Geometric dynamics of a harmonic oscillator, arbitrary minimal uncertainty states and the smallest step 3 nilpotent Lie group

Abstract

The paper presents a new method of geometric solution of a Schrödinger equation by constructing 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 group G (also known under many names, e.g. quartic group). 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 a specific fiducial vector to produce a geometric solution. A technical aspect considered in this paper is that a certain modification of a coherent state transform is required: although the irreducible representation of the group G is square-integrable modulo a subgroup H, the obtained dynamic is transverse to the homogeneous space G/H.

Nonlinear dynamics from linear quantum evolutions

“…These solutions coincides with the solutions obtained in [3] if one performs the following substitutions:…”

“…Therefore, starting with a normalized vector lying on the unit sphere S(H), equation (1.3) defines an immersion of V into S(H) which is, again, injective. However, it is worth noticing that even though V is a vector space, the immersion does not respect linearity, i.e., ψ v 1 + ψ v 2 = ψ v 1 +v 2 in general, and i(V) is not a linear subspace of H. In summary, generalized coherent states allow us to define injective maps from a given set to the Hilbert space H associated with a quantum system, every point of the set labelling a given vector in H. This is the main property we will exploit in the first part of the paper in order to study how to induce one parameter groups of transformations on subsets of states starting from unitary maps on H. Using coherent states it is possible to interpret these induced maps as classical-like dynamics in the framework of classical-to quantum transition (see [2,3,4]). However, it is worth stressing that the following discussion can be extended to submanifolds which do not possess classical-like properties, representing, therefore, generic constraints.…”

Solving the Schrödinger equation by reduction to a first-order differential operator through a coherent states transform

“…Thereafter, the corresponding first order PDE can be explicitly solved and the geometrised evolution is realised by a time-dependent coordinate transformations. The method is based in the development [7,35,38,39] of the CS from group representations [4,20,45], notably an analyticity-type condition in the image space of CST as explained below.…”

“…(1) Chose a fiducial vector |0 which is annihilated by certain form of ρ, cf. (16); (2) Calculate [7,35,38,39] the respective operator C, cf. (18), which annihilates the image space of the CST (4).…”

“…Here we briefly introduce the necessary background. It was already used in [7], which contains a detailed presentation and further references.…”