In linear algebra (and its application to 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. Wellknown applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.
Main article: Creation and annihilation operators 
There is a relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field theory which lies in representation theory. The creation operator a_{i}^{†} increments the number of particles in state i, while the corresponding annihilation operator a_{i} decrements the number of particles in state i. This clearly satisfies the requirements of the above definition of a ladder operator: the incrementing or decrementing of the eigenvalue of another operator (in this case the particle number operator).
Confusion arises because the term ladder operator is typically used to describe an operator that acts to increment or decrement a quantum number describing the state of a system. To change the state of a particle with the creation/annihilation operators of QFT requires the use of both annihilation and creation operators. 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. 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 particular, the highest weight is annihilated by the raising operators; the rest of the positive root space is obtained by repeatedly applying the lowering operators (one set of ladder operators per subalgebra).
From a representation theory standpoint a linear representation of a semisimple Lie group in continuous real parameters induces a set of generators for the Lie algebra. A complex linear combination of those are the ladder operators.^{[clarification needed]} For each parameter there is a set of ladder operators; these are then a standardized way to navigate one dimension of the root system and root lattice.^{[2]} 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.
Suppose that two operators X and N have the commutation relation
In other words, if is an eigenstate of N with eigenvalue n, then is an eigenstate of N with eigenvalue n + c or is zero. The operator X is a raising operator for N if c is real and positive, and a lowering operator for N if c is real and negative.
If N is a Hermitian operator, then c must be real, and the Hermitian adjoint of X obeys the commutation relation
In particular, if X is a lowering operator for N, then X^{†} is a raising operator for N and conversely.^{[dubious – discuss]}
Main article: Angular momentum operator 
A particular application of the ladder operator concept is found in the quantummechanical treatment of angular momentum. For a general angular momentum vector J with components J_{x}, J_{y} and J_{z} one defines the two ladder operators^{[3]}
The commutation relation between the cartesian components of any angular momentum operator is given by
From this, the commutation relations among the ladder operators and J_{z} are obtained:
The properties of the ladder operators can be determined by observing how they modify the action of the J_{z} operator on a given state:
Compare this result with
Thus, one concludes that is some scalar multiplied by :
This illustrates the defining feature of ladder operators in quantum mechanics: the incrementing (or decrementing) of a quantum number, thus mapping one quantum state onto another. This is the reason that they are often known as raising and lowering operators.
To obtain the values of α and β, first take the norm of each operator, recognizing that J_{+} and J_{−} are a Hermitian conjugate pair ():
The product of the ladder operators can be expressed in terms of the commuting pair J^{2} and J_{z}:
Thus, one may express the values of α^{2} and β^{2} in terms of the eigenvalues of J^{2} and J_{z}:
The phases of α and β are not physically significant, thus they can be chosen to be positive and real (Condon–Shortley phase convention). We then have^{[4]}
Confirming that m is bounded by the value of j (), one has
The above demonstration is effectively the construction of the Clebsch–Gordan coefficients.
Many terms in the Hamiltonians of atomic or molecular systems involve the scalar product of angular momentum operators. An example is the magnetic dipole term in the hyperfine Hamiltonian:^{[5]}
The angular momentum algebra can often be simplified by recasting it in the spherical basis. Using the notation of spherical tensor operators, the "−1", "0" and "+1" components of J^{(1)} ≡ J are given by^{[6]}
From these definitions, it can be shown that the above scalar product can be expanded as
The significance of this expansion is that it clearly indicates which states are coupled by this term in the Hamiltonian, that is those with quantum numbers differing by m_{i} = ±1 and m_{j} = ∓1 only.
Main article: Quantum harmonic oscillator 
Another application of the ladder operator concept is found in the quantummechanical treatment of the harmonic oscillator. We can define the lowering and raising operators as
They provide a convenient means to extract energy eigenvalues without directly solving the system's differential equation.
Main article: Hydrogenlike atom 
There are two main approaches given in the literature using ladder operators, one using the Laplace–Runge–Lenz vector, another using factorization of the Hamiltonian.
Another application of the ladder operator concept is found in the quantum mechanical treatment of the electronic energy of hydrogenlike atoms and ions. The Laplace–Runge–Lenz vector commutes with the Hamiltonian for an inverse square spherically symmetric potential and can be used to determine ladder operators for this potential.^{[7]}^{[8]} We can define the lowering and raising operators (based on the classical Laplace–Runge–Lenz vector)
The commutators needed to proceed are
Given the Pauli equations^{[9]}^{[10]} IV:
The Hamiltonian for a hydrogenlike potential can be written in spherical coordinates as
Suppose is an eigenvector of the Hamiltonian, where is the angular momentum, and represents the energy, so , and we may label the Hamiltonian as :
The factorization method was developed by Infeld and Hull^{[11]} for differential equations. Newmarch and Golding^{[12]} applied it to spherically symmetric potentials using operator notation.
Suppose we can find a factorization of the Hamiltonian by operators as

(1) 
and
For the hydrogenic atom, setting
Whenever there is degeneracy in a system, there is usually a related symmetry property and group. The degeneracy of the energy levels for the same value of but different angular momenta has been identified as the SO(4) symmetry of the spherically symmetric Coulomb potential.^{[13]}^{[14]}
The 3D isotropic harmonic oscillator has a potential given by
It can similarly be managed using the factorization method.
A suitable factorization is given by^{[12]}
It then follows the so that
There is degeneracy caused from angular momentum; there is additional degeneracy caused by the oscillator potential. Consider the states and apply the lowering operators : giving the sequence with the same energy but with decreasing by 2. In addition to the angular momentum degeneracy, this gives a total degeneracy of ^{[15]}
The degeneracies of the 3D isotropic harmonic oscillator are related to the special unitary group SU(3)^{[15]}^{[16]}
Many sources credit Paul Dirac with the invention of ladder operators.^{[17]} Dirac's use of the ladder operators shows that the total angular momentum quantum number needs to be a nonnegative halfinteger multiple of ħ.