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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
:\left(-\frac \frac + \fracm \omega^2 x^2\right) \psi(x) = E \psi(x)
Make a coordinate substitution to nondimensionalize the differential equation
:x \ = \ \sqrt} q.
and the Schrödinger equation for the oscillator becomes
: \frac \left(-\frac + q^2 \right) \psi(q) = E \psi(q).
Note that the quantity \hbar \omega = h \nu is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as
: -\frac + q^2 = \left(-\frac+q \right) \left(\frac+ q \right) + \frac q - q \frac
The last two terms can be simplified by considering their effect on an arbitrary differentiable function f(q),
:\left(\frac q- q \frac \right)f(q) = \frac(q f(q)) - q \frac = f(q)
which implies,
:\frac q- q \frac = 1
Therefore
: -\frac + q^2 = \left(-\frac+q \right) \left(\frac+ q \right) + 1
and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,
: \hbar \omega \left(: a \ \ = \ \frac + q\right) as the "annihilation operator" or the "lowering operator"
then the Schrödinger equation for the oscillator becomes
: \hbar \omega \left( a^\dagger a + \frac \right) \psi(q) = E \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 \frac, where "p" is the nondimensionalized momentum operator
then we have
: (p ) = i \,
and
:a = \frac\right)
:a^\dagger = \frac\right).
Note that these imply that
: (a^\dagger ) = \frac (q + ip , q-i p ) = \frac (() + (q )) = \frac ((p ) + (p )) = 1 .
The operators a and a^\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
:\hat H = \hbar \omega \left( a \, a^\dagger - \frac\right) = \hbar \omega \left( a^\dagger \, a + \frac\right).\qquad\qquad(
*)
One can compute the commutation relations between the a and a^\dagger operators and the Hamiltonian:
:(H, a ) = (\omega( a a^\dagger - \frac),a ) = \hbar \omega (a a^\dagger, a ) = \hbar \omega ( a () + () a^\dagger) = -\hbar \omega a.
:(H, a^\dagger ) = \hbar \omega \, a^\dagger .
These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator. Assuming that \psi_n is an eigenstate of the Hamiltonian \hat H \psi_n = E_n\, \psi_n. Using these commutation relations, it follows that〔
:\hat H\, a\psi_n = (E_n - \hbar \omega)\, a\psi_n .
:\hat H\, a^\dagger\psi_n = (E_n + \hbar \omega)\, a^\dagger\psi_n .
This shows that a\psi_n and a^\dagger\psi_n are also eigenstates of the Hamiltonian, with eigenvalues E_n - \hbar \omega and E_n + \hbar \omega respectively. This identifies the operators a and a^\dagger as "lowering" and "raising" operators between the eigenstates. The energy difference between adjacent eigenstates is \Delta E = \hbar \omega.
The ground state can be found by assuming that the lowering operator possesses a nontrivial kernel, a\, \psi_0 = 0 with \psi_0\ne0. Using the formula above for the Hamiltonian,
one obtains
:0=\hbar \omega\, a^\dagger a \psi_0 = \left(\hat H - \frac \right) \,\psi_0.
so \psi_0 is an eigenfunction of the Hamiltonian. This gives the ground state energy E_0 = \hbar \omega /2. This allows one to identify the energy eigenvalue of any eigenstate \psi_n as〔
:E_n = \left(n + \frac\right)\hbar \omega.
Furthermore, it turns out that the first-mentioned operator in (
*), the number operator N=a^\dagger a\,, plays a most important role in applications, while the second one, a \,a^\dagger\,, can simply be replaced by N +1\,. So one simply gets
: \hbar\omega \,\left(N+\frac\right)\,\psi (q) =E\,\psi (q).

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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