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| '''Creation and annihilation operators''' are [[Operator (mathematics)|mathematical operators]] that have widespread applications in [[quantum mechanics]], notably in the study of [[quantum harmonic oscillator]]s and many-particle systems.<ref name='Feynman1998p151'>{{harv|Feynman|1998|p=151}}</ref> 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 [[Hermitian adjoint|adjoint]] of the annihilation operator. In many subfields of [[physics]] and [[chemistry]], the use of these operators instead of [[wavefunction]]s is known as [[second quantization]]. | |
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| 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.
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| 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]].
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| The [[mathematics]] for the creation and annihilation operators for [[bosons]] is the same as for the [[ladder operators]] of the [[quantum harmonic oscillator]].<ref name='Feynman1998p167'>{{harv|Feynman|1998|p=167}}</ref> 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 [[Commutator#Anticommutator|anticommutators]] instead of commutators.<ref name='Feynman1998p174-5'>{{harv|Feynman|1998|pp=174–5}}</ref>
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| ==Ladder operators for quantum harmonic oscillator==
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| In the context of the quantum harmonic oscillator, we reinterpret the ladder operators as creation and annihilation operators, adding or subtracting fixed [[quantum|quanta]] of energy to the oscillator system. Creation/annihilation operators are different for [[boson]]s (integer spin) and [[fermion]]s (half-integer spin). This is because their [[wavefunction]]s have different [[identical particles|symmetry properties]].
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| First consider the simpler bosonic case of the phonons of the quantum harmonic oscillator.
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| Start with the [[Schrödinger equation]] for the one dimensional time independent [[quantum harmonic oscillator]]
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| :<math>\left(-\frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{1}{2}m \omega^2 x^2\right) \psi(x) = E \psi(x)</math>
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| Make a coordinate substitution to [[nondimensionalization|nondimensionalize]] the differential equation
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| :<math>x \ = \ \sqrt{ \frac{\hbar}{m \omega}} q</math>.
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| and the Schrödinger equation for the oscillator becomes
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| :<math> \frac{\hbar \omega}{2} \left(-\frac{d^2}{d q^2} + q^2 \right) \psi(q) = E \psi(q)</math>.
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| Note that the quantity <math> \hbar \omega = h \nu </math> is the same energy as that found for light [[quantum|quanta]] and that the parenthesis in the [[Hamiltonian (quantum mechanics)|Hamiltonian]] can be written as
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| :<math> -\frac{d^2}{dq^2} + q^2 = \left(-\frac{d}{dq}+q \right) \left(\frac{d}{dq}+ q \right) + \frac {d}{dq}q - q \frac {d}{dq} </math>
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| The last two terms can be simplified by considering their effect on an arbitrary differentiable function f(q),
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| :<math>\left(\frac{d}{dq} q- q \frac{d}{dq} \right)f(q) = \frac{d}{dq}(q f(q)) - q \frac{df(q)}{dq} = f(q) </math>
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| which implies,
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| :<math>\frac{d}{dq} q- q \frac{d}{dq} = 1 </math>
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| Therefore
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| :<math> -\frac{d^2}{dq^2} + q^2 = \left(-\frac{d}{dq}+q \right) \left(\frac{d}{dq}+ q \right) + 1 </math>
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| and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,
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| :<math> \hbar \omega \left[\frac{1}{\sqrt{2}} \left(-\frac{d}{dq}+q \right)\frac{1}{\sqrt{2}} \left(\frac{d}{dq}+ q \right) + \frac{1}{2} \right] \psi(q) = E \psi(q)</math>.
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| If we define
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| :<math>a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)</math> as the "creation operator" or the "raising operator" and
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| :<math> a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \frac{d}{dq} + q\right)</math> as the "annihilation operator" or the "lowering operator"
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| then the Schrödinger equation for the oscillator becomes
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| :<math> \hbar \omega \left( a^\dagger a + \frac{1}{2} \right) \psi(q) = E \psi(q)</math>
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| This is ''significantly'' simpler than the original form. Further simplifications of this equation enables one to derive all the properties listed above thus far.
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| Letting <math>p = - i \frac{d}{dq}</math>, where "p" is the nondimensionalized [[momentum operator]]
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| then we have
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| :<math> [q, p] = i \,</math>
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| and
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| :<math>a = \frac{1}{\sqrt{2}}(q + i p) = \frac{1}{\sqrt{2}}\left( q + \frac{d}{dq}\right) </math>
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| :<math>a^\dagger = \frac{1}{\sqrt{2}}(q - i p) = \frac{1}{\sqrt{2}}\left( q - \frac{d}{dq}\right)</math>.
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| Note that these imply that
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| :<math> [a, a^\dagger ] = \frac{1}{2} [ q + ip , q-i p] = \frac{1}{2} ([q,-ip] + [ip, q]) = \frac{-i}{2} ([q, p] + [q, p]) = 1 </math>
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| in contrast to the so-called "[[normal operators]]" of mathematics, which have a similar representation (e.g. <math>A= W_1 + i\, W_2)\,,</math> with self-adjoint <math>W_i\,.</math> But in the case of normal operators one would be dealing with [[commuting]] <math> W_i\,,</math> i.e. with <math>W_1W_2=W_2W_1\,,</math> so that the 1 at the extreme r.h.s. of the previous equation would be replaced by 0, which would have the consequence of one-and-the-same set of eigenfunctions (and/or eigendistributions) for both <math> W_1</math> and <math> W_2</math>, whereas here common eigenfunctions or eigendistributions of the operators p and q don't exist.
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| Thus, although in the present case one is explicitly dealing with non-normal operators, by the commutation relation given above, the Hamiltonian operator can be expressed as
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| :<math>\hat H = \hbar \omega \left( a \, a^\dagger - \frac{1}{2}\right).</math>
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| :<math>\hat H = \hbar \omega \left( a^\dagger \, a + \frac{1}{2}\right).</math>
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| And <math>a</math> and <math>a^\dagger\,,</math> operators give the following commutation relations with the Hamiltonian<ref>{{cite web|last=Branson|first=Jim|title=Quantum Physics at UCSD|url=http://quantummechanics.ucsd.edu/ph130a/130_notes/node170.html#section:HOraise|accessdate=16 May 2012}}</ref>
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| :<math>[\hat H, a ] = -\hbar \omega \, a.</math>
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| :<math>[\hat H, a^\dagger ] = \hbar \omega \, a^\dagger .</math>
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| These relations can be used to find the energy eigenstates of the quantum harmonic oscillator. Assuming that <math>\psi_n</math> is an eigenstate of the Hamiltonian <math>\hat H \psi_n = E_n\, \psi_n</math>. Using these commutation relations it can be shown that<ref>{{cite web|last=Branson|first=Jim|title=Quantum Physics at UCSD|url=http://quantummechanics.ucsd.edu/ph130a/130_notes/node170.html#section:HOraise|accessdate=16 May 2012}}</ref>
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| :<math>\hat H\, a\psi_n = (E_n - \hbar \omega)\, a\psi_n .</math>
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| :<math>\hat H\, a^\dagger\psi_n = (E_n + \hbar \omega)\, a^\dagger\psi_n .</math>
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| This shows that <math>a\psi_n</math> and <math>a^\dagger\psi_n</math> are also eigenstates of the Hamiltonian with eigenvalues <math>E_n - \hbar \omega</math> and <math>E_n + \hbar \omega</math>. This identifies the operators <math>a</math> and <math>a^\dagger</math> as lowering and rising operators between the eigenstates. Energy difference between two eigenstates is <math>\Delta E = \hbar \omega</math>.
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| The ground state can be found by assuming that the lowering operator will collapse it, <math>a\, \psi_0 = 0</math>. And then using the Hamiltonian in terms of rising and lowering operators,
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| :<math>a^\dagger a \psi_0 = 0 = \left(\frac{\hat H}{\hbar \omega} - \frac{1}{2} \right) \,\psi_0 = \left(\frac{E_0}{\hbar \omega} - \frac{1}{2} \right) \,\psi_0.</math>
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| the wave-function on the right is non-zero, thus term in brackets must be. This gives the ground state energy <math>E_0 = \hbar \omega /2</math>. This allows to identify the energy eigenvalue of any eigenstate <math>\psi_n</math> as<ref>{{cite web|last=Branson|first=Jim|title=Quantum Physics at UCSD|url=http://quantummechanics.ucsd.edu/ph130a/130_notes/node170.html#section:HOraise|accessdate=16 May 2012}}</ref>
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| :<math>E_n = \left(n + \frac{1}{2}\right)\hbar \omega</math>
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| Furthermore it can be shown that the first-mentioned operator, the '''number operator''' <math>N=a^\dagger a\,,</math> plays a most-important role in applications, while the second one, <math>a \,a^\dagger\,,</math> can simply be replaced by <math>N +1\,.</math> So one simply gets
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| : <math>\hbar\omega \,\left(N+\frac{1}{2}\right)\,\psi (q) =E\,\psi (q)</math>.
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| === Applications ===
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| The ground state <math>\ \psi_0(q)</math> of the [[quantum harmonic oscillator]] can be found by imposing the condition that
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| :<math> a \ \psi_0(q) = 0</math>.
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| Written out as a differential equation, the wavefunction satisfies
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| :<math>q \psi_0 + \frac{d\psi_0}{dq} = 0</math>
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| which has the solution
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| :<math>\psi_0(q) = C \exp\left(-{q^2 \over 2}\right).</math>
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| The normalization constant ''C'' can be found to be <math>1\over \sqrt[4]{\pi}</math> from <math>\int_{-\infty}^\infty \psi_0^* \psi_0 \,dq = 1</math>, using the [[Gaussian integral]].
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| === Matrix representation ===
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| The matrix counterparts of the creation and annihilation operators obtained from the quantum harmonic oscillator model are
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| :<math>{a}^{\dagger}=\begin{pmatrix}
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| 0 & 0 & 0 & \dots & 0 &\dots \\
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| \sqrt{1} & 0 & 0 & \dots & 0 & \dots\\
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| 0 & \sqrt{2} & 0 & \dots & 0 & \dots\\
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| 0 & 0 & \sqrt{3} & \dots & 0 & \dots\\
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| \vdots & \vdots & \vdots & \ddots & \vdots & \dots\\
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| 0 & 0 & 0 & 0 & \sqrt{n} &\dots & \\
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| \vdots & \vdots & \vdots & \vdots & \vdots &\ddots \end{pmatrix}
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| </math>
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| :<math>{a}=\begin{pmatrix}
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| 0 & \sqrt{1} & 0 & 0 & \dots & 0 & \dots \\
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| 0 & 0 & \sqrt{2} & 0 & \dots & 0 & \dots \\
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| 0 & 0 & 0 & \sqrt{3} & \dots & 0 & \dots \\
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| 0 & 0 & 0 & 0 & \ddots & \vdots & \dots \\
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| \vdots & \vdots & \vdots & \vdots & \ddots & \sqrt{n} & \dots \\
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| 0 & 0 & 0 & 0 & \dots & 0 & \ddots \\
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| \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}</math>
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| Substituting backwards, the laddering operators are recovered. They can be obtained via the relationships
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| <math>a^\dagger_{ij} = \langle\psi_i | {a}^\dagger | \psi_j\rangle</math> and
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| <math>a_{ij} = \langle\psi_i | {a} | \psi_j\rangle</math>. The wavefunctions are those of the quantum harmonic oscillator, and are sometimes called the "number basis".
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| === Mathematical details ===
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| The operators derived above are actually a specific instance of a more generalized class of creation and annihilation operators. The more abstract form of the operators satisfy the properties below.
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| Let ''H'' be the one-particle [[Hilbert space]]. To get the [[boson]]ic [[CCR algebra]], look at the algebra generated by ''a''(''f'') for any ''f'' in ''H''. The operator ''a''(''f'') is called an annihilation operator and the map ''a''(.) is [[antilinear]]. Its [[adjoint]]{{dn|date=December 2013}} is ''a''<sup>†</sup>(''f'') which is [[linear]] in ''H''.
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| For a boson,
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| :<math>[a(f),a(g)]=[a^\dagger(f),a^\dagger(g)]=0</math>
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| :<math>[a(f),a^\dagger(g)]=\langle f|g \rangle</math>,
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| where we are using [[bra-ket notation]].
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| For a fermion, the [[anticommutator]]s are
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| :<math>\{a(f),a(g)\}=\{a^\dagger(f),a^\dagger(g)\}=0 </math>
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| :<math>\{a(f),a^\dagger(g)\}=\langle f|g \rangle</math>.
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| A [[CAR algebra]].
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| Physically speaking, ''a''(''f'') removes (i.e. annihilates) a particle in the state {{Dket|''f''}} whereas ''a''<sup>†</sup>(''f'') creates a particle in the state {{Dket|''f''}}.
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| The [[free field]] [[vacuum state]] is the state with no particles. In other words,
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| :<math>a(f)|0\rangle=0</math>
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| where {{Dket|0}} is the vacuum state.
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| If {{Dket|''f''}} is normalized so that {{Dbraket|''f''|''f''}} = 1, then ''a''<sup>†</sup>(''f'') ''a''(''f'') gives the number of particles in the state {{Dket|''f''}}.
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| === Creation and annihilation operators for reaction-diffusion equations ===
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| The annihilation and creation operator description has also been useful to analyze classical reaction diffusion equations, such as the situation when a gas of molecules ''A'' diffuse and interact on contact, forming an inert product: {{nowrap|''A'' + ''A'' → ∅ .}} To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider <math>n_{i}</math> particles at a site <math>i</math> on a 1-d lattice. Each particle diffuses independently, so that the probability that one of them leaves the site for short times <math>dt</math> is proportional to <math>n_{i}dt</math>, say <math>\alpha n_{i}dt</math> to hop left and <math>\alpha n_{i}dt</math> to hop right. All <math>n</math> particles will stay put with a probability <math>1-2\alpha n_{i}dt</math>.
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| We can now describe the occupation of particles on the lattice as a `ket' of the form {{Dket|''n<sub>1</sub>, n<sub>2</sub>, ...''}}. A slight modification of the annihilation and creation operators is needed so that
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| :<math>a|n\rangle= \sqrt{n} \ |n-1\rangle</math>
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| and
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| :<math>a^{\dagger}|n\rangle= \sqrt{n+1} \ |n+1\rangle</math>.
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| This modification preserves the commutation relation
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| :<math>[a,a^{\dagger}]=1</math>,
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| but allows us to write the pure diffusive behaviour of the particles as
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| :<math>\partial_{t}|\psi\rangle=-\alpha\sum(2a_{i}^{\dagger}a_{i}-a_{i-1}^{\dagger}a_{i}-a_{i+1}^{\dagger}a_{i})|\psi\rangle=-\alpha\sum(a_{i}^{\dagger}-a_{i-1}^{\dagger})(a_{i}-a_{i-1})|\psi\rangle </math>
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| The reaction term can be deduced by noting that <math>n</math> particles can interact in <math>n(n-1)</math> different ways, so that the probability that a pair annihilates is <math>\lambda n(n-1)dt</math> and the probability that no pair annihilates is <math>1-\lambda n(n-1)dt</math> leaving us with a term
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| :<math>\lambda\sum(a_{i}a_{i}-a_{i}^{\dagger}a_{i}^{\dagger}a_{i}a_{i})</math>
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| yielding
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| :<math>\partial_{t}|\psi\rangle=-\alpha\sum(a_{i}^{\dagger}-a_{i-1}^{\dagger})(a_{i}-a_{i-1})|\psi\rangle+\lambda\sum(a_{i}^{2}-a_{i}^{\dagger 2}a_{i}^{2})|\psi\rangle </math>
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| Other kinds of interactions can be included in a similar manner.
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| This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.
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| ==Creation and annihilation operators in quantum field theories==
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| {{Main|Second quantization|Quantum_field_theory#Second_quantization}}
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| In [[Quantum field theory|quantum field theories]] and [[many-body problem]]s one works with creation and annihilation operators of quantum states, <math>a^\dagger_i</math> and <math>a^{\,}_i</math>. These operators change the eigenvalues of the [[number operator]],
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| : <math>N = \sum_i n_i = \sum_i a^\dagger_i a^{\,}_i</math>,
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| by one, in analogy to the harmonic oscillator. The indices (such as <math>i</math>) represent [[quantum numbers]] that label the single-particle states of the system; hence, they are not necessarily single numbers. For example, a [[tuple]] of quantum numbers <math>(n, l, m, s)</math> is used to label states in the [[hydrogen atom]].
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| The commutation relations of creation and annihilation operators in a multiple-[[boson]] system are,
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| : <math>[a^{\,}_i, a^\dagger_j] \equiv a^{\,}_i a^\dagger_j - a^\dagger_ja^{\,}_i = \delta_{i j},</math>
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| : <math>[a^\dagger_i, a^\dagger_j] = [a^{\,}_i, a^{\,}_j] = 0,</math>
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| where <math>[\ \ , \ \ ]</math> is the [[commutator]] and <math>\delta_{i j}</math> is the [[Kronecker delta]].
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| For [[fermion]]s, the commutator is replaced by the [[anticommutator]] <math>\{\ \ , \ \ \}</math>,
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| : <math>\{a^{\,}_i, a^\dagger_j\} \equiv a^{\,}_i a^\dagger_j +a^\dagger_j a^{\,}_i = \delta_{i j},</math>
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| : <math>\{a^\dagger_i, a^\dagger_j\} = \{a^{\,}_i, a^{\,}_j\} = 0.</math>
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| Therefore, exchanging disjoint (i.e. <math>i \ne j</math>) operators in a product of creation of annihilation operators will reverse the sign in fermion systems, but not in boson systems.
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| ==See also==
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| * [[Bogoliubov transformation]]s - arises in the theory of quantum optics.
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| * [[Optical Phase Space]]
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| * [[Fock space]]
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| * [[Canonical commutation relation]]s
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| ==References==
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| *{{cite book | last = Feynman | first = Richard P. | authorlink = Richard Feynman | coauthors = | title = Statistical Mechanics: A Set of Lectures | publisher = Addison-Wesley | year = 1998 | origyear = 1972 | edition=2nd | location = Reading, Massachusetts | page = | url = http://books.google.com/books?id=Ou4ltPYiXPgC&pg=Front | doi = | id = | isbn = 978-0-201-36076-9| ref=harv }}
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| ==Footnotes==
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| <references/>
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| {{Physics operator}}
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum field theory]]
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