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.
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.
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.
From a representation theory standpoint a linear representation of a semi-simple Lie group in continuous real parameters induces a set of generators for the Lie algebra.
[2] The ladder operators of the quantum harmonic oscillator or the "number representation" of second quantization are just special cases of this fact.
[dubious – discuss] A particular application of the ladder operator concept is found in the quantum-mechanical treatment of angular momentum.
For a general angular momentum vector J with components Jx, Jy and Jz 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 Jz are obtained:
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 J2 and Jz:
Many terms in the Hamiltonians of atomic or molecular systems involve the scalar product of angular momentum operators.
The angular momentum algebra can often be simplified by recasting it in the spherical basis.
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 mi = ±1 and mj = ∓1 only.
Another application of the ladder operator concept is found in the quantum-mechanical treatment of the harmonic oscillator.
They provide a convenient means to extract energy eigenvalues without directly solving the system's differential equation.
Ladder operator applying to harmonic oscillator's energy levels:
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 hydrogen-like atoms and ions.
[7][8] We can define the lowering and raising operators (based on the classical Laplace–Runge–Lenz vector)
is the maximum value of the angular momentum quantum number consonant with all other conditions)
The Hamiltonian for a hydrogen-like potential can be written in spherical coordinates 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.
There is an upper bound to the ladder operator if the energy is negative (so
Whenever there is degeneracy in a system, there is usually a related symmetry property and group.
but different angular momenta has been identified as the SO(4) symmetry of the spherically symmetric Coulomb potential.
[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