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

TL;DR

This paper introduces a geometric method for solving the Schrödinger equation for harmonic oscillators using minimal nilpotent Lie groups, allowing arbitrary minimal uncertainty states without the need for specific fiducial vectors.

Contribution

It develops a new geometric solution approach applicable to arbitrary minimal uncertainty states via a modified coherent state transform associated with a step three nilpotent Lie group.

Findings

Derived a geometric solution for harmonic oscillator states using the shear group.

Showed that traditional transforms require specific fiducial vectors, unlike the new method.

Demonstrated the necessity of modifying the coherent state transform for this approach.

Abstract

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.