The State Vector and the Schrödinger Equation

The main question that the AtomicDensityMatrix package is intended to address is

Given the initial state of an atomic system subject to light and/or other external fields and relaxation mechanisms, what is the subsequent evolution of the system and what effect does this evolution have on the interacting light fields?

In the absence of relaxation, the evolution of a quantum-mechanical state vector |ψ is governed by the Schrödinger equation i t|ψ(t)=H|ψ(t), where H is the Hamiltonian. (In the ADM package, we generally use units in which H has units of frequency and the reduced Planck's constant is equal to one.)

As a simple example, we will find the evolution of system consisting of two opposite-parity states with no Zeeman or hyperfine substructure, separated by an energy Δ and subject to a static electric field.

Load the package.
Define an atomic system and form its ket vector.
The Hamiltonian describing the interaction with an electric field. We normalize the electric field to frequency units by dividing by the reduced dipole matrix element between states 1 and 2.
The Schrödinger equation.
Create the initial conditions.
A list of the dependent variables.
Solve the Schrödinger equation.
The real and imaginary parts of the elements of the state vector.
Plot the trajectory of the elements of the state vector in the complex plane.
The probability of finding each state as a function of time.
Plot the probabilities.

The state vector and Schrödinger equation describe the evolution of a single atom (or a completely polarized atomic ensemble) when there are no relaxation processes at work. However, we are often interested in the evolution, including relaxation mechanisms, of partially polarized or unpolarized ensembles, for which the experimental observables are averages over the ensemble. For this we use a formalism that generalizes the state vector: the density matrix.