We describe a computational method for simulating time-dependent quantum mechanical systems interacting with external fields. In this method, the Schrödinger equation is solved by expanding the wave function in the basis set of the unperturbed Hamiltonian. The expansion yields a set of coupled ordinary differential equations for the expansion coefficients. This coupled channel method can be applied to many time-dependent problems. We apply the coupled channel method to a particle in a box interacting with a laser pulse. We show that if only two states are involved, the method leads naturally to the solutions for Rabi flopping, and that the system exhibits Rabi flopping behavior even with realistic, nonmonochromatic laser pulses. We also discuss instances where simulations could improve the understanding or address misconceptions of beginning students in quantum mechanics.
We describe a computational method for simulating time-dependent quantum mechanical systems interacting with external fields. In this method, the Schrödinger equation is solved by expanding the wave function in the basis set of the unperturbed Hamiltonian. The expansion yields a set of coupled ordinary differential equations for the expansion coefficients. This coupled channel method can be applied to many time-dependent problems. We apply the coupled channel method to a particle in a box interacting with a laser pulse. We show that if only two states are involved, the method leads naturally to the solutions for Rabi flopping, and that the system exhibits Rabi flopping behavior even with realistic, nonmonochromatic laser pulses. We also discuss instances where simulations could improve the understanding or address misconceptions of beginning students in quantum mechanics.
We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E⃗(t) through the Stark Hamiltonian H=λ(J⃗⋅E⃗)2.
This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.
Abstract This chapter begins the discussion of the time evolution of an active quantum system. From the earlier chapters, time dependent physics has been observed through the presence of the time evolution of the phase of each eigenstate. The Hamiltonian itself is time independent. This represents the same kind of evolution of a classical system like the vibration of a tuning fork when it has been struck or the oscillation of an LC circuit if the capacitor is charged to some voltage and then the switch is closed. In the quantum case, the Hamiltonian has also been time independent. The time evolution evolves according the full-time dependent Schrödinger equation, depending only on a single initial condition of the state vector or wave function and the corresponding time evolution of the phase factor for each eigenstate. However in this chapter, we consider the case of when there is a time dependent Hamiltonian such as a sinewave generator or laser. As in the case of resonant tunneling, we see the importance in dynamics of resonant coupling. With an oscillating potential energy term, we see the presence of Rabi oscillations in the probability amplitude of a two-state system on resonance, which can be viewed as a quantum flip-flop between two states of a quantum bit (qubit).
We propose a method of solution of the many-body Schr\"odinger equation that involves an expansion of the wave function in terms of a finite, time-dependent basis of a relevant subspace. The equations of motion for the expansion coefficients generalize previous proposals of approximate dynamics. The method is illustrated in the case of an N-particle system with an SU(2) Hamiltonian, and it is shown that it improves the approximation that disregards off-diagonal elements of the dynamical matrices.
This research paper presents a simulation of a particle in a quantum mechanical oscillator, modeled as a multidimensional wave function. The simulation was created by solving the time-dependent Schrodinger equation, which was discretized in both the spatial and time domains, and then approximated using a finite difference method. The objective of this research was to gain a deeper understanding of the behavior of a particle in a quantum mechanical oscillator and to provide visualizations and calculations of relevant physical observables. Initial conditions for the wave function, such as a Gaussian wave packet, were chosen and parameters for the simulation, such as the spatial step size and total simulation time, were set. The results of the simulation showed the evolution of the wave function over time, providing a clear picture of the particles behavior within the potential well of the oscillator. Additionally, physical observables, such as the average position, average kinetic energy, and average potential energy, were calculated and used to validate the simulation and the predictions of quantum mechanics. In conclusion, this research provides a comprehensive understanding of the behavior of a particle in a quantum mechanical oscillator and highlights the importance of simulations in advancing our knowledge of quantum mechanics. The results of this research can contribute to various fields, such as quantum computation and quantum control, and have the potential to drive further breakthroughs in these fields. KEYWORDS: Quantum mechanical oscillator, particle behavior, multidimensional wave function, Schrödinger equation, finite difference method, physical observables, simulation, quantum mechanics, quantum computation, quantum control.
The general, N state matrix representation of the time-dependent Schrödinger equation is equivalent to an N degree of freedom classical Hamiltonian system. We describe how classical mechanical methods and ideas can be applied towards understanding and modeling exact quantum dynamics. Two applications are presented. First, we illustrate how qualitative insights may be gained by treating the two state problem with a time-dependent coupling. In the case of periodic coupling, Poincaré surfaces of section are used to view the quantum dynamics, and features such as the Floquet modes take on interesting interpretations. The second application illustrates computational implications by showing how Liouville’s theorem, or more generally the symplectic nature of classical Hamiltonian dynamics, provides a new perspective for carrying out numerical wave packet propagation. We show how certain simple and explicit symplectic integrators can be used to numerically propagate wave packets. The approach is illustrated with an application to the problem of a diatomic molecule interacting with a laser, although it and related approaches may be useful for describing a variety of problems.