"Quantum Liouville", Moyal's Equation
The density matrix operator may also be realized in phase space. Under the Wigner map, the density matrix transforms into the equivalent Wigner function,
The equation for the time-evolution of the Wigner function is then the Wigner-transform of the above von Neumann equation,
where H(q,p) is the Hamiltonian, and { { •,• } } is the Moyal bracket, the transform of the quantum commutator.
The evolution equation for the Wigner function is then analogous to that of its classical limit, the Liouville equation of classical physics. In the limit of vanishing Planck's constant ħ, W(q,p,t) reduces to the classical Liouville probability density function in phase space.
The classical Liouville equation can be solved using the method of characteristics for partial differential equations, the characteristic equations being Hamilton's equations. The Moyal equation in quantum mechanics similarly admits formal solutions in terms of quantum characteristics, predicated on the ∗−product of phase space, although, in actual practice, solution-seeking follows different methods.
Read more about this topic: Density Matrix
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