The quantum Arnold transformation

Abstract

By a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg-Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schrödinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wave functions and closed analytic expressions for time-evolution operators.

19 pages, minor changes, references updated

References in corpus (4)

• The Magnus expansion and some of its applications
• Time-Dependent Diffeomorphisms as Quantum Canonical Transformations and the Time-Dependent Harmonic Oscillator
• Inverting Time-Dependent Harmonic Oscillator Potential by a Unitary Transformation and a New Class of Exactly Solvable Oscillators
• Unitary transformation for the system of a particle in a linear potential

Cited by in corpus (15)

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• Exactly solvable time-dependent non-Hermitian quantum systems from point transformations
• Two Cold Atoms in a Time-Dependent Harmonic Trap in One Dimension
• Exact solutions for time-dependent non-Hermitian oscillators: classical and quantum pictures
• Time-dependent conformal transformations and the propagator for quadratic systems
• Unitary equivalence of twisted quantum states
• Quantum and classical dissipation of charged particles
• Solving the Schrodinger Equation by Reduction to a First-order Differential Operator through a Coherent States Transform
• Temporal factorization of a non-stationary electromagnetic cavity field
• Twisted charged particles in the uniform magnetic field with broken symmetry
• Optical realization of the dissipative quantum oscillator
• Universal Analytic Solution for the Quantum Transport of Structured Matter-Waves in Magnetic Optics
• Geometry-Enabled Radiation from Structured Paraxial Electrons
• Projective Time, Cayley Transformations and the Schwarzian Geometry of the Free Particle--Oscillator Correspondence