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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Incoherent superposition ： ウィキペディア英語版 Density matrix A density matrix is a matrix that describes a quantum system in a ''mixed state'', a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a ''pure state''. The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics. Explicitly, suppose a quantum system may be found in state $|\; \backslash psi\_1\; \backslash rangle$ with probability ''p''1, or it may be found in state $|\; \backslash psi\_2\; \backslash rangle$ with probability ''p''2, or it may be found in state $|\; \backslash psi\_3\; \backslash rangle$ with probability ''p''3, and so on. The density operator for this system is : $\backslash hat\backslash rho\; =\; \backslash sum\_i\; p\_i\; |\backslash psi\_i\; \backslash rangle\; \backslash langle\; \backslash psi\_i|,$ where $\backslash$ need not be orthogonal and $\backslash sum\_i\; p\_i=1$. By choosing an orthonormal basis $\backslash$, one may resolve the density operator into the density matrix, whose elements are〔 : $\backslash rho\_\; =\; \backslash sum\_i\; p\_i\; \backslash langle\; u\_m\; |\; \backslash psi\_i\; \backslash rangle\; \backslash langle\; \backslash psi\_i\; |\; u\_n\; \backslash rangle$ = \langle u_m |\hat \rho | u_n \rangle. The density operator can also be defined in terms of the density matrix, : $\backslash hat\backslash rho\; =\; \backslash sum\_\; |u\_m\backslash rangle\; \backslash rho\_\; \backslash langle\; u\_n|\; .$ For an operator $\backslash hat\; A$ (which describes an observable $A$ of the system), the expectation value $\backslash langle\; A\; \backslash rangle$ is given by〔 : $\backslash langle\; A\; \backslash rangle\; =\; \backslash sum\_i\; p\_i\; \backslash langle\; \backslash psi\_i\; |\; \backslash hat\; |\; \backslash psi\_i\; \backslash rangle$ = \sum_ \langle u_m | \hat\rho | u_n \rangle \langle u_n | \hat | u_m \rangle = \sum_ \rho_ A_ = \operatorname(\rho A). In words, the expectation value of ''A'' for the mixed state is the sum of the expectation values of ''A'' for each of the pure states $|\backslash psi\_i\backslash rangle$ weighted by the probabilities ''pi'' and can be computed as the trace of the product of the density matrix with the matrix representation of $A$ in the same basis. Mixed states arise in situations where the experimenter does not know which particular states are being manipulated. Examples include a system in thermal equilibrium (or additionally chemical equilibrium) or a system with an uncertain or randomly varying preparation history (so one does not know which pure state the system is in). Also, if a quantum system has two or more subsystems that are entangled, then each subsystem must be treated as a mixed state even if the complete system is in a pure state. The density matrix is also a crucial tool in quantum decoherence theory. The density matrix is a representation of a linear operator called the ''density operator''. The close relationship between matrices and operators is a basic concept in linear algebra. In practice, the terms ''density matrix'' and ''density operator'' are often used interchangeably. Both matrix and operator are self-adjoint (or Hermitian), positive semi-definite, of trace one, and may be infinite-dimensional. The formalism was introduced by John von Neumann in 1927 and independently, but less systematically by Lev Landau and Felix Bloch in 1927 and 1946 respectively. ==Pure and mixed states== In quantum mechanics, a quantum system is represented by a state vector (or ket) $|\; \backslash psi\; \backslash rangle$. A quantum system with a state vector $|\; \backslash psi\; \backslash rangle$ is called a ''pure state''. However, it is also possible for a system to be in a statistical ensemble of different state vectors: For example, there may be a 50% probability that the state vector is $|\; \backslash psi\_1\; \backslash rangle$ and a 50% chance that the state vector is $|\; \backslash psi\_2\; \backslash rangle$. This system would be in a ''mixed state''. The density matrix is especially useful for mixed states, because any state, pure or mixed, can be characterized by a single density matrix. A mixed state is different from a quantum superposition. In fact, a quantum superposition of pure states is another pure state, for example $|\; \backslash psi\; \backslash rangle\; =\; (|\; \backslash psi\_1\; \backslash rangle\; +\; |\; \backslash psi\_2\; \backslash rangle)/\backslash sqrt$. A state is pure if and only if its density matrix $\backslash rho$ satisfies $tr(\backslash rho^2)=1$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Density matrix」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.