Minimum uncertainty wavelets in non-relativistic super-symmetric quantum mechanics

Abstract

We consider the connection to the harmonic oscillator, super-symmetric quantum mechanics (SUSY-QM) and coherent states of the recently derived constrained Heisenberg “minimum uncertainty” (μ-) wavelets [Phys Rev Lett 85:5263 (2000); Phys Rev A65: 052106-1 (2002); J Phys Chem A107:7318 (2003)]. We analyze several new types of raising and lowering operators,which also can be viewed as arising from a (non-unitary) similarity transformation of the Harmonic Oscillator Hamiltonian and ladder operators. We show that these new ladder operators lead to a new SUSY formalism for harmonic oscillation, so that the μ-wavelets naturally manifest SUSY properties. Using these new ladder operators, we construct coherent and supercoherent states for the oscillator. In the discussion, we consider possible implications of similarity transformations for quantum mechanics. In an appendix we consider the classical limit of the μ-wavelet oscillator.