Solving the Schrodinger Equation by Reduction to a First-order Differential Operator through a Coherent States Transform

Abstract

The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This generalises the geometric dynamics of a harmonic oscillator in the Fock space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and write down explicit solutions for such systems.

LaTeX, 7 page, 5 PDF graphics in three figures; v3: several minor improvements, references added

References in corpus (5)

• Geometrization of Quantum Mechanics
• The quantum Arnold transformation
• p-Mechanics as a Physical Theory. An Introduction
• Geometrization of some quantum mechanics formalism
• Basics of Quantum Mechanics, Geometrization and some Applications to Quantum Information

Cited by in corpus (4)

• Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group
• Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration
• Metamorphism as a covariant transform for the SSR group
• Metamorphism -- an Integral Transform Reducing the Order of a Differential Equation