There is some confusion regarding the relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field 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 an annihilation operator to remove a particle from the initial state and a creation operator to add a particle to the final state.
The term "ladder operator" is also sometimes used in mathematics, in the context of the theory of Lie algebras and in particular the affine Lie algebras, to describe the su(2) subalgebras, from which 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).
for some scalar c. If is an eigenstate of N with eigenvalue equation,
then the operator X acts on in such a way as to shift the eigenvalue by c:
In other words, if is an eigenstate of N with eigenvalue n then is an eigenstate of N with eigenvalue n + c or it 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.
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, J+ and J–,
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 are obtained,
(Technically, this is the Lie algebra of ).
The properties of the ladder operators can be determined by observing how they modify the action of the Jz 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 J2 and Jz,
Thus, one may express the values of |α|2 and |β|2 in terms of the eigenvalues of J2 and Jz,
Another application of the ladder operator concept is found in the quantum mechanical treatment of the electronic energy of hydrogen-like atoms and ions
We can define the lowering and raising operators (based on the classical Laplace–Runge–Lenz vector)
where is the angular momentum, is the linear momentum, is the reduced mass of the system, is the electronic charge, and is the atomic number of the nucleus.
Analogous to the angular momentum ladder operators, one has and .
The commutators needed to proceed are:
where the "?" indicates a nascent quantum number which emerges from the discussion.
author=David, C. W.,
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Burkhardt, C. E., and Levanthal, J.,
"Lenz vector operations on spherical hydrogen atom eigenfunctions",
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Wolfgang Pauli,"Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik",Z. Physik,
36, 336 (1926);
B. L. Van der Waerden,
Sources of Quantum Mechanics, Dover, New York,1968