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Creation operators and annihilation operators are
mathematical operators Mathematical Operators is a Unicode block containing characters for mathematical, logical, and set notation. Notably absent are the plus sign (+), greater than sign (>) and less than sign (<), due to them already appearing in the Basi ...
that have widespread applications in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually denoted \hat) lowers the number of particles in a given state by one. A creation operator (usually denoted \hat^\dagger) increases the number of particles in a given state by one, and it is the
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
of the annihilation operator. In many subfields of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and chemistry, the use of these operators instead of
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s is known as
second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
. They were introduced by
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
. Creation and annihilation operators can act on states of various types of particles. For example, in quantum chemistry and
many-body theory The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
the creation and annihilation operators often act on
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
states. They can also refer specifically to the
ladder operators In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
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 In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanic ...
. Constructing Hamiltonians using these operators has the advantage that the theory automatically satisfies the
cluster decomposition theorem In physics, the cluster decomposition property states that experiments carried out far from each other cannot influence each other. Usually applied to quantum field theory, it requires that vacuum expectation values of operators localized in bound ...
. The mathematics for the creation and annihilation operators for
bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
is the same as for the
ladder operators In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
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 In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
the mathematics is different, involving anticommutators instead of commutators.


Ladder operators for the quantum harmonic oscillator

In the context of the quantum harmonic oscillator, one reinterprets 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
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
s (integer spin) and fermions (half-integer spin). This is because their
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s have different symmetry properties. First consider the simpler bosonic case of the photons of the quantum harmonic oscillator. Start with the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
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. 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 Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified 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 , coinciding with the usual canonical commutation relation -i ,p1 , in position space representation: p:=-i\frac. 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 frac \left(-\frac+q \right)\frac \left(\frac+ q \right) + \frac \right\psi(q) = E \psi(q). If one defines :a^\dagger \ = \ \frac \left(-\frac + q\right) as the "creation operator" or the "raising operator" and : a \ \ = \ \frac \left(\ \ \ \!\frac + q\right) as the "annihilation operator" or the "lowering operator", the Schrödinger equation for the oscillator reduces to : \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 enable one to derive all the properties listed above thus far. Letting p = - i \frac, where p is the nondimensionalized momentum operator one has : , p= i \, and :a = \frac(q + i p) = \frac\left( q + \frac\right) :a^\dagger = \frac(q - i p) = \frac\left( q - \frac\right). Note that these imply : , a^\dagger = \frac q + ip , q-i p= \frac ( ,-ip+ p, q = \frac ( , p+ , p = 1. The operators a\, and a^\dagger\, may be contrasted to
normal operators In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal operat ...
, which commute with their adjoints. 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 may compute the commutation relations between the a\, and a^\dagger\, operators and the Hamiltonian: : hat H, a = hbar \omega \left ( a a^\dagger - \frac\right ) ,a= \hbar \omega a a^\dagger, a= \hbar \omega ( a ^\dagger,a+ ,aa^\dagger) = -\hbar \omega a. : hat H, a^\dagger = \hbar \omega \, a^\dagger . These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator as follows. 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 adjacent 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. Applying the Hamiltonian to the ground state, :\hat H\psi_0 = \hbar\omega\left(a^\dagger a+\frac\right)\psi_0 = \hbar\omega a^\dagger a \psi_0 + \frac\psi_0=0+\frac\psi_0=E_0\psi_0. So \psi_0 is an eigenfunction of the Hamiltonian. This gives the ground state energy E_0 = \hbar \omega /2, which 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 the most important role in applications, while the second one, a a^\dagger \, can simply be replaced by N+1. Consequently, : \hbar\omega \,\left(N+\frac\right)\,\psi (q) =E\,\psi (q)~. The
time-evolution operator Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be dis ...
is then :U(t)=\exp ( -it \hat/\hbar) = \exp (-it\omega (a^\dagger a+1/2)) ~, := e^ ~ \sum_^ a^ a^k ~.


Explicit eigenfunctions

The ground state \ \psi_0(q) of the quantum harmonic oscillator can be found by imposing the condition that : a \ \psi_0(q) = 0. Written out as a differential equation, the wavefunction satisfies :q \psi_0 + \frac = 0 with the solution :\psi_0(q) = C \exp\left(-\right). The normalization constant is found to be 1/ \sqrt /math> from \int_^\infty \psi_0^* \psi_0 \,dq = 1,  using the
Gaussian integral The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is \int_^\infty e^\,dx = \s ...
. Explicit formulas for all the eigenfunctions can now be found by repeated application of a^\dagger to \psi_0.


Matrix representation

The matrix expression of the creation and annihilation operators of the quantum harmonic oscillator with respect to the above orthonormal basis is : a^\dagger = \begin 0 & 0 & 0 & 0 & \dots & 0 & \dots \\ \sqrt & 0 & 0 & 0 & \dots & 0 & \dots \\ 0 & \sqrt & 0 & 0 & \dots & 0 & \dots \\ 0 & 0 & \sqrt & 0 & \dots & 0 & \dots \\ \vdots & \vdots & \vdots & \ddots & \ddots & \dots & \dots \\ 0 & 0 & 0 & \dots & \sqrt & 0 & \dots & \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots \end : a =\begin 0 & \sqrt & 0 & 0 & \dots & 0 & \dots \\ 0 & 0 & \sqrt & 0 & \dots & 0 & \dots \\ 0 & 0 & 0 & \sqrt & \dots & 0 & \dots \\ 0 & 0 & 0 & 0 & \ddots & \vdots & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots & \sqrt & \dots \\ 0 & 0 & 0 & 0 & \dots & 0 & \ddots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end These can be obtained via the relationships a^\dagger_ = \langle\psi_i \mid a^\dagger \mid \psi_j\rangle and a_ = \langle\psi_i \mid a \mid \psi_j\rangle. The eigenvectors \psi_i are those of the quantum harmonic oscillator, and are sometimes called the "number basis".


Generalized creation and annihilation operators

The operators derived above are actually a specific instance of a more generalized notion of creation and annihilation operators. The more abstract form of the operators are constructed as follows. Let H be a one-particle Hilbert space (that is, any Hilbert space, viewed as representing the state of a single particle). The (
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
ic) CCR algebra over H is the algebra-with-conjugation-operator (called ''*'') abstractly generated by elements a(f), where f\,ranges freely over H, subject to the relations : (f),a(g) ^\dagger(f),a^\dagger(g)0 : (f),a^\dagger(g)\langle f\mid g \rangle, in bra–ket notation. The map a: f \to a(f) from H to the bosonic CCR algebra is required to be complex
antilinear In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y ...
(this adds more relations). Its
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
is a^\dagger(f), and the map f\to a^\dagger(f) is complex linear in . Thus H embeds as a complex vector subspace of its own CCR algebra. In a representation of this algebra, the element a(f) will be realized as an annihilation operator, and a^\dagger(f) as a creation operator. In general, the CCR algebra is infinite dimensional. If we take a Banach space completion, it becomes a C*-algebra. The CCR algebra over H is closely related to, but not identical to, a
Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More prec ...
. For fermions, the (fermionic)
CAR algebra A car or automobile is a motor vehicle with wheels. Most definitions of ''cars'' say that they run primarily on roads, seat one to eight people, have four wheels, and mainly transport people instead of goods. The year 1886 is regarded as t ...
over ''H'' is constructed similarly, but using
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
relations instead, namely :\=\=0 :\=\langle f\mid g \rangle. The CAR algebra is finite dimensional only if H is finite dimensional. If we take a Banach space completion (only necessary in the infinite dimensional case), it becomes a C^* algebra. The CAR algebra is closely related to, but not identical to, a Clifford algebra. Physically speaking, a(f) removes (i.e. annihilates) a particle in the state , f\rangle whereas a^\dagger(f) creates a particle in the state , f\rangle. The
free field In physics a free field is a field without interactions, which is described by the terms of motion and mass. Description In classical physics, a free field is a field whose equations of motion are given by linear partial differential equat ...
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
is the state with no particles, characterized by :a(f) \left, 0\right\rangle=0. If , f\rangle is normalized so that \langle f, f\rangle = 1, then N=a^\dagger(f)a(f) gives the number of particles in the state , f\rangle.


Creation and annihilation operators for reaction-diffusion equations

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: ''A+A\to \empty''. To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider n_ particles at a site on a one dimensional lattice. Each particle moves to the right or left with a certain probability, and each pair of particles at the same site annihilates each other with a certain other probability. The probability that one particle leaves the site during the short time period is proportional to n_i \, dt, let us say a probability \alpha n_dt to hop left and \alpha n_i \, dt to hop right. All n_i particles will stay put with a probability 1-2\alpha n_i \, dt. (Since is so short, the probability that two or more will leave during is very small and will be ignored.) We can now describe the occupation of particles on the lattice as a `ket' of the form , \dots, n_, n_0, n_1, \dots\rangle. It represents the juxtaposition (or conjunction, or tensor product) of the number states \dots, , n_\rangle , n_\rangle, , n_\rangle, \dots located at the individual sites of the lattice. Recall that :a\mid\! n\rangle= \sqrt \ , n-1\rangle and :a^\dagger \mid\! n\rangle= \sqrt\mid\! n+1\rangle, for all  ≥ 0, while : ,a^\mathbf 1 This definition of the operators will now be changed to accommodate the "non-quantum" nature of this problem and we shall use the following definition: :an\rangle = (n), n1\rangle :a^\daggern\rangle = , n1\rangle note that even though the behavior of the operators on the kets has been modified, these operators still obey the commutation relation : ,a^\mathbf 1 Now define '' a_i'' so that it applies '' a'' to , n_i\rangle. Correspondingly, define a^\dagger_i as applying a^\dagger to , n_i\rangle. Thus, for example, the net effect of a_ a^\dagger_i is to move a particle from the ''(i-1)^'' to the ''i^''site while multiplying with the appropriate factor. This allows writing the pure diffusive behavior of the particles as :\partial_\mid\! \psi\rangle=-\alpha\sum(2a_i^\dagger a_i-a_^\dagger a_i-a_^\dagger a_i)\mid\!\psi\rangle=-\alpha\sum(a_i^\dagger-a_^\dagger)(a_i-a_) \mid\! \psi\rangle, where the sum is over i. The reaction term can be deduced by noting that n particles can interact in n(n-1) different ways, so that the probability that a pair annihilates is \lambda n(n-1)dt, yielding a term :\lambda\sum(a_i a_i-a_i^\dagger a_i^\dagger a_i a_i) where number state ''n'' is replaced by number state ''n'' − 2 at site i at a certain rate. Thus the state evolves by :\partial_t\mid\!\psi\rangle=-\alpha\sum(a_i^\dagger-a_^\dagger)(a_i-a_) \mid\!\psi\rangle+\lambda\sum(a_i^2-a_i^a_i^2)\mid\!\psi\rangle Other kinds of interactions can be included in a similar manner. This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.


Creation and annihilation operators in quantum field theories

In quantum field theories and
many-body problem The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
s one works with creation and annihilation operators of quantum states, a^\dagger_i and a^_i. These operators change the eigenvalues of the number operator, : N = \sum_i n_i = \sum_i a^\dagger_i a^_i, by one, in analogy to the harmonic oscillator. The indices (such as i) represent
quantum numbers In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be k ...
that label the single-particle states of the system; hence, they are not necessarily single numbers. For example, a
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
of quantum numbers (n, l, m, s) is used to label states in the hydrogen atom. The commutation relations of creation and annihilation operators in a multiple-
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
system are, : ^_i, a^\dagger_j\equiv a^_i a^\dagger_j - a^\dagger_ja^_i = \delta_, : ^\dagger_i, a^\dagger_j= ^_i, a^_j= 0, where \ , \ \ /math> is the commutator and \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
. For fermions, the commutator is replaced by the
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
\, : \ \equiv a^_i a^\dagger_j +a^\dagger_j a^_i = \delta_, : \ = \ = 0. Therefore, exchanging disjoint (i.e. i \ne j) operators in a product of creation or annihilation operators will reverse the sign in fermion systems, but not in boson systems. If the states labelled by ''i'' are an orthonormal basis of a Hilbert space ''H'', then the result of this construction coincides with the CCR algebra and CAR algebra construction in the previous section but one. If they represent "eigenvectors" corresponding to the continuous spectrum of some operator, as for unbound particles in QFT, then the interpretation is more subtle.


Normalization

While Zee obtains the
momentum space In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
normalization hat a_,\hat a_^\dagger= \delta(\mathbf - \mathbf) via the symmetric convention for Fourier transforms, Tong and Peskin & Schroeder use the common asymmetric convention to obtain hat a_,\hat a_^\dagger= (2\pi)^3\delta(\mathbf - \mathbf). Each derives hat \phi(\mathbf x), \hat \pi(\mathbf x')= i\delta(\mathbf x - \mathbf x'). Srednicki additionally merges the Lorentz-invariant measure into his asymmetric Fourier measure, \tilde=\frac, yielding hat a_,\hat a_^\dagger= (2\pi)^3 2\omega\,\delta(\mathbf - \mathbf').


See also

*
Fock space The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intr ...
*
Segal–Bargmann space In mathematics, the Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions ''F'' in ''n'' complex variables satisfying the square-integr ...
*
Optical phase space In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an ''optical system''. For any such system, a plot o ...
* Bogoliubov–Valatin transformation * Holstein–Primakoff transformation *
Jordan–Wigner transformation The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators. It was proposed by Pascual Jordan and Eugene Wigner for one-dimensional lattice models, but now two-dimensional ana ...
* Jordan–Schwinger transformation * Klein transformation * Canonical commutation relations


References

* *
Albert Messiah Albert Messiah (23 September 1921, Nice – 17 April 2013, Paris) was a French physicist. He studied at the Ecole Polytechnique. He spent the Second World War in the Free France forces: he embarked on 22 June 1940 at Saint-Jean-de-Luz for Engla ...
, 1966. ''Quantum Mechanics'' (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Ch. XII
online


Footnotes

{{Physics operator Quantum mechanics Quantum field theory