Wikipedia / Ladder operator
Wikipedia / Ladder operator
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 and lowering operators are commonly known as the creation and annihilation operators, respectively. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.
Terminology
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 ai† increments the number of particles in state i, while the corresponding annihilation operator ai 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. 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).
Motivation from mathematics
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. A complex linear combination of those are the ladder operators. 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. 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.
General formulation
Suppose that two operators X and N have the commutation relation for some scalar c is an eigenstate of N then the operator X in such a way as to shift the eigenvalue by c
is an eigenstate of N with eigenvalue n 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
In particular, if X is a lowering operator for N, then X† is a raising operator for N and conversely.
Angular momentum
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 where i is the imaginary unit.
The commutation relation between the cartesian components of any where εijk is the Levi-Civita symbol, and each of i, j and k can take any of the values x, y and z.
From this, the commutation relations among the ladder operators and Jz).
The properties of the ladder operators can be determined by observing how they modify the action of the Jz
is some scalar
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
The product of the ladder operators can be expressed in terms of the commuting pair J2 and Jz
Thus, one may express the values of |α|2 and |β|2 in terms of the eigenvalues of J2 and Jz
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
Confirming that m is bounded by the value of j
The above demonstration is effectively the construction of the Clebsch–Gordan coefficients.
Applications in atomic and molecular physics
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: where I is the nuclear spin.
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
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.
Harmonic oscillator
They provide a convenient means to extract energy eigenvalues without directly solving the system's differential equation.
Hydrogen-like 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.
Laplace–Runge–Lenz vector
Another application of the ladder operator concept is found in the quantum mechanical treatment of the electronic energy of hydrogen-like 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. We can define the lowering and raising operators (based on the classical Laplace–Runge–Lenz vector is the reduced mass is the atomic number.
where the "?" indicates a nascent quantum number which emerges from the discussion.
Given the Pauli equations which leads to the Rydberg formula is the traditional quantum number.
Factorization of the Hamiltonian
which is real and self-conjugate.
The factorization method was developed by Infeld and Hull for differential equations. Newmarch and Golding applied it to spherically symmetric potentials using operator notation.
as have the same energy.
), then if follows from equation (
Relation to group theory
but different angular momenta has been identified as the SO(4) symmetry of the spherically symmetric Coulomb potential.
3D isotropic harmonic oscillator
The 3D isotropic harmonic oscillator
It can similarly be managed using the factorization method.
Factorization method
A suitable factorization is given by
Relation to group theory
The degeneracies of the 3D isotropic harmonic oscillator are related to the special unitary group SU(3)
History
Many sources credit Paul Dirac with the invention of ladder operators. Dirac's use of the ladder operators shows that the total angular momentum quantum number needs to be a non-negative half-integer multiple of .