Density Matrix and Statistical Tensors

Abstract

The states of physical systems in quantum mechanics can be classified in two categories: pure states and mixed states. A pure state is characterized by a certain wave function or a state vector |ψ > in an abstract Hilbert space. No vector |ψ > can be related to the mixed state. The necessity of introducing mixed states arises for quantum systems that are not closed. For example, a subsystem that is part of a larger closed system, cannot be characterized by its own wave function, which depends on coordinates of the subsystem, and therefore it is in a mixed state. The system cannot be considered closed even when it is isolated at present if it was not isolated in the past due to interaction with another system. This is a typical situation in the analysis of different characteristics of reaction products, when only part of the products (or part of the characteristics of the product) are observed. Mixed states are described by a statistical or density operator ρ, which operates in the Hilbert space and in general has the form: $$\rho = \mathop \sum \limits_n {W_n}\left| {{\psi _n} > < {\psi _n}} \right|$$ (1.1) where |ψ > n ) is a complete set of state vectors and the weight coefficients, Wn, are real positive numbers which, as we will see later, characterize the probability of finding the system in a particular pure state. The density operator is a generalization of the concept of the state vector. Every mixed state is characterized by its density operator.