The time-dependent quantum harmonic oscillator: a pedagogical approach via the Lewis-Riesenfeld dynamical invariant method
arXiv:2411.12894 · doi:10.1088/1361-6404/add972
Abstract
In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis-Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis-Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.
arXiv comment24 pages, 4 figures
References in corpus (39)
- Quantum sensing
- Quantum metrology with nonclassical states of atomic ensembles
- Shortcuts to adiabaticity: concepts, methods, and applications
- Fast optimal frictionless atom cooling in harmonic traps
- Stimulating uncertainty: Amplifying the quantum vacuum with superconducting circuits
- Transient energy excitation in shortcuts to adiabaticity for the time dependent harmonic oscillator
- Frictionless atom cooling in harmonic traps: a time-optimal approach
- Dynamical critical scaling of long-range interacting quantum magnets
- Fast and robust spin manipulation in a quantum dot by electric fields
- On the Time Dependence of Adiabatic Particle Number
- Classical and Quantum Action-Phase Variables for Time-Dependent Oscillators
- A Single Trapped Ion as a Time-Dependent Harmonic Oscillator
- Rapid Quantum Squeezing by Jumping the Harmonic Oscillator Frequency
- The physics and applications of strongly coupled plasmas levitated in electrodynamic traps
- Applications of Lie systems in dissipative Milne--Pinney equations
- Particle Dynamics in Damped Nonlinear Quadrupole Ion Traps
- Squeezing and robustness of frictionless cooling strategies
- Fast and stable manipulation of a charged particle in a Penning trap
- A unified approach to exact solutions of time-dependent Lie-algebraic quantum systems
- A time-dependent harmonic oscillator with two frequency jumps: an exact algebraic solution
- Exact Delta Kick Cooling, Time-Optimal Control of Scale-Invariant Dynamics, and Shortcuts to Adiabaticity Assisted by Kicks
- Efficient algebraic solution for a time-dependent quantum harmonic oscillator
- A bead on a hoop rotating about a horizontal axis: a 1-D ponderomotive trap
- Ehrenfest Dynamics and Frictionless Cooling Methods
- Shortcuts to adiabaticity for an interacting Bose-Einstein condensate via exact solutions of the generalized Ermakov equation
- Exact solution of a time-dependent quantum harmonic oscillator with two frequency jumps via the Lewis-Riesenfeld dynamical invariant method
- A Comment on "Semiquantum Chaos"
- Nonadiabatic energy fluctuations of scale-invariant quantum systems in a time-dependent trap
- Quasienergy operators and generalized squeezed states for systems of trapped ions
- Exact solutions of a damped harmonic oscillator in a time dependent noncommutative space
- Exact solution of damped harmonic oscillator with a magnetic field in a time dependent noncommutative space
- Continuous-time quantum harmonic oscillator state engineering
- Adiabatic Shortcuts Completion in Quantum Field Theory: Annihilation of Created Particles
- Steen-Ermakov-Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation
- Lewis and Berry phases for a gravitational wave interacting with a quantum harmonic oscillator
- The time-dependent harmonic oscillator revisited
- Quantum harmonic oscillator in a time dependent noncommutative background
- The Mechanical Paul Trap: Introducing the Concept of Ion Trapping
- Shortcut-to-adiabaticity for coupled harmonic oscillators