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An annihilation operator is used to remove a particle from the initial state and a creation operator is used to add a particle to the final state. The term "ladder operator" or "raising and lowering operators" is also sometimes used in mathematics, in the context of the theory of Lie algebras and in particular the affine Lie algebras.
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 ...
For this reason, a is called an annihilation operator ("lowering operator"), and a † a creation operator ("raising operator"). The two operators together are called ladder operators . Given any energy eigenstate, we can act on it with the lowering operator, a , to produce another eigenstate with ħω less energy.
In 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 raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator.
A ladder of quantized energy levels, called the Jaynes–Cummings ladder, ... are the bosonic creation and annihilation operators and is the ...
That is, the l-th creation operator creates a particle in the l-th state k l, and the vacuum state is a fixed point of annihilation operators as there are no particles to annihilate. We can generate any Fock state by operating on the vacuum state with an appropriate number of creation operators :
The terms antinormal order and antinormal ordering are analogously defined, where the annihilation operators are placed to the left of the creation operators. Normal ordering of a product of quantum fields or creation and annihilation operators can also be defined in many other ways .
Using the notation for multi-photon states, Glauber characterized the state of complete coherence to all orders in the electromagnetic field to be the eigenstate of the annihilation operator—formally, in a mathematical sense, the same state as found by Schrödinger. The name coherent state took hold after Glauber's work.