Annihilation, creation, and ladder operators

These are some notes, mostly for my own benefit, on annihilation, creation, and ladder operators in quantum mechanics, with a few remarks towards the end on angular momentum, spin and Clebsch–Gordan coefficients.

First, the abstract definition: if T, L: V → V are linear operators on a vector space V over a field K, then L is said to be a ladder operator for T if there is a scalar c ∈ K such that the commutator of T and L satisfies

The operator L is called a raising operator for T if c is real and positive, and a lowering operator for T if c is real and negative.

The motivation behind this definition is that if (λ, v) ∈ K × V is an eigenpair for T (i.e. Tv = λv), then a quick calculation reveals that (λ + c, Lv) is an eigenpair for T:

Ladder operators come up in quantum mechanics because many of the elementary operations on quantum systems act as ladder operators and increase or decrease the eigenvalues of other operators. Those eigenvalues often encode important information about the system, and the increments and decrements provided by the ladder operators often come in discrete, rather than continuous, values. Annihilation and creation operators are a prime example of this phenomenon.