# The State Vector and the Schrödinger Equation

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

In the absence of relaxation, the evolution of a quantum-mechanical state vector ψ〉 is governed by the Schrödinger equation i ℏ∂_{t}ψ(*t*)〉=Hψ(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 ℏ 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.

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.