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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 Creation and annihilation operators ： ウィキペディア英語版 Creation and annihilation operators Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one. A creation operator increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization. Creation and annihilation operators can act on states of various types of particles. For example, in quantum chemistry and many-body theory the creation and annihilation operators often act on electron states. They can also refer specifically to the ladder operators for the quantum harmonic oscillator. In the latter case, the raising operator is interpreted as a creation operator, adding a quantum of energy to the oscillator system (similarly for the lowering operator). They can be used to represent phonons. The mathematics for the creation and annihilation operators for bosons is the same as for the ladder operators of the quantum harmonic oscillator. For example, the commutator of the creation and annihilation operators that are associated with the same boson state equals one, while all other commutators vanish. However, for fermions the mathematics is different, involving anticommutators instead of commutators. ==Ladder operators for the quantum harmonic oscillator== In the context of the quantum harmonic oscillator, we reinterpret the ladder operators as creation and annihilation operators, adding or subtracting fixed quanta of energy to the oscillator system. Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin). This is because their wavefunctions have different symmetry properties. First consider the simpler bosonic case of the phonons of the quantum harmonic oscillator. Start with the Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator :$\backslash left(-\backslash frac\; \backslash frac\; +\; \backslash fracm\; \backslash omega^2\; x^2\backslash right)\; \backslash psi(x)\; =\; E\; \backslash psi(x)$ Make a coordinate substitution to nondimensionalize the differential equation :$x\; \backslash \; =\; \backslash \; \backslash sqrt\}\; q$. and the Schrödinger equation for the oscillator becomes :$\backslash frac\; \backslash left(-\backslash frac\; +\; q^2\; \backslash right)\; \backslash psi(q)\; =\; E\; \backslash psi(q)$. Note that the quantity $\backslash hbar\; \backslash omega\; =\; h\; \backslash nu$ is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as :$-\backslash frac\; +\; q^2\; =\; \backslash left(-\backslash frac+q\; \backslash right)\; \backslash left(\backslash frac+\; q\; \backslash right)\; +\; \backslash frac\; q\; -\; q\; \backslash frac$ The last two terms can be simplified by considering their effect on an arbitrary differentiable function f(q), :$\backslash left(\backslash frac\; q-\; q\; \backslash frac\; \backslash right)f(q)\; =\; \backslash frac(q\; f(q))\; -\; q\; \backslash frac\; =\; f(q)$ which implies, :$\backslash frac\; q-\; q\; \backslash frac\; =\; 1$ Therefore :$-\backslash frac\; +\; q^2\; =\; \backslash left(-\backslash frac+q\; \backslash right)\; \backslash left(\backslash frac+\; q\; \backslash right)\; +\; 1$ and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2, : then the Schrödinger equation for the oscillator becomes :$\backslash hbar\; \backslash omega\; \backslash left(\; a^\backslash dagger\; a\; +\; \backslash frac\; \backslash right)\; \backslash psi(q)\; =\; E\; \backslash psi(q)$ This is ''significantly'' simpler than the original form. Further simplifications of this equation enables one to derive all the properties listed above thus far. Letting $p\; =\; -\; i\; \backslash frac$, where "p" is the nondimensionalized momentum operator then we have :$(p)\; =\; i\; \backslash ,$ and :$a\; =\; \backslash frac\backslash right)$ :$a^\backslash dagger\; =\; \backslash frac\backslash right)$. Note that these imply that :$(a^\backslash dagger)\; =\; \backslash frac\; (q\; +\; ip\; ,\; q-i\; p)\; =\; \backslash frac\; (()\; +\; (q))\; =\; \backslash frac\; ((p)\; +\; (p))\; =\; 1$. The operators $a$ and $a^\backslash dagger$ may be contrasted with normal operators, which commute with their adjoints. A normal operator has a representation $A=\; B\; +\; i\; C$ where $B,C$ are self-adjoint and commute, i.e. $BC=CB$. By contrast, $a$ has the representation $a=q+ip$ where $p,q$ are self-adjoint but $()=1$. Then $B$ and $C$ have a common set of eigenfunctions (and are simultaneously diagonalizable), whereas p and q famously don't and aren't. Despite this, we go on. Using the commutation relations given above, the Hamiltonian operator can be expressed as :$\backslash hat\; H\; =\; \backslash hbar\; \backslash omega\; \backslash left(\; a\; \backslash ,\; a^\backslash dagger\; -\; \backslash frac\backslash right)\; =\; \backslash hbar\; \backslash omega\; \backslash left(\; a^\backslash dagger\; \backslash ,\; a\; +\; \backslash frac\backslash right).\backslash qquad\backslash qquad($ *) One can compute the commutation relations between the $a$ and $a^\backslash dagger$ operators and the Hamiltonian: :$(H,\; a)\; =\; (\backslash omega(\; a\; a^\backslash dagger\; -\; \backslash frac),a)\; =\; \backslash hbar\; \backslash omega\; (a\; a^\backslash dagger,\; a)\; =\; \backslash hbar\; \backslash omega\; (\; a\; ()\; +\; ()\; a^\backslash dagger)\; =\; -\backslash hbar\; \backslash omega\; a.$ :$(H,\; a^\backslash dagger)\; =\; \backslash hbar\; \backslash omega\; \backslash ,\; a^\backslash dagger\; .$ These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator. Assuming that $\backslash psi\_n$ is an eigenstate of the Hamiltonian $\backslash hat\; H\; \backslash psi\_n\; =\; E\_n\backslash ,\; \backslash psi\_n$. Using these commutation relations, it follows that〔 :$\backslash hat\; H\backslash ,\; a\backslash psi\_n\; =\; (E\_n\; -\; \backslash hbar\; \backslash omega)\backslash ,\; a\backslash psi\_n\; .$ :$\backslash hat\; H\backslash ,\; a^\backslash dagger\backslash psi\_n\; =\; (E\_n\; +\; \backslash hbar\; \backslash omega)\backslash ,\; a^\backslash dagger\backslash psi\_n\; .$ This shows that $a\backslash psi\_n$ and $a^\backslash dagger\backslash psi\_n$ are also eigenstates of the Hamiltonian, with eigenvalues $E\_n\; -\; \backslash hbar\; \backslash omega$ and $E\_n\; +\; \backslash hbar\; \backslash omega$ respectively. This identifies the operators $a$ and $a^\backslash dagger$ as "lowering" and "raising" operators between the eigenstates. The energy difference between adjacent eigenstates is $\backslash Delta\; E\; =\; \backslash hbar\; \backslash omega$. The ground state can be found by assuming that the lowering operator possesses a nontrivial kernel, $a\backslash ,\; \backslash psi\_0\; =\; 0$ with $\backslash psi\_0\backslash ne0$. Using the formula above for the Hamiltonian, one obtains :$0=\backslash hbar\; \backslash omega\backslash ,\; a^\backslash dagger\; a\; \backslash psi\_0\; =\; \backslash left(\backslash hat\; H\; -\; \backslash frac\; \backslash right)\; \backslash ,\backslash psi\_0.$ so $\backslash psi\_0$ is an eigenfunction of the Hamiltonian. This gives the ground state energy $E\_0\; =\; \backslash hbar\; \backslash omega\; /2$. This allows one to identify the energy eigenvalue of any eigenstate $\backslash psi\_n$ as〔 :$E\_n\; =\; \backslash left(n\; +\; \backslash frac\backslash right)\backslash hbar\; \backslash omega.$ Furthermore, it turns out that the first-mentioned operator in ( *), the number operator $N=a^\backslash dagger\; a\backslash ,,$ plays a most important role in applications, while the second one, $a\; \backslash ,a^\backslash dagger\backslash ,,$ can simply be replaced by $N\; +1\backslash ,.$ So one simply gets : $\backslash hbar\backslash omega\; \backslash ,\backslash left(N+\backslash frac\backslash right)\backslash ,\backslash psi\; (q)\; =E\backslash ,\backslash psi\; (q)$. 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「Creation and annihilation operators」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.