Geometric Dynamics of a Harmonic Oscillator, Arbitrary Minimal Uncertainty States and the Smallest Step 3 Nilpotent Lie Group

The paper presents a new method of geometric solution of a Schrodinger equation by a construction of an equivalent first-order partial differential equation with a bigger number of variables. The equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform. The method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three Lie group—the shear group (also known as quartic group in literature). We obtain a geometric solution for an arbitrary minimal uncertainty state used as a fiducial vector. In contrast, it is shown that the well-known Fock–Segal–Bargmann transform and the Heisenberg group require the specific fiducial vector to produce a geometric solution. A technical aspect considered in this paper is that some modification of a coherent state transform is required: although representations of the group G square-integrability modulo a subgroup H, the obtained dynamic is transverse to the homogeneous space G/H.