[Paper Review] Geometric Dynamics of a Harmonic Oscillator, Non-Admissible Mother Wavelets and Squeezed States

Fadhel Almalki, Vladimir V. Kisil|arXiv (Cornell University)|May 3, 2018

Quantum Mechanics and Non-Hermitian Physics3 citations

TL;DR

This paper introduces a geometric method to solve the Schrödinger equation for the harmonic oscillator by reformulating it as a first-order PDE in higher dimensions, constrained via coherent state transforms from the shear group. It demonstrates that arbitrary minimal uncertainty states can serve as fiducial vectors for geometric solutions—unlike the Heisenberg group, which requires a specific fiducial vector—while addressing non-square-integrable representations through a modified coherent state transform.

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 the representation of the shear group is not square-integrable and a respective modification of a coherent state transform is required.

Motivation & Objective

• To develop a geometric approach to solving the Schrödinger equation using higher-dimensional first-order PDEs.
• To extend geometric solutions to arbitrary minimal uncertainty states, overcoming limitations of standard coherent state transforms.
• To address the challenge of non-square-integrable representations in the shear group by modifying the coherent state transform.
• To clarify the conditions under which fiducial vectors yield geometric solutions, contrasting the shear group with the Heisenberg group.

Proposed method

• Construct an equivalent first-order partial differential equation in a higher-dimensional space by extending the original Schrödinger equation.
• Impose auxiliary conditions derived from the coherent state transform associated with the minimal nilpotent step-three Lie group (the shear group).
• Restrict the solution to a subspace defined by these auxiliary conditions to recover the original physical solution.
• Utilize the shear group's structure to define a modified coherent state transform suitable for non-square-integrable representations.
• Apply the method to the harmonic oscillator with an arbitrary minimal uncertainty state as the fiducial vector.
• Demonstrate that the method yields geometric solutions for all such states, unlike the Fock–Segal–Bargmann transform which requires a specific fiducial vector.

Experimental results

Research questions

• RQ1 Can a geometric solution of the Schrödinger equation be constructed for arbitrary minimal uncertainty states using a higher-dimensional first-order PDE?
• RQ2 How does the choice of fiducial vector affect the existence of geometric solutions in coherent state frameworks?
• RQ3 What modifications are required to the coherent state transform when the group representation is not square-integrable?
• RQ4 Why does the Heisenberg group require a specific fiducial vector for geometric solutions, while the shear group does not?
• RQ5 What is the role of the shear group’s structure in enabling geometric solutions for general squeezed states?

Key findings

• The method successfully constructs a geometric solution for the harmonic oscillator using an equivalent first-order PDE in higher dimensions.
• Arbitrary minimal uncertainty states can serve as fiducial vectors in the geometric solution framework when using the shear group’s coherent state transform.
• The Fock–Segal–Bargmann transform and the Heisenberg group require a specific fiducial vector to produce geometric solutions, unlike the shear group.
• A modified coherent state transform is necessary for the shear group due to its non-square-integrable representation, which is explicitly constructed in the paper.
• The geometric solution is valid for all squeezed states generated by the shear group, demonstrating broader applicability than standard approaches.
• The approach reveals a structural distinction between the shear group and the Heisenberg group in the context of coherent state transforms and geometric quantization.