Creation and annihilation operators

Creation and annihilation operators

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Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics
Quantum mechanics
Quantum 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 oscillators
Quantum harmonic oscillator
The 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 adjoint
Hermitian adjoint
In 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 physics
Physics
Physics 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 chemistry
Chemistry
Chemistry 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 wavefunction
Wavefunction
Not 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 chemistry
Quantum chemistry
Quantum 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 theory
Many-body theory
The 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 electron
Electron
The 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 oscillator
Quantum harmonic oscillator
The 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 mathematics
Mathematics
Mathematics 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 oscillator
Quantum harmonic oscillator
The 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 commutator
Commutator
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:...

 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 quanta
Quantum
In 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 boson
Boson
In 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 fermion
Fermion
In 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 wavefunction
Wavefunction
Not 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 properties
Identical particles
Identical 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 equation
Schrödinger equation
The 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 oscillator
Quantum harmonic oscillator
The 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 nondimensionalize
Nondimensionalization
Nondimensionalization 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 quanta
Quantum
In 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 Hamiltonian
Hamiltonian (quantum mechanics)
In 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 operator
Momentum operator
In 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 oscillator
Quantum harmonic oscillator
The 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 integral
Gaussian integral
The 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 space
Hilbert space
The 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 boson
Boson
In 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 adjoint
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 = .Specifically, adjoint may mean:...

 is a(f) which is linear
Linear
In 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 notation
Bra-ket notation
Bra-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 field
Free field
In 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 state
Vacuum state
In 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
    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
    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
    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
    Canonical commutation relation
    In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:[x,p_x] = i\hbar...

    s