The Von Neumann Equation For Time Evolution
See also: Liouville's theorem (Hamiltonian)#Quantum Liouville equationJust 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 (in fact, the two equations are equivalent, in the sense that either can be derived from the other.) The von Neumann equation dictates that
where the brackets denote a commutator.
Note that 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:
where 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 comes out the same as in the Schrödinger picture.
Taking the density operator to be in the Schrödinger picture makes sense, since it is composed of 'Schrödinger' kets and bras evolved in time, as per the Schrödinger picture. If the Hamiltonian is time-independent, this differential equation can be easily solved to yield
Read more about this topic: Density Matrix
Famous quotes containing the words von, neumann, equation, time and/or evolution:
“The true, prescriptive artist strives after artistic truth; the lawless artist, following blind instinct, after an appearance of naturalness. The one leads to the highest peaks of art, the other to its lowest depths.”
—Johann Wolfgang Von Goethe (17491832)
“What a lesson here for our world. One blast, thousands of years of civilization wiped out.”
—Kurt Neumann (19061958)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)
“There is too much sour grapes for my taste in the present American attitude. The time to denounce the bankers was when we were all feeding off their gold plate; not now! At present they have not only my sympathy but my preference. They are the last representatives of our native industries.”
—Edith Wharton (18621937)
“Historians will have to face the fact that natural selection determined the evolution of cultures in the same manner as it did that of species.”
—Konrad Lorenz (19031989)