Editing Density matrix (section)

== Von Neumann equation for time evolution ==
{{See also|Liouville's theorem (Hamiltonian)#Quantum Liouville equation}}
Just as the [[Schrödinger equation]] describes how pure states evolve in time, the '''von Neumann equation''' (also known as the '''Liouville–von Neumann equation''') describes how a density operator evolves in time. The von Neumann equation dictates that<ref>{{citation |title=The theory of open quantum systems|last1= Breuer |first1=Heinz|last2= Petruccione|first2=Francesco|page=110|isbn=978-0-19-852063-4|url=https://books.google.com/books?id=0Yx5VzaMYm8C&pg=PA110 |year=2002|publisher= Oxford University Press }}</ref><ref>{{Citation|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><ref>{{citation|title=Classical Mechanics and Relativity|last=Müller-Kirsten|first=Harald J.W.|pages=175–179|publisher=World Scientific|year=2008|isbn=978-981-283-251-1}}</ref>
: <math> i \hbar \frac{d}{dt} \rho = [H, \rho]~, </math>
where the brackets denote a [[commutator]].

This equation only holds when the density operator is taken to be in the [[Schrödinger picture]], even though this equation seems at first look to emulate the Heisenberg equation of motion in the [[Heisenberg picture]], with a crucial sign difference:
: <math> i \hbar \frac{d}{dt} A_\text{H} = -[H, A_\text{H}]~,</math>
where <math>A_\text{H}(t)</math> is some ''Heisenberg picture'' operator; but in this picture the density matrix is ''not time-dependent'', and the relative sign ensures that the time derivative of the expected value <math>\langle A \rangle</math> comes out ''the same as in the Schrödinger picture''.<ref name=Hall2013pp419-440/>

If the Hamiltonian is time-independent, the von Neumann equation can be easily solved to yield
: <math>\rho(t) = e^{-i H t/\hbar} \rho(0) e^{i H t/\hbar}.</math>

For a more general Hamiltonian, if <math>G(t)</math> is the wavefunction propagator over some interval, then the time evolution of the density matrix over that same interval is given by
: <math> \rho(t) = G(t) \rho(0) G(t)^\dagger.</math>
If one enters the [[interaction picture]], choosing to focus on some component <math>H_1</math> of the Hamiltonian <math>H = H_0 + H_1</math>, the equation for the evolution of the interaction-picture density operator <math>\rho_{\,\mathrm{I}}(t)</math> possesses identical structure to the von Neumann equation, except the Hamiltonian must also be transformed into the new picture:

:<math>{\displaystyle i \hbar {\frac {d }{d t}}\rho _{\text{I}}(t)=[H_{1,{\text{I}}}(t),\rho _{\text{I}}(t)],}</math>

where <math>{\displaystyle H_{1,{\text{I}}}(t)=e ^{i H_{0}t/\hbar }H_{1}e ^{-i H_{0}t/\hbar } }</math>.