Related papers: Operator Methods for the Time Evolution of Wave Fu…
In quantum mechanics courses, students often solve the Schr\"odinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as,…
We discuss a method to follow step-by-step time evolution of atomic and molecular systems based on QED (Quantum Electrodynamics). Our strategy includes expanding the electron field operator by localized wavepackets to define creation and…
A quantum system is described, whose wave function has a complexity which increases exponentially with time. Namely, for any fixed orthonormal basis, the number of components required for an accurate representation of the wave function…
An open system is not conservative because energy can escape to the outside. An open system by itself is thus not conservative. As a result, the time-evolution operator is not hermitian in the usual sense and the eigenfunctions (factorized…
We characterize good clocks, which are naturally subject to fluctuations, in statistical terms. We also obtain the master equation that governs the evolution of quantum systems according to these clocks and find its general solution. This…
Efficient time integration methods based on operator splitting are introduced for the Westervelt equation, a nonlinear damped wave equation that arises in nonlinear acoustics as mathematical model for the propagation of sound waves in high…
We examine several numerical techniques for the calculation of the dynamics of quantum systems. In particular, we single out an iterative method which is based on expanding the time evolution operator into a finite series of Chebyshev…
The classical and quantum evolution of a generic probability distribution is analyzed. To that end, a formalism based on the decomposition of the distribution in terms of its statistical moments is used, which makes explicit the differences…
Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can…
We propose the construction of equations of motion based on symmetries in quantum-mechanical systems, using Heisenberg's uncertainty principle as a minimal foundation. From canonical operators, two spaces of conjugate operators are…
We present an outline of a technique to associate certain methods from time optimal quantum control with various transforms on SU(3). Unitary operators are taken from certain time dependent Hamiltonians and transformation laws are derived.…
We define the action operator in the consistent histories formalism, as the quantum analogue of the classical action functional, for the simple harmonic oscillator case. The action operator is shown to be the generator of time…
We apply the method of flow equations to describe quantum systems subject to a time-periodic drive with a time-dependent envelope. The driven Hamiltonian is expressed in terms of its constituent Fourier harmonics with amplitudes that may…
For a time-dependent classical quadratic oscillator we introduce pairs of real and complex invariants that are linear in position and momentum. Each pair of invariants realize explicitly a canonical transformation from the phase space to…
Identifying the real and imaginary parts of wave functions with coordinates and momenta, quantum evolution may be mapped onto a classical Hamiltonian system. In addition to the symplectic form, quantum mechanics also has a positive-definite…
In this paper, we mainly consider large time behavior for the classical free wave equation $u_{tt}-\Delta u=0$ in $\mathbb{R}^n$. We derive some large time optimal estimates for the quantity of solution $\|u(t,\cdot)\|_{L^2}$ with initial…
It is shown that evolution of wave functions in nonintegrable quantum systems is unpredictable for a long time T because of rapid growth of number of elementary computational operations $\mathcal O(T)\sim T^\alpha$. On the other hand, the…
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator splitting decomposition technique that allows for a mapping of the…
The time-dependent variational principle using generalized Gaussian trial functions yields a finite dimensional approximation to the full quantum dynamics and is used in many disciplines. It is shown how these 'semi-quantum' dynamics may be…
Operators in ergodic spin-chains are found to grow according to hydrodynamical equations of motion. The study of such operator spreading has aided our understanding of many-body quantum chaos in spin-chains. Here we initiate the study of…