Why is the density operator used
In physics one describes with one Density matrix or a Density operator (also statistical matrix or. statistical operator) a quantum state about which one does not have the maximum possible knowledge. The density matrix contains the maximum information about results that can be obtained through measurements (exclusively) on the system described with the density matrix.
Let us consider a quantum mechanical system which is based on a Hilbert spaceH is modeled. A bounded linear operator ρ on H is a density operator if:
- it is positive semi-definite,
- it is track class with track equal to 1.
- Each density operator is self-adjoint (or hermitesch), since positive operators are always self-adjoint.
- The set of all density operators is a convex set whose boundary is the set of pure (quantum mechanical) states. In contrast to classical theories, the set is not a simplex, i. H. In general, a density operator cannot be represented unambiguously as a convex combination of pure states.
- The probability of measuring an observableA. on a system that is described by the density operator ρ, the measured valuea to get is through
- given, where the orthonormalized eigenvectors to the eigenvaluea are and Pa the projector on the corresponding eigenspace.
- The mean of the measured values when measuring an observableA. is
Although the terms Density matrix and Density operator are often used synonymously, there is a mathematical difference. Just as in linear algebra a matrix is the basic representation of a linear operator, in quantum mechanics you can choose between abstract Density operator and a concrete one Density matrix can be distinguished in a certain representation. If ρ is a density operator, then denoted
z. B. the density matrix in position representation. The Density matrix in the position representation, however, there is no real matrix, since the position representation over a continuum of improper Basis vectors | x > is defined, but a so-called integral kernel. In finite-dimensional Hilbert spaces (e.g. in spin systems), on the other hand, we get a real matrix if we have an orthonormal basis choose:
In the following, however, the two terms are again used synonymously, as the specific meaning is usually derived from the context.
Construction of density matrices
If | Ψ> is a pure state, then is the projection operator
the associated density operator. The following applies here: . Conversely, it already follows from this condition that ρΨ describes a pure state. The following always applies to mixed states .
A N-Level system, about which one has no knowledge, can be via the density matrix
are described, where the N-dimensional identity matrix called.
The quantum mechanical Concept of state is by its nature statistical to see. Even with the best possible preparation of a quantum mechanical system (i.e. maximum knowledge in the above sense) the system can be described by a pure state, but even then can be described in general no statements about the outcome of a single experiment to meet. Rather, all statements about the outcome of an experiment are statistical, i. H. mean values, standard deviations and other moments of probability distributions are predicted. The origin of this character of quantum mechanics can be seen in Heisenberg's uncertainty relation.
Quantum mechanics therefore does not make any statements about individual measurements on a system, but rather about the measurement repeated as often as possible under the same preparation conditions. Therefore it is also doubtful whether the term Status than the description of the properties of a concrete single quantum system is permissible. The modern interpretation quantum mechanics understands by this term the entirety of very many (in the ideal case any number) under the same conditions independently of each other prepared systems of the same type. This whole is called an ensemble.
The concept of the ensemble now suggests expanding the concept of state in the same way as in statistical mechanics. We see that although every pure state defines an ensemble, not all ensembles of quantum mechanical states can be characterized by pure states. The reason for this lies in the fact that no one can do this with the taxidermist in the above scheme forcesto prepare as precisely as possible. The simplest case is e.g. B. that during the preparation there is a chance, with certain probabilities, to switch between different "pure" preparations. Such a preparation is called a mixture. Is z. B. before a mixture of different states in which with probability pi the pure state has been prepared, this ensemble is represented by the density matrix
described. Note that the individual terms of this sum do not provide any information about the global phase of the states included, so it is a mixed state Not is a coherent superposition of pure states.
In general it can be shown that each Ensemble, especially those through bad Preparation arise through mixed states can be defined. However, in contrast to classical statistical mechanics, such states cannot be clearly broken down into pure states. Various Mixtures can do the same Define state. This is another reason why it is inadmissible to regard mixed states as an ensemble of pure states that all describe a single system.
From the Schrödinger equation, which describes the time development (dynamics) of pure quantum states, the time development of mixed states can be derived directly. For this one uses any decomposition of the density matrix into pure states, the dynamics of which satisfy the Schrödinger equation, and from this one calculates the dynamics of the mixed state
in which H is the Hamilton operator of the system. This equation is as von Neumann's equation of motion known ( Heisenberg's equation of motion).
One can easily integrate this differential equation for time-independent Hamilton operators and one obtains with the unitary time expansion operator the equation
It is noteworthy that for this operator the usual Heisenberg's equation of motion does not hold because ρ is not an observable. The transformation with the time expansion operator is also not in accordance with the usual time expansion equation for operators.
One possible entropy definition of a density matrix ρ is the Von Neumann entropy. This is:
in which kB. is the Boltzmann constant, and the trace over space H is taken in which ρ operates.
The entropy of any pure state is zero because the eigenvalues of the density matrix are zero and one. This agrees with the heuristic argument that there is no uncertainty about the preparation of the condition.
It can be shown that unitary operators applied to a state (like the time evolution operator obtained from the Schrödinger equation) do not change the entropy of the system. This combines the reversibility of a process with its change in entropy - a fundamental result that connects quantum mechanics with information theory and thermodynamics.
Category: quantum physics
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