Reflects the statistical nature of quantum mechanics. Specifically, the density matrix, which is sometimes also called the statistical matrix, illustrates that any knowledge about a quantum mechanical system stems from the observation of many identically prepared systems, i.e., the ensemble average. For a system in a well-defined state ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '|Psi angle' 'utf-8' math_unknown_error given by ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '|Psi( heta) angle =Sigma_n c_n( heta)|Psi_n angle' 'utf-8' math_unknown_error, where ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '|Psi_n angle' 'utf-8' math_unknown_error forms a complete basis, the density matrix elements are defined as

where ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'hat{ ho}=|Psi angle langleoldsymbol{Psi}|' 'utf-8' math_unknown_error. It follows that the individual matrix elements ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' ' ho_{mn}' 'utf-8' math_unknown_error can also be calculated through

For statistical mixtures of states, the definition for the density matrix must be generalized to account for the uncertainties of the different admixtures of pure states:

where ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'p( heta)' 'utf-8' math_unknown_error is the probability distribution of

finding the state ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '|Psi( heta) angle' 'utf-8' math_unknown_error in the mixed state.

The density matrix contains information about the specific preparation of a quantum system. This is in contrast to the matrix elements ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'O_{nm}=langlepsi_n|hat{O}|psi_m angle' 'utf-8' math_unknown_error of an observable ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'hat{O}.~O_{nm}' 'utf-8' math_unknown_error depends only on the specific operator ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'hat{O}' 'utf-8' math_unknown_error and the

basis set, but contains no information about the quantum state ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '|Psi angle' 'utf-8' math_unknown_error itself.

The diagonal elements ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' ' ho_{nn}' 'utf-8' math_unknown_error are called populations, as ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' ' ho_{nn}' 'utf-8' math_unknown_error give the populations, i.e., the probability of finding the system in state ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'psi_n( ho_{nn}=P_n)' 'utf-8' math_unknown_error which leads to the condition ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' ' ho_{nn}ge 0' 'utf-8' math_unknown_error. This terminology is also justified by the property of the density matrix:

The off-diagonal elements ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' ' ho_{nm}' 'utf-8' math_unknown_error. are termed the coherences, as they are measures for the coherences between states ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '|Psi_n angle' 'utf-8' math_unknown_error and ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '|Psi_m angle' 'utf-8' math_unknown_error. In the case that a particular density matrix ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' ' ho' 'utf-8' math_unknown_error represents a pure state, as opposed to a statistical mixture, the density matrix is idempotent, i.e.,

Consequently, also,

In contrast, for the density matrix of mixed states, we find:

Finally, the density matrix is Hermitian, i.e.,

The density matrix allows a straightforward calculation of expectation values ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'langlehat{O} angle' 'utf-8' math_unknown_error for an observable ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'hat{O}:' 'utf-8' math_unknown_error

With the help of the density matrix, one solves for ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'langlehat{O} angle:' 'utf-8' math_unknown_error

As an example, the density state for the simplest coherent state ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '|Psi_{mbox{coh}} angle' 'utf-8' math_unknown_error given by

yields

For the special case of ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' ' heta = pi/4' 'utf-8' math_unknown_error, we find:

In contrast, for a completely incoherent state or mixed state, where states with values of all different ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' ' heta' 'utf-8' math_unknown_error are mixed with an equal probability, we solve for the density matrix: