Editing Schrödinger equation (section)

== Density matrices==
{{main|Density matrix }}
Wave functions are not always the most convenient way to describe quantum systems and their behavior. When the preparation of a system is only imperfectly known, or when the system under investigation is a part of a larger whole, [[density matrix|density matrices]] may be used instead.<ref name=":0" />{{rp|74}} A density matrix is a [[Positive-semidefinite matrix|positive semi-definite operator]] whose [[Trace class|trace]] is equal to 1. (The term "density operator" is also used, particularly when the underlying Hilbert space is infinite-dimensional.) The set of all density matrices is [[convex set|convex]], and the extreme points are the operators that project onto vectors in the Hilbert space. These are the density-matrix representations of wave functions; in Dirac notation, they are written <math display="block"> \hat{\rho} = |\Psi\rangle\langle \Psi|.</math>

The density-matrix analogue of the Schrödinger equation for wave functions is<ref>{{cite book |title=The theory of open quantum systems| last1= Breuer |first1=Heinz|last2= Petruccione|first2=Francesco|page=110|isbn=978-0-19-852063-4 |year=2002 | publisher= Oxford University Press | url=https://books.google.com/books?id=0Yx5VzaMYm8C&pg=PA110}}</ref><ref>{{cite book|url=https://books.google.com/books?id=o-HyHvRZ4VcC&pg=PA16 |title=Statistical mechanics|last=Schwabl|first=Franz|page=16|isbn=978-3-540-43163-3|year=2002|publisher=Springer }}</ref>
<math display="block"> i \hbar \frac{\partial \hat{\rho}}{\partial t} = [\hat{H}, \hat{\rho}],</math>
where the brackets denote a [[commutator]]. This is variously known as the von Neumann equation, the Liouville–von Neumann equation, or just the Schrödinger equation for density matrices.<ref name=":0" />{{rp|312}} If the Hamiltonian is time-independent, this equation can be easily solved to yield
<math display="block">\hat{\rho}(t) = e^{-i \hat{H} t/\hbar} \hat{\rho}(0) e^{i \hat{H} t/\hbar}.</math>

More generally, if the unitary operator <math>\hat{U}(t)</math> describes wave function evolution over some time interval, then the time evolution of a density matrix over that same interval is given by
<math display="block"> \hat{\rho}(t) = \hat{U}(t) \hat{\rho}(0) \hat{U}(t)^\dagger.</math>

Unitary evolution of a density matrix conserves its [[von Neumann entropy]].<ref name=":0" />{{rp|267}}