Angular momentum operator

In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Such an operator is applied to a mathematical representation of the physical state of a system and yields an angular momentum value if the state has a definite value for it. In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion. 
There are several angular momentum operators: total angular momentum (usually denoted J), orbital angular momentum (usually denoted L), and spin angular momentum (spin for short, usually denoted S). The term "angular momentum operator" can (confusingly) refer to either the total or the orbital angular momentum. Total angular momentum is always conserved, see Noether's theorem.
Overview.
In quantum mechanics, angular momentum can refer to one of three different, but related things.
Orbital angular momentum.
The classical definition of angular momentum is formula_1. The quantum-mechanical counterparts of these objects share the same relationship:
where r is the quantum position operator, p is the quantum momentum operator, × is cross product, and L is the "orbital angular momentum operator". L (just like p and r) is a "vector operator" (a vector whose components are operators), i.e. formula_3 where "L"x, "L"y, "L"z are three different quantum-mechanical operators.
In the special case of a single particle with no electric charge and no spin, the orbital angular momentum operator can be written in the position basis as:
where ∇ is the vector differential operator, del.
Spin angular momentum.
There is another type of angular momentum, called "spin angular momentum" (more often shortened to "spin"), represented by the spin operator formula_5. Spin is often depicted as a particle literally spinning around an axis, but this is only a metaphor: spin is an intrinsic property of a particle, unrelated to any sort of (yet experimentally observable) motion in space. All elementary particles have a characteristic spin, which is usually nonzero. For example, electrons always have "spin 1/2" while photons always have "spin 1" (details below).
Total angular momentum.
Finally, there is total angular momentum formula_6, which combines both the spin and orbital angular momentum of a particle or system:
Conservation of angular momentum states that J for a closed system, or J for the whole universe, is conserved. However, L and S are "not" generally conserved. For example, the spin–orbit interaction allows angular momentum to transfer back and forth between L and S, with the total J remaining constant.
Commutation relations.
Commutation relations between components.
The orbital angular momentum operator is a vector operator, meaning it can be written in terms of its vector components formula_3. The components have the following commutation relations with each other:
where [ , ] denotes the commutator
This can be written generally as
where "l", "m", "n" are the component indices (1 for "x", 2 for "y", 3 for "z"), and "εlmn" denotes the Levi-Civita symbol.
A compact expression as one vector equation is also possible:
The commutation relations can be proved as a direct consequence of the canonical commutation relations formula_13, where "δlm" is the Kronecker delta.
There is an analogous relationship in classical physics:
where "L""n" is a component of the "classical" angular momentum operator, and formula_15 is the Poisson bracket.
The same commutation relations apply for the other angular momentum operators (spin and total angular momentum):
These can be "assumed" to hold in analogy with L. Alternatively, they can be "derived" as discussed below.
These commutation relations mean that L has the mathematical structure of a Lie algebra, and the "εlmn" are its structure constants. In this case, the Lie algebra is SU(2) or SO(3) in physics notation (formula_17 or formula_18 respectively in mathematics notation), i.e. Lie algebra associated with rotations in three dimensions. The same is true of J and S. The reason is discussed below. These commutation relations are relevant for measurement and uncertainty, as discussed further below.
In molecules the total angular momentum F is the sum of the 
rovibronic (orbital) angular momentum N, the electron spin angular 
momentum S, and the nuclear spin angular momentum I. For electronic 
singlet states the rovibronic angular momentum is denoted J rather
than N. As explained by Van Vleck, 
the components of the molecular rovibronic angular momentum referred to 
molecule-fixed axes have different commutation relations from those
given above which are for the components about space-fixed axes.
Commutation relations involving vector magnitude.
Like any vector, the square of a magnitude can be defined for the orbital angular momentum operator,
formula_20 is another quantum operator. It commutes with the components of formula_21,
One way to prove that these operators commute is to start from the ["L"ℓ, "L""m"] commutation relations in the previous section:
Mathematically, formula_20 is a Casimir invariant of the Lie algebra SO(3) spanned by formula_21.
As above, there is an analogous relationship in classical physics:
where formula_26 is a component of the "classical" angular momentum operator, and formula_15 is the Poisson bracket.
Returning to the quantum case, the same commutation relations apply to the other angular momentum operators (spin and total angular momentum), as well,
Uncertainty principle.
In general, in quantum mechanics, when two observable operators do not commute, they are called complementary observables. Two complementary observables cannot be measured simultaneously; instead they satisfy an uncertainty principle. The more accurately one observable is known, the less accurately the other one can be known. Just as there is an uncertainty principle relating position and momentum, there are uncertainty principles for angular momentum.
The Robertson–Schrödinger relation gives the following uncertainty principle:
where formula_30 is the standard deviation in the measured values of "X" and formula_31 denotes the expectation value of "X". This inequality is also true if "x, y, z" are rearranged, or if "L" is replaced by "J" or "S".
Therefore, two orthogonal components of angular momentum (for example Lx and Ly) are complementary and cannot be simultaneously known or measured, except in special cases such as formula_32.
It is, however, possible to simultaneously measure or specify "L"2 and any one component of "L"; for example, "L"2 and "L"z. This is often useful, and the values are characterized by the azimuthal quantum number ("l") and the magnetic quantum number ("m"). In this case the quantum state of the system is a simultaneous eigenstate of the operators "L"2 and "L"z, but "not" of "L"x or "L"y. The eigenvalues are related to "l" and "m", as shown in the table below.
Quantization.
In quantum mechanics, angular momentum is "quantized" – that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where formula_33 is reduced Planck constant:
Derivation using ladder operators.
A common way to derive the quantization rules above is the method of "ladder operators". The ladder operators for the total angular momentum formula_6 are defined as:
Suppose formula_36 is a simultaneous eigenstate of formula_37 and formula_38 (i.e., a state with a definite value for formula_37 and a definite value for formula_38). Then using the commutation relations for the components of formula_41, one can prove that each of the states formula_42 and formula_43 is either zero or a simultaneous eigenstate of formula_37 and formula_38, with the same value as formula_36 for formula_37 but with values for formula_38 that are increased or decreased by formula_33 respectively. The result is zero when the use of a ladder operator would otherwise result in a state with a value for formula_38 that is outside the allowable range. Using the ladder operators in this way, the possible values and quantum numbers for formula_37 and formula_38 can be found.
! Derivation of the possible values and quantum numbers for formula_53 and formula_54. Click [show] on the right
From formula_60 is obtained,
Applying both sides of the above equation to formula_55,
Since formula_64 and formula_65 are real observables, formula_66 is not negative and formula_67
where exp is matrix exponential.
In simpler terms, the total angular momentum operator characterizes how a quantum system is changed when it is rotated. The relationship between angular momentum operators and rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics, as discussed further below.
Just as J is the generator for rotation operators, L and S are generators for modified partial rotation operators. The operator
rotates the position (in space) of all particles and fields, without rotating the internal (spin) state of any particle. Likewise, the operator
rotates the internal (spin) state of all particles, without moving any particles or fields in space. The relation J = L + S comes from:
i.e. if the positions are rotated, and then the internal states are rotated, then altogether the complete system has been rotated.
SU(2), SO(3), and 360° rotations.
Although one might expect formula_71 (a rotation of 360° is the identity operator), this is "not" assumed in quantum mechanics, and it turns out it is often not true: When the total angular momentum quantum number is a half-integer (1/2, 3/2, etc.), formula_72, and when it is an integer, formula_73. Mathematically, the structure of rotations in the universe is "not" SO(3), the group of three-dimensional rotations in classical mechanics. Instead, it is SU(2), which is identical to SO(3) for small rotations, but where a 360° rotation is mathematically distinguished from a rotation of 0°. (A rotation of 720° is, however, the same as a rotation of 0°.)
On the other hand, formula_74 in all circumstances, because a 360° rotation of a "spatial" configuration is the same as no rotation at all. (This is different from a 360° rotation of the "internal" (spin) state of the particle, which might or might not be the same as no rotation at all.) In other words, the formula_75 operators carry the structure of SO(3), while formula_76 and formula_77 carry the structure of SU(2).
From the equation formula_78, one picks an eigenstate formula_79 and draws
which is to say that the orbital angular momentum quantum numbers can only be integers, not half-integers.
Connection to representation theory.
Starting with a certain quantum state formula_81, consider the set of states formula_82 for all possible formula_83 and formula_84, i.e. the set of states that come about from rotating the starting state in every possible way. The linear span of that set is a vector space, and therefore the manner in which the rotation operators map one state onto another is a "representation" of the group of rotation operators.
From the relation between J and rotation operators,
The ladder operator derivation above is a method for classifying the representations of the Lie algebra SU(2).
Connection to commutation relations.
Classical rotations do not commute with each other: For example, rotating 1° about the "x"-axis then 1° about the "y"-axis gives a slightly different overall rotation than rotating 1° about the "y"-axis then 1° about the "x"-axis. By carefully analyzing this noncommutativity, the commutation relations of the angular momentum operators can be derived.
Conservation of angular momentum.
The Hamiltonian "H" represents the energy and dynamics of the system. In a spherically-symmetric situation, the Hamiltonian is invariant under rotations:
where "R" is a rotation operator. As a consequence, formula_88, and then formula_89 due to the relationship between J and "R". By the Ehrenfest theorem, it follows that J is conserved.
To summarize, if "H" is rotationally-invariant (spherically symmetric), then total angular momentum J is conserved. This is an example of Noether's theorem.
If "H" is just the Hamiltonian for one particle, the total angular momentum of that one particle is conserved when the particle is in a central potential (i.e., when the potential energy function depends only on formula_90). Alternatively, "H" may be the Hamiltonian of all particles and fields in the universe, and then "H" is "always" rotationally-invariant, as the fundamental laws of physics of the universe are the same regardless of orientation. This is the basis for saying conservation of angular momentum is a general principle of physics.
For a particle without spin, J = L, so orbital angular momentum is conserved in the same circumstances. When the spin is nonzero, the spin–orbit interaction allows angular momentum to transfer from L to S or back. Therefore, L is not, on its own, conserved.
Angular momentum coupling.
Often, two or more sorts of angular momentum interact with each other, so that angular momentum can transfer from one to the other. For example, in spin–orbit coupling, angular momentum can transfer between L and S, but only the total J = L + S is conserved. In another example, in an atom with two electrons, each has its own angular momentum J1 and J2, but only the total J = J1 + J2 is conserved.
In these situations, it is often useful to know the relationship between, on the one hand, states where formula_91 all have definite values, and on the other hand, states where formula_92 all have definite values, as the latter four are usually conserved (constants of motion). The procedure to go back and forth between these bases is to use Clebsch–Gordan coefficients.
One important result in this field is that a relationship between the quantum numbers for formula_93:
For an atom or molecule with J = L + S, the term symbol gives the quantum numbers associated with the operators formula_95.
Orbital angular momentum in spherical coordinates.
Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. The angular momentum in the spatial representation is
In spherical coordinates the angular part of the Laplace operator can be expressed by the angular momentum. This leads to the relation
When solving to find eigenstates of the operator formula_98, we obtain the following
where
are the spherical harmonics.