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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 [20] [21] [22]
For Liouville's equation in quantum mechanics, see Von Neumann equation. For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelfand equation . In differential geometry , Liouville's equation , named after Joseph Liouville , [ 1 ] [ 2 ] is the nonlinear partial differential equation satisfied by the conformal factor f of a ...
The Schrödinger equation or, actually, the von Neumann equation, is a special case of the GKSL equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation. [2]
The analog of Liouville equation in quantum mechanics describes the time evolution of a mixed state. Canonical quantization yields a quantum-mechanical version of this theorem, the von Neumann equation. This procedure, often used to devise quantum analogues of classical systems, involves describing a classical system using Hamiltonian mechanics.
This is variously known as the von Neumann equation, the Liouville–von Neumann equation, or just the Schrödinger equation for density matrices. [ 25 ] : 312 If the Hamiltonian is time-independent, this equation can be easily solved to yield ρ ^ ( t ) = e − i H ^ t / ℏ ρ ^ ( 0 ) e i H ^ t / ℏ . {\displaystyle {\hat {\rho }}(t)=e^{-i ...
The time evolution of the phase space distribution is given by a quantum modification of Liouville flow. [2] [9] [19] This formula results from applying the Wigner transformation to the density matrix version of the quantum Liouville equation, the von Neumann equation.
In differential algebra, see Liouville's theorem (differential algebra) In differential geometry, see Liouville's equation; In coarse-grained modelling, see Liouville's equation in coarse graining phase space in classical physics and fine graining of states in quantum physics (von Neumann density matrix)
The Neumann series is used in functional analysis. It is closely connected to the resolvent formalism for studying the spectrum of bounded operators and, applied from the left to a function, it forms the Liouville-Neumann series that formally solves Fredholm integral equations.