Density matrix
In physics the density operator (also statistical operator) describes an ensemble of similar systems, the probability that a chosen at system is in a particular state. The density operator is represented by the density of the matrix (or random matrix).
The density operator was originally in the framework of classical physics of Stokes for the polarization state of a light beam developed ( Stokes parameters ). Hereinafter, the density operator is represented in the context of systems in quantum mechanically defined states.
The density operator occurs in the context of quantum statistics.
- 5.1 density matrix for pure states
- 5.2 density matrix for a uniformly distributed ensemble
Construction
The density operator describes a mixed state. In the considered ensemble there are several identical systems with the probabilities in the orthogonal states. The states are not orthogonal, the respective weights is no longer present, the probability of the mixture in the respective state. The weights are normalized to 1: Then ( in Bra- Ket notation) of the density operator is given by
It is
The projection operator, which projected out applied to an arbitrary state vector whose component "parallel" to the state:
The factor is the probability amplitude to be found in the present state system in the state.
With the help of the projection operators can also write the density operator as
For an ensemble in which all systems are prepared in the same state, the density operator, therefore, is simply the projection operator itself:
The density operator for the canonical ensemble is:
In the eigenbasis of the Hamiltonian has the form ( 1) assumes. The same is obtained for the density operator of the grand canonical ensemble.
Readings
For each component of state mixture of the average of the measured results of a physical quantity is given by the expectation value It is the operator associated to (see quantum mechanics, observables ).
Since the ensemble is a mixture of systems within the various participating states, the average of all measurements on the individual systems is the weighted sum of the individual expected values :
This is equal to the track
As you can see by the orthonormal basis vectors with the help of a complete system: Because of ( unit operator) is
Are the just the eigenstates of observables (ie with eigenvalues ), then further
This is the weighted about the ensemble means for the probability of finding a chosen at system eigenstate. So, the probability of obtaining the eigenvalue, as a result in a single measurement. Characteristic that is represented by a non-coherent sum which is independent of the relative phase of the states involved in the ensemble.
Example: density operator and the density matrix for the electron polarization
The density of the matrix is the matrix with which the operator can be displayed with respect to an orthonormal basis:
Basis states
In the following, the symbol " " means that a Bra, Ket or operator is represented with respect to a basis (see also Bra- Ket # representation). The states " spin up" ( bezgl z -axis) and are presented as ket vectors by columns "spin off ". The corresponding bra vectors are then row vectors: and. The projection operators ( by matrix multiplication ):
These are also the density matrices for fully - or - direction polarized electrons.
Polarization in the z- direction
The z- component of the spin, the diagonal matrix formed by eigenvalues for the predicted measurement result obtained properly for the ensemble
For the ensemble results
Other Polar Activision direction
The states of in - and - direction polarized electrons are the projection operators to have ( in the basis of eigenstates! ) The matrices is characteristic that these are not diagonal matrices, and that the various stages by which the eigenstates as a chain- vectors were superimposed here again refer to the matrix elements outside the main diagonal. This is an expression of the coherent superposition of the eigenstates are formed by the out - eigenstates.
Unpolarized Ensemble
Are the electrons polarized half each in direction, ie, the density matrix:
The same density matrix is obtained for a mixture of electrons, which are polarized to 50 % in direction. Thus, all possible measurement results are identical to those on the ensemble, which was formed from polarized electrons. The original polarization directions are physically (and thus conceptual ) can not be distinguished: It is a both times and become the same ensemble.
Mixture of different polarization directions
For a mixture of electrons with spin in direction and direction ( or shares ), ie the density matrix
The expectation value of the spin in direction is then
The polarized in the ( ) direction carry electrons thus expected not to the expected value at.
Formal definition
Given a quantum-mechanical system, which is modeled in a Hilbert space. A bounded linear operator on a density operator if:
Although the terms density matrix density operator and are often used interchangeably, there is a mathematical difference. Just like in linear algebra a matrix is the basic representation of a linear operator, can be distinguished in quantum mechanics between abstract density operator and a specific density matrix in a particular representation. If a density operator, so called
The density matrix in the coordinate representation. However, it is not a real matrix, since the coordinate representation is defined over a continuum of improper basis vectors, but a so-called integral kernel.
In finite-dimensional Hilbert spaces (eg, spin systems ) results, however, then a positive semidefinite matrix with trace 1, ie, a true density matrix, if an orthonormal basis is chosen:
Properties
- Each density operator is self-adjoint (or Hermitian ) positive because operators are always self-adjoint.
- The set of density operators is a convex set, the edge of which is the amount of pure ( quantum ) states. The amount is in contrast to the classical theory is not simplex, i.e. a density operator is generally not clearly convex pure states represented.
- The probability obtained in the measurement of an observable in a system will be described by the density operator, the measurement value is given by
- The mean of the measured values ( expected value ) for measurement of an observable
Density matrix for pure states
Is the ensemble considered a pure ensemble, the system thus consists only of a pure state, then for the density matrix.
Is always valid for mixed states.
Density matrix for a uniformly distributed ensemble
A level system in which all states are equally likely, the density matrix
Where the- dimensional identity matrix.
Time development
From the Schrödinger equation, the time evolution ( dynamics) describes pure quantum states, one can immediately derive the time evolution of mixed states. This is done using an arbitrary decomposition of the density matrix into pure states whose dynamics satisfies the Schrödinger equation, and calculates the dynamics of the mixed state to
Where the system of Hamiltonian. This equation is as of Neumann'sche - known equation of motion (not to be confused with the Heisenberg equation of motion).
This differential equation can be solved for time-independent Hamiltonians and receives with the unitary time evolution operator equation
This solution can be easily checked by inserting.
It is remarkable that the usual Heisenberg equation of motion does not apply to the operator, since the time evolution operator of the derived directly from the Schrödinger equation dynamics obeys. And the time evolution of the Opertors by the time evolution operator is not carried out according to the usual time evolution equation of operators ( for ordinary observable A ), but this is understandable since
Entropy
By means of the density matrix of the entropy of a system can be defined as follows (see also the Von Neumann entropy):
The Boltzmann constant, and the trace is taken over the space in which operates.
The entropy of each pure state is zero, since the eigenvalues of the density matrix are zero and one. This is consistent with the heuristic argument that there is no uncertainty about the preparation of the state prevails.
It can be shown that in a state applied to unitary operators (such as derived from the Schrodinger equation, time evolution operator ) does not change, the entropy of the system. Connecting the reversibility of a process having its entropy - a fundamental result, which connects the quantum mechanics and the information theory and thermodynamics.