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The formalism of density operators and matrices was introduced in 1927 by John von Neumann [29] and independently, but less systematically, by Lev Landau [30] and later in 1946 by Felix Bloch. [31] Von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements.
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]
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 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.
The quantum Liouville equation is the Weyl–Wigner transform of the von Neumann evolution equation for the density matrix in the Schrödinger representation. The quantum Hamilton equations are the Weyl–Wigner transforms of the evolution equations for operators of the canonical coordinates and momenta in the Heisenberg representation.
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 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.