In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
(and its application to
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
), a raising or lowering operator (collectively known as ladder operators) is an
operator that increases or decreases the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of another operator. In quantum mechanics, the raising operator is sometimes called the
creation operator
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...
, and the lowering operator the
annihilation operator
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...
. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the
and
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
.
Terminology
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
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. 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
In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles.
The number operator acts on Fock space. Let
:, \Psi\rangle_\nu=, \phi_1,\phi_2 ...
).
Confusion arises because the term ''ladder operator'' is typically used to describe an operator that acts to increment or decrement a
quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
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 algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s and in particular the
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
s, 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).
General formulation
Suppose that two operators ''X'' and ''N'' have the
commutation relation
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
,
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.
If ''N'' is a
Hermitian operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
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
vice versa.
Angular momentum
A particular application of the ladder operator concept is found in the
quantum mechanical treatment of
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. For a general angular momentum
vector, J, with components, ''J
x'', ''J
y'' and ''J
z'' one defines the two ladder operators, ''J
+'' and ''J
–'',
where ''i'' is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
.
The
commutation relation
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
between the
cartesian Cartesian means of or relating to the French philosopher René Descartes—from his Latinized name ''Cartesius''. It may refer to:
Mathematics
*Cartesian closed category, a closed category in category theory
*Cartesian coordinate system, modern ...
components of ''any'' angular momentum operator is given by
where ''ε
ijk'' 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 ''J
z'' 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 ''J
z'' operator on a given state,
Compare this result with
Thus one concludes that
is some
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
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:
Confirming that ''m'' is bounded by the value of ''j'' (
), one has
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
In pure mathematics, pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis (linear algebra), basis used to express spherical tensors. The spherical basis closel ...
. Using the notation of
spherical tensor operator
In pure mathematics, pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operator (mathematics), operators which are scalar (mathematics), scalars and vector (mathematics and physics) ...
s, the "-1", "0" and "+1" components of J
(1) ≡ J are given by,
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''.
Harmonic oscillator
Another application of the ladder operator concept is found in the quantum mechanical 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.
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)
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:
and
.
Therefore,
and
so
where the "?" indicates a nascent quantum number which emerges from the discussion.
Given the Pauli equations
Pauli Equation IV:
and Pauli Equation III:
and starting with the equation
and expanding,
one obtains (assuming
is the maximum value of the angular momentum quantum number consonant with all other conditions),
which leads to the
Rydberg formula:
implying that
, where
is the traditional quantum number.
Factorization of the Hamiltonian
The Hamiltonian for a hydrogen-like potential can be written in spherical coordinates as
where
and
is the radial momentum
which is real and self-conjugate.
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 for differential equations.
Newmarch and Golding
[
]
applied it to spherically symmetric potentials using operator notation.
Suppose we can find a factorization of the Hamiltonian by operators
as
and
for scalars
and
. The vector
may be evaluated in two different ways as
which can be re-arranged as
showing that
is an eigenstate of
with eigenvalue
If
then
and the states
and
have the same energy.
For the hydrogenic atom, setting
with
a suitable equation for
is
with
There is an upper bound to the ladder operator if the energy is negative, (so
for some
) then
from Equation ()
and
can be identified with
Relation to group theory
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)
In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4.
In this article ''rotation'' means ''rotational dis ...
symmetry of the spherically symmetric Coulomb potential.
3D isotropic harmonic oscillator
The
3D isotropic harmonic oscillator has a potential given by
It can similarly be managed using the factorization method.
Factorization method
A suitable factorization is given by
with
and
Then
and continuing this,
Now the Hamiltonian only has positive energy levels as can be seen from
This means that for some value of
the series must terminate with
and then
This is decreasing in energy by
unless for some value of
,
. Identifying this value as
gives
It then follows the
so that
giving a recursion relation on
with solution
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
Relation to group theory
The degeneracies of the 3D isotropic harmonic oscillator are related to the special unitary group
SU(3)[
, D. M. "." 33 (3) (1965) 207–211.
]
History
Many sources credit
Dirac
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety of ...
with the invention of ladder operators.
[https://www.fisica.net/mecanica-quantica/quantum_harmonic_oscillator_lecture.pdf ] 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 .
See also
*
Creation and annihilation operators
*
*
Chevalley basis In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite ...
References
{{Physics operator
Quantum mechanics
de:Erzeugungs- und Vernichtungsoperator