )‐model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken‐type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE‐condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.