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The ladder operators of the quantum harmonic oscillator or the "number representation" of second quantization are just special cases of this fact. Ladder operators then become ubiquitous in quantum mechanics from the angular momentum operator, to coherent states and to discrete magnetic translation operators.
Another type of operator in quantum field theory, discovered in the early 1970s, is known as the anti-symmetric operator.This operator, similar to spin in non-relativistic quantum mechanics is a ladder operator that can create two fermions of opposite spin out of a boson or a boson from two fermions.
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.
The mathematics for the creation and annihilation operators for bosons is the same as for the ladder operators of the quantum harmonic oscillator. [4] 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.
The mathematical formulation of quantum mechanics (QM) is built upon the concept of an operator. Physical pure states in quantum mechanics are represented as unit-norm vectors (probabilities are normalized to one) in a special complex Hilbert space. Time evolution in this vector space is given by the application of the evolution operator.
In quantum mechanics, the quantum analog G is now a Hermitian matrix, and the equations of motion are given by commutators, = [,]. The infinitesimal canonical motions can be formally integrated, just as the Heisenberg equation of motion were integrated, A ′ = U † A U {\displaystyle A'=U^{\dagger }AU} where U = e iGs and s is an arbitrary ...
In the quantum mechanics study of optical phase space, the displacement operator for one mode is the shift operator in quantum optics, ^ = (^ † ^), where is the amount of displacement in optical phase space, is the complex conjugate of that displacement, and ^ and ^ † are the lowering and raising operators, respectively.
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry .