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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. For example to describe the su(2) subalgebras, the root system and the highest weight modules can be constructed by means of the ladder operators. [1]
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.
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. Well-known applications of ladder ...
That is, the resulting spin operators for higher-spin systems in three spatial dimensions can be calculated for arbitrarily large s using this spin operator and ladder operators. For example, taking the Kronecker product of two spin- 1 / 2 yields a four-dimensional representation, which is separable into a 3-dimensional spin-1 ( triplet ...
Each additional boson then corresponds to a decrease of ħ in the spin projection. Thus, the spin raising and lowering operators + = + and =, so that [+,] =, correspond (in the sense detailed below) to the bosonic annihilation and creation operators, respectively. The precise relations between the operators must be chosen to ensure the correct ...
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 raising and lowering operators can be used to alter the value of m, ... addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s, ...
(The operator ð is effectively a covariant derivative operator in the sphere.) An important property of the new function ðη is that if η had spin weight s, ðη has spin weight s + 1. Thus, the operator raises the spin weight of a function by 1. Similarly, we can define an operator ð which will lower the spin weight of a function by 1: