**Creation and annihilation operators** are mathematical operators that have widespread applications in

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, notably in the study of

quantum harmonic oscillatorsThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

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

adjointIn mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator.Adjoints of operators generalize conjugate transposes of square matrices to infinite-dimensional situations...

of the annihilation operator. In many subfields of

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and

chemistryChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

, the use of these operators instead of

wavefunctionNot to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

s is known as second quantization.

Creation and annihilation operators can act on states of various types of particles. For example, in

quantum chemistryQuantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...

and

many-body theoryThe many-body theory is an area of physics which provides the framework for understanding the collective behavior of vast assemblies of interacting particles. In general terms, the many-body theory deals with effects that manifest themselves only in systems containing large numbers of constituents...

the creation and annihilation operators often act on

electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

states.

They can also refer specifically to the ladder operators for the

quantum harmonic oscillatorThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

. 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

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

for the creation and annihilation operators for bosons is the same as for the ladder operators of the

quantum harmonic oscillatorThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

. For example, the

commutatorIn 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:...

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.

## Derivation for 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

quantaIn physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...

of energy to the oscillator system. Creation/annihilation operators are different for

bosonIn particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s (integer spin) and

fermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s (half-integer spin). This is because their

wavefunctionNot to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

s have different

symmetry propertiesIdentical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...

.

For now let's just consider the case of the phonons of the quantum harmonic oscillator, which are bosons, because fermions are more complicated.

Start with the

Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

for the one dimensional time independent

quantum harmonic oscillatorThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

Make a coordinate substitution to

nondimensionalizeNondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis...

the differential equation

.

and the Schrödinger equation for the oscillator becomes

.

Notice that the quantity

is the same energy as that found for light

quantaIn physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...

and that the parenthesis in the

HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

can be written as

The last two terms can be simplified by considering their effect on an arbitrary differentiable function f(q),

which implies,

Therefore

and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,

.

If we define

as the "creation operator" or the "raising operator" and

as the "annihilation operator" or the "lowering operator"

then the Schrödinger equation for the oscillator becomes

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

, where "p" is the nondimensionalized

momentum operatorIn quantum mechanics, momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once...

then we have

and

.

Note that these imply that

### Applications

The ground state

of the

quantum harmonic oscillatorThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

can be found by imposing the condition that

.

Written out as a differential equation, the wavefunction satisfies

which has the solution

The normalization constant

*C* can be found to be

from

, using the

Gaussian integralThe Gaussian integral, also known as the Euler-Poisson integral or Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.It is named after the German mathematician and...

.

## Matrix representation

The matrix counterparts of the creation and annihilation operators obtained from the quantum harmonic oscillator model are

Substituting backwards, the laddering operators are recovered. They can be obtained via the relationships

and

. The wavefunctions are those of the quantum harmonic oscillator, and are sometimes called the "number basis".

## Mathematical details

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.

Let

*H* be the one-particle

Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

. To get the

bosonIn particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

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

adjointIn 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 = .Specifically, adjoint may mean:...

is

*a*^{†}(

*f*) which is

linearIn mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

in

*H*.

For a boson,

,

where we are using

bra-ket notationBra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

.

For a fermion, the anticommutators are

.

A CAR algebra.

Physically speaking,

*a*(

*f*) removes (i.e. annihilates) a particle in the state whereas

*a*^{†}(

*f*) creates a particle in the state .

The

free fieldIn classical physics, a free field is a field whose equations of motion are given by linear partial differential equations. Such linear PDE's have a unique solution for a given initial condition....

vacuum stateIn quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...

is the state with no particles. In other words,

where is the vacuum state.

If is normalized so that = 1, then

*a*^{†}(

*f*)

*a*(

*f*) gives the number of particles in the state .

## 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: To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider

particles at a site

on a 1-d lattice. Each particle diffuses independently, so that the probability that one of them leaves the site for short times

is proportional to

, say

to hop left and

to hop right. All

particles will stay put with a probability

.

We can now describe the occupation of particles on the lattice as a `ket' of the form . A slight modification of the annihilation and creation operators is needed so that

and

.

This modification preserves the commutation relation

,

but allows us to write the pure diffusive behaviour of the particles as

The reaction term can be deduced by noting that

particles can interact in

different ways, so that the probability that a pair annihilates is

and the probability that no pair annihilates is

leaving us with a term

yielding

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.

## See also

- Bogoliubov transformation
In theoretical physics, the Bogoliubov transformation, named after Nikolay Bogolyubov, is a unitary transformation from a unitary representation of some canonical commutation relation algebra or canonical anticommutation relation algebra into another unitary representation, induced by an...

s - arises in the theory of quantum optics.
- 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 of the quadratures against each other, possibly as...

- Fock space
The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...

- Canonical commutation relation
In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:[x,p_x] = i\hbar...

s