Creation operators 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.) 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. They were introduced by Paul Dirac.

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 creation operator is interpreted as a raising operator, adding a quantum of energy to the oscillator system (similarly for the lowering operator). They can be used to represent phonons. Constructing Hamiltonians using these operators has the advantage that the theory automatically satisfies the cluster decomposition theorem.

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, 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 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 photons of the quantum harmonic oscillator. Start with the Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator

Make a coordinate substitution to nondimensionalize

is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian

The last two terms can be simplified by considering their effect on an arbitrary differentiable function

.

as the "creation operator" or the "raising operator" as the "annihilation operator" or the "lowering operator" This is significantly simpler than the original form. Further simplifications of this equation enable one to derive all the properties listed above thus far.

is the nondimensionalized momentum operator, one has

may be contrasted to normal operators, which commute with their adjoints.

operators and the Hamiltonian:

These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator as follows.

. Using these commutation relations, it follows that

.

. Applying the Hamiltonian to the ground state,

is an eigenfunction of the Hamiltonian.

as

Furthermore, it turns out that the first-mentioned operator in (*), the number operator.

The time-evolution operator

Explicit eigenfunctions

of the quantum harmonic oscillator

Written out as a differential equation

The normalization constant , using the Gaussian integral.

Matrix representation

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

Generalized creation and annihilation operators

Thanks to representation theory and C*-algebras the operators derived above are actually a specific instance of a more generalized notion of creation and annihilation operators in the context of CCR and CAR algebras. Mathematically and even more generally ladder operators can be understood in the context of a root system of a semisimple Lie group and the associated semisimple Lie algebra without the need of realizing the representation as operators on a functional Hilbert space.

In the Hilbert space be a one-particle Hilbert space (that is, any Hilbert space, viewed as representing the state of a single particle).

Bosonic CCR algebra

The (bosonic) CCR algebra is the algebra-with-conjugation-operator (called *, subject to the relations

in bra–ket notation.

to the bosonic CCR algebra is required to be complex antilinear (this adds more relations). Its adjoint is complex linear in as a creation operator.

In general, the CCR algebra is infinite dimensional. If we take a Banach space completion, it becomes a C*-algebra is closely related to, but not identical to, a Weyl algebra.

Fermionic CAR algebra

For fermions, the (fermionic) CAR algebra is constructed similarly, but using anticommutator relations instead, namely

algebra. The CAR algebra is closely related, but not identical to, a Clifford algebra.

.

The free field vacuum state

.

In reaction-diffusion equations

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 . (Since is so short, the probability that two or more will leave during is very small and will be ignored.)

located at the individual sites of the lattice. Recall that

for all

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:

is to move a particle from the to the -th site while multiplying with the appropriate factor.

where number state is replaced by number state at a certain rate.

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.

In quantum field theories

In quantum field theories and many-body problems. These operators change the eigenvalues of the number operator) represent quantum numbers that label the single-particle states of the system; hence, they are not necessarily single numbers. For example, a tuple is used to label states in the hydrogen atom.

The commutation relations of creation and annihilation operators in a multiple-boson is the commutator is the Kronecker delta.

For fermions, the commutator is replaced by the anticommutator ) 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 conventions

While Zee obtains the momentum space via the symmetric convention for Fourier transforms, Tong and Peskin & Schroeder.

.

See also

• Fock space

• Segal–Bargmann space

• Optical phase space

• Coherent state

• Bogoliubov–Valatin transformation

• Holstein–Primakoff transformation

• Jordan–Wigner transformation

• Jordan–Schwinger transformation

• Klein transformation

• Canonical commutation relations

Notes

References

• Albert Messiah, 1966. Quantum Mechanics (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Ch. XII. online