Pure state

The terms of pure and mixed state described in quantum statistics, the branch of quantum mechanics for many-body systems, certain quantum mechanical states.

Reiner condition

A pure state exists when the system under consideration is in a defined state, which is described by a state vector of the Hilbert space. Then it is found with a probability in this state. Thus, the density operator

Just the projection onto the state. This is idempotent, that is, it applies.

An alternative definition of a pure state can be extended also to the more general state term for C *-algebras of operators. A state on a C * - algebra is a positive linear functional with norm 1, ie, an image with and. The set of states forms a convex set. A pure state is a state which is extremal in. That a pure state can not as a convex combination of ( a linear combination with positive coefficients whose sum to 1 ) describe two other states.

Mixed States

The equivalent to a pure state is a state mixture. It is when there is no single pure state, in which the system is located with a probability of 1. This common situation arises, for example, if one prepares the system repeatedly, creating chances with one of the pure states. The states do not have to be orthogonal to each other. Then the density operator is

The expected value of any operator ( with eigenvalues ​​and eigenstates ) is then associated with the weighted sum of the expected values ​​of each of the individual states (" incoherent superposition " ):

Are the pure states orthogonal to each other, then the weight is the probability of finding the system in the pure state. If they are not orthogonal, it is nothing. Rather, the probability of finding the mixture in a certain state:

Different compositions can produce the same density matrix:

Here, the states of the eigenstates of Spin-1/2-Systems. To illustrate the matrix you get beyond the usual (but arbitrary ) choice of representation and, where the character is to be read as "is represented by ", see Bra- Ket notation # representation.

The density operator of a mixed state can be recognized that applies to him.

Swell

  • H. Lin, An Introduction to the Classification of Amenable C * - algebras, World Scientific (2001)
  • Quantum mechanics
  • Quantum field theory
Particle statistics C*-Algebra
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