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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 first quantization ： ウィキペディア英語版 first quantization A first quantization of a physical system is a semi-classical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well or a bulk electromagnetic field or gravitational field) is treated classically. First quantization is appropriate for studying a single quantum-mechanical system being controlled by a laboratory apparatus that is itself large enough that classical mechanics is applicable to most of the apparatus. ==Theoretical background== The starting point is the notion of quantum states and the observables of the system under consideration. Quantum theory postulates that all quantum states are represented by state vectors in a Hilbert space, and that all observables are represented by Hermitian operators acting on that space. Parallel state vectors represent the same physical state, and therefore one mostly deals with normalized state vectors. Any given Hermitan operator $\backslash hat$ has a number of eigenstates $|\backslash psi\_\backslash alpha\backslash rangle$ that are left invariant by the action of the operator up to a real scale factor $\backslash alpha$, i. e., $\backslash hat|\backslash psi\_\backslash alpha\backslash rangle=\backslash alpha|\backslash psi\_\backslash alpha\backslash rangle$. The scale factors are denoted the eigenvalues of the operator. It is a fundamental theorem of Hilbert space theory that the set of all eigenvectors of any given Hermitian operator forms a complete basis set of the Hilbert space. In general the eigenstates $|\backslash psi\_\backslash alpha\backslash rangle$ and $|\backslash psi\_\backslash beta\backslash rangle$ of two different Hermitian operators $\backslash hat$ and $\backslash hat$ are not the same. By measurement of the type $\backslash hat$ the quantum state can be prepared to be in an eigenstate $|\backslash psi\_\backslash beta\backslash rangle$. This state can also be expressed as a superposition of eigenstates $|\backslash psi\_\backslash alpha\backslash rangle$ as $|\backslash psi\_\backslash beta\backslash rangle=\backslash sum\_\backslash alpha|\backslash psi\_\backslash alpha\backslash rangle\; C\_$. If one measures the dynamical variable associated with the operator $\backslash hat$ in this state, one cannot in general predict the outcome with certainty. It is only described in probabilistic terms. The probability of having any given $|\backslash psi\_\backslash alpha\backslash rangle$ as the outcome is given as the absolute square $|C\_|^2$ of the associated expansion coefficient. This non-causal element of quantum theory is also known as the wave function collapse. However, between collapse events the time evolution of quantum states is perfectly deterministic. The time evolution of a state vector $|\backslash psi\; (t)\backslash rangle$ is governed by the central operator in quantum mechanics, the Hamiltonian $\backslash hat$ (the operator associated with the total energy of the system), through Schrödinger's equation: $i\; \backslash hbar\; \backslash frac|\backslash psi\; (t)\backslash rangle\; =\; \backslash hat\; H\; |\backslash psi\; (t)\backslash rangle$ Each state vector $|\backslash psi\backslash rangle$ is associated with an adjoint state vector $(|\backslash psi\backslash rangle)^\backslash dagger\; =\; \backslash langle\; \backslash psi\; |$ and can form inner products, "bra(c)kets" $\backslash langle\; \backslash psi\; |\backslash phi\backslash rangle$ between adjoint "bra" states $\backslash langle\; \backslash psi|$ and "ket" states $|\backslash phi\backslash rangle$. The standard geometrical terminology is used; e.g. the norm squared of $|\backslash psi\backslash rangle$ is given by $\backslash langle\; \backslash psi\; |\backslash psi\backslash rangle$ and $|\backslash psi\backslash rangle$ and $|\backslash phi\backslash rangle$ are said to be orthogonal if $\backslash langle\; \backslash psi\; |\backslash phi\backslash rangle\; =\; 0$. If $$ is an orthonormal basis of the Hilbert space, the above-mentioned expansion coefficient $C\_$ is found forming inner products: $C\_=\backslash langle\; \backslash psi\_\backslash alpha\; |\backslash psi\_\backslash beta\backslash rangle$. A further connection between the direct and the adjoint Hilbert space is given by the relation $\backslash langle\; \backslash psi\; |\backslash phi\backslash rangle\; =\; \backslash langle\; \backslash phi\; |\backslash psi\backslash rangle^$ *, which also leads to the definition of adjoint operators. For a given operator $\backslash hat$ the adjoint operator $\backslash hat^\backslash dagger$ is defined by demanding $\backslash langle\; \backslash psi\; |\backslash hat|\backslash phi\backslash rangle\; =\; \backslash langle\; \backslash phi\; |\backslash hat^\backslash dagger|\backslash psi\backslash rangle^$ * for any $|\backslash psi\backslash rangle$ and $|\backslash phi\backslash rangle$. 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「first quantization」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.