Quantum mechanical density operator

THE QUANTUM MECHANICAL DENSITY OPERATOR AND ITS TIME EVOLUTION QUANTUM DYNAMICS USING THE QUANTUM LIOUVILLE EQUATION 

Surely the atoms never began by forming A conscious pact, a treaty with each other, 

Where they should stay apart, where they come together. 

More hkely, being so many, in many ways Harassed and driven through the universe From an infinity of time, hy trying All kind of motion, every combination. 

They came at last into such disposition As now estahhshes the sum of things... 

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This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenberg-type uncertainty principles, the Schrodinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (r, p f) is now the quantum mechanical density operator (often referred to as the density matrix ), whose time evolution is detennined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or... 

As soon as large ensembles of particles with statistical populations of the eigenstates and incoherent exchange and relaxation processes between these states are investigated, quantum statistical tools are necessary to describe the system. In this situation the quantum mechanical density operator p has to be employed. For the coherent evolution of the density operator under the influence of a Hamiltonian H, the following differential equation is found [80]... 

This contribution deals with the description of molecular systems electronically excited by light or by collisions, in terms of the statistical density operator. The advantage of using the density operator instead of the more usual wavefunction is that with the former it is possible to develop a consistent treatment of a many-atom system in contact with a medium (or bath), and of its dissipative dynamics. A fully classical calculation is usually suitable for a many-atom system in its ground electronic state, but is not acceptable when the system gets electronically excited, so that a quantum treatment must then be introduced initially. The quantum mechanical density operator (DOp) satisfies the Liouville-von Neumann (L-vN) equation [1-3], which involves the Hamiltonian operator of the whole system. When the system of interest, or object, is only part of the whole, the treatment can be based on the reduced density operator (RDOp) of the object, which satisfies a modified L-vN equation including dissipative rates [4-7]. 

The starting point for the description of laser-driven multistate dynamics is the semiclassical limit of the Liouville-von Neumann (LvN) equation for the quantum mechanical density operator p. 

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