Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes
arXiv:1405.0445 · doi:10.1140/epjp/i2014-14114-3
Abstract
In quantum physics the free particle and the harmonically trapped particle are arguably the most important systems a physicist needs to know about. It is little known that, mathematically, they are one and the same. This knowledge helps us to understand either from the viewpoint of the other. Here we show that all general time-dependent solutions of the free-particle Schroedinger equation can be mapped to solutions of the Schroedinger equation for harmonic potentials, both the trapping oscillator and the inverted 'oscillator'. This map is fully invertible and therefore induces an isomorphism between both types of system, they are equivalent. A composition of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant and different center position. The map is independent of the state of the system, consisting only of a coordinate transformation and multiplication by a form factor, and can be chosen such that the state is identical in both systems at one point in time. This transition point in time can be chosen freely, the wave function of the particle evolving in time in one system before the transition point can therefore be linked up smoothly with the wave function for the other system and its future evolution after the transition point. Such a cut-and-paste procedure allows us to describe the instantaneous changes of the environment a particle finds itself in. Transitions from free to trapped systems, between harmonic traps of different spring constants or center positions, or, from harmonic binding to repulsive harmonic potentials are straightforwardly modelled. This includes some time-dependent harmonic potentials. The mappings introduced here are computationally more efficient than either state-projection or harmonic oscillator propagator techniques conventionally employed when describing instantaneous (non-a 1,920 characters
arXiv comment8 pages, 5 Figures
References in corpus (13)
- Laser cooling of a diatomic molecule
- Coherent Diabatic Ion Transport and Separation in a Multi-Zone Trap Array
- Controlling fast transport of cold trapped ions
- Morse potential derived from first principles
- Rovibrational cooling of molecules by optical pumping
- Wigner flow reveals topological order in quantum phase space dynamics
- Inverted Oscillator
- Coherent states for time dependent harmonic oscillator: the step function
- Harmonic states for the free particle
- Equivalence between focussed paraxial beams and the quantum harmonic oscillator
- Recent Applications of the Theory of Lie Systems in Ermakov Systems
- Amplitude and phase representation of quantum invariants for the time dependent harmonic oscillator
- Generalized Adiabatic Product Expansion: A nonperturbative method of solving time-dependent Schroedinger equation
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- Quantum inverted harmonic potential
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- Non-Hermitian inverted Harmonic Oscillator-Type Hamiltonians Generated from Supersymmetry with Reflections
- Free to Harmonic Unitary Transformations in Quantum and Koopman Dynamics
- Equivalence of a harmonic oscillator to a free particle and Eisenhart lift
- On the Formation of Lines in Quantum Phase Space
- Evolution of piecewise polynomial wave functions