Solution To Quantum Liouville Equation In Phase Space Via A Stochastic Process
Sarah John
A numerical technique for the solution of the Quantum Liouville equation is discussed which is expected to be useful for mixed states describing highly excited many body systems.;The equation is represented in phase-space, where the density operator becomes the Wigner function. The Liouville operator which determines its time evolution, separates into a classical and quantum term. When the Wigner function varies slowly in momentum, the quasi-classical approximation is valid in which one keeps only the first order quantum term in the Liouville equation.;The Wigner function is represented by a sample set of gaussian smeared points. In an iterative time development scheme, the points are propagated first along a classical phase-space trajectory, which is then followed by a quantum development. The latter is represented by a stochastic process incorporated into the numerical procedure as a generalized Markovian process.;This method is numerically tested for an anharmonic quartic potential. The contours of the Wigner function at different time intervals are compared with the purely classical and exact results. It is seen that it provides a definite improvement over the purely classical approximation. Various averages are also compared and sources of error discussed.;In addition it was discovered that intermediate mixed states show a 'quantum focusing' effect, wherein the function peaks to values greater than the original maximum. This quantum-mechanical effect did not appear in the pure case, and disappears because of Liouville's theorem for highly mixed states which are essentially classical. There are other numerical schemes, which treat the points as moving in an effective potential, wherein the points retain their identity. These again by Liouville's theorem cannot show this focusing effect.
•Jan 1, 1986•2 citations