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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 Translation operator (quantum mechanics) ： ウィキペディア英語版 Translation operator (quantum mechanics) In quantum mechanics, a translation operator is defined as an operator which shifts particles and fields by a certain amount in a certain direction. More specifically, for any displacement vector $\backslash mathbf\; x$, there is a corresponding translation operator $\backslash hat(\backslash mathbf)$ that shifts particles and fields by the amount $\backslash mathbf\; x$. For example, if $\backslash hat(\backslash mathbf)$ acts on a particle located at position $\backslash mathbf\; r$, the result is a particle at position $\backslash mathbf$. Translation operators are linear and unitary. They are closely related to the momentum operator; for example, a translation operator that moves by an infinitesimal amount in the $y$ direction has a simple relationship to the $y$-component of the momentum operator. Because of this, conservation of momentum holds when the translation operators commute with the Hamiltonian, i.e. when laws of physics are translation-invariant. This is an example of Noether's theorem. ==Action on position eigenkets and wavefunctions== The translation operator $\backslash hat(\backslash mathbf)$ moves particles and fields by the amount x. Therefore, if a particle is in an eigenstate of the position operator $|\backslash mathbf\backslash rangle$ (i.e., precisely located at the position r), then after $\backslash hat(\backslash mathbf)$ acts on it, the particle is at the position (r+x): :$\backslash hat(\backslash mathbf)|\backslash mathbf\backslash rangle\; =\; |\backslash mathbf+\backslash mathbf\backslash rangle$ An alternative (and equivalent) way to describe what the translation operator does is based on position-space wavefunctions. If a particle has a position-space wavefunction $\backslash psi(\backslash mathbf)$, and $\backslash hat\; T(\backslash mathbf)$ acts on the particle, the new position-space wavefunction is $\backslash psi\text{'}(\backslash mathbf)$ defined by $\backslash psi\text{'}(\backslash mathbf)\; =\; \backslash psi(\backslash mathbf\; -\; \backslash mathbf)$. This relation is easier to remember as $\backslash psi\text{'}(\backslash mathbf+\backslash mathbf)\; =\; \backslash psi(\backslash mathbf)$: "The value of the new wavefunction at the new point equals the value of the old wavefunction at the old point".〔(Lecture notes by Robert Littlejohn )〕 Here is an example showing that these two descriptions are equivalent. The state $|\backslash mathbf\backslash rangle$ corresponds to the wavefunction $\backslash psi(\backslash mathbf)\; =\; \backslash delta(\backslash mathbf\; -\; \backslash mathbf)$ (where $\backslash delta$ is the Dirac delta function), while the state $\backslash hat(\backslash mathbf)|\backslash mathbf\backslash rangle\; =\; |\backslash mathbf+\backslash mathbf\backslash rangle$ corresponds to the wavefunction $\backslash psi\text{'}(\backslash mathbf)\; =\; \backslash delta(\backslash mathbf\; -\; (\backslash mathbf\; +\; \backslash mathbf))$. These indeed satisfy $\backslash psi\text{'}(\backslash mathbf)\; =\; \backslash psi(\backslash mathbf\; -\; \backslash mathbf)$. : constitutes a basis of state space.〔Page no.-108, Chapter-2,Volume-1, Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë〕 Therefore it is possible to characterize each ket by its " wave function in the $\backslash left\backslash$ representation": : $\backslash psi(\backslash mathbf\; r)\; =\; \backslash langle\; \backslash mathbf\; r|\backslash psi\backslash rangle$〔Page no. 68, Section 1.10, R. Shankar, Principles of Quantum Mechanics〕 Consider the wave function associated with the ket $\backslash mathbf|\backslash psi\backslash rangle$ in the $\backslash left\backslash$ representation: Using the fact that the position operator $\backslash mathbf$ is Hermitian and $|\backslash mathbf\; r\backslash rangle$ is the eigenvector of $\backslash mathbf$ with eigenvalue $\backslash mathbf\; r$, : $\backslash langle\; \backslash mathbf\; r|\backslash mathbf|\backslash psi\backslash rangle\; =\; \backslash mathbf\; r\backslash langle\; \backslash mathbf\; r|\backslash psi\backslash rangle\; =\; \backslash mathbf\; r\backslash psi(\backslash mathbf\; r)$ The action of $\backslash mathbf$ in the $\backslash left\backslash$ representation is therefore simply a multiplication by $\backslash mathbf\; r$. As explained in a later section, when a translation operator acts on a bra in the position eigenbasis, the result is: :$\backslash langle\; \backslash mathbf\; r|\backslash hat\; T(\backslash mathbf\; x)\; =\; \backslash langle\; \backslash mathbf\; r\; -\; \backslash mathbf\; x|$ Therefore, the wave function associated with the ket $\backslash hat\; T(\backslash mathbf\; x)|\backslash psi\backslash rangle$ in the $\backslash left\backslash$ representation is written as: : $\backslash langle\; \backslash mathbf\; r|\backslash hat\; T(\backslash mathbf\; x)|\backslash psi\backslash rangle\; =\; \backslash langle\; \backslash mathbf\; r-\backslash mathbf\; x|\backslash psi\backslash rangle\; =\; \backslash psi(\backslash mathbf\; r-\backslash mathbf\; x)$ |} 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「Translation operator (quantum mechanics)」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.