) 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.
[2] 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.
Constructing Hamiltonians using these operators has the advantage that the theory automatically satisfies the cluster decomposition theorem.
[5] 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.
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,
is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as
The last two terms can be simplified by considering their effect on an arbitrary differentiable function
These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator as follows.
The ground state can be found by assuming that the lowering operator possesses a nontrivial kernel:
of the quantum harmonic oscillator can be found by imposing the condition that
Explicit formulas for all the eigenfunctions can now be found by repeated application of
[7] The matrix expression of the creation and annihilation operators of the quantum harmonic oscillator with respect to the above orthonormal basis is
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.
[8] In the Hilbert space representation case the operators are constructed as follows: Let
to the bosonic CCR algebra is required to be complex antilinear (this adds more relations).
embeds as a complex vector subspace of its own CCR algebra.
If we take a Banach space completion (only necessary in the infinite dimensional case), it becomes a
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
To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider
The probability that one particle leaves the site during the short time period dt is proportional to
It represents the juxtaposition (or conjunction, or tensor product) of the number states
This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.
The commutation relations of creation and annihilation operators in a multiple-boson system are,
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.