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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 physics and other quantum problems involving rotational symmetry. In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion.
Contents
- Spin orbital and total angular momentum
- Orbital angular momentum operator
- Commutation relations between components
- Commutation relations involving vector magnitude
- Uncertainty principle
- Quantization
- Derivation using ladder operators
- Visual interpretation
- Quantization in macroscopic systems
- Angular momentum as the generator of rotations
- SU2 SO3 and 360 rotations
- Connection to representation theory
- Connection to commutation relations
- Conservation of angular momentum
- Angular momentum coupling
- Orbital angular momentum in spherical coordinates
- References
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.
Spin, orbital, and total angular momentum
The classical definition of angular momentum is
However, there is another type of angular momentum, called spin angular momentum (more often shortened to spin), represented by the spin operator S. Almost all elementary particles have spin. 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 motion in space. All elementary particles have a characteristic spin, for example electrons always have "spin 1/2" while photons always have "spin 1" (details below).
Finally, there is total angular momentum J, 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.
Orbital angular momentum operator
The orbital angular momentum operator L is mathematically defined as the cross product of a wave function's position operator (r) and momentum operator (p):
This is analogous to the definition of angular momentum in classical physics.
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 a single vector equation:
where ∇ is the vector differential operator, del.
Commutation relations between components
The orbital angular momentum operator is a vector operator, meaning it can be written in terms of its vector components
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
There is an analogous relationship in classical physics:
where Ln is a component of the classical angular momentum operator, and
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), the rotation group 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.
Commutation relations involving vector magnitude
Like any vector, a magnitude can be defined for the orbital angular momentum operator,
L2 is another quantum operator. It commutes with the components of L,
One way to prove that these operators commute is to start from the [Lℓ, Lm] commutation relations in the previous section:
Mathematically, L2 is a Casimir invariant of the Lie algebra SO(3) spanned by L.
As above, there is an analogous relationship in classical physics:
where Li is a component of the classical angular momentum operator, and
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
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
It is, however, possible to simultaneously measure or specify L2 and any one component of L; for example, L2 and Lz. 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 L2 and Lz, but not of Lx or Ly. 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
Derivation using ladder operators
A common way to derive the quantization rules above is the method of ladder operators. The ladder operators are defined:
Suppose a state
By manipulating these ladder operators and using the commutation rules, it is possible to prove almost all of the quantization rules above.
Since S and L have the same commutation relations as J, the same ladder analysis works for them.
The ladder-operator analysis does not explain one aspect of the quantization rules above: the fact that L (unlike J and S) cannot have half-integer quantum numbers. This fact can be proven (at least in the special case of one particle) by writing down every possible eigenfunction of L2 and Lz, (they are the spherical harmonics), and seeing explicitly that none of them have half-integer quantum numbers. An alternative derivation is below.
Visual interpretation
Since the angular momenta are quantum operators, they cannot be drawn as vectors like in classical mechanics. Nevertheless, it is common to depict them heuristically in this way. Depicted on the right is a set of states with quantum numbers
Quantization in macroscopic systems
The quantization rules are technically true even for macroscopic systems, like the angular momentum L of a spinning tire. However they have no observable effect. For example, if
Angular momentum as the generator of rotations
The most general and fundamental definition of angular momentum is as the generator of rotations. More specifically, let
where 1 is the identity operator. Also notice that R is an additive morphism :
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
On the other hand,
From the equation
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
From the relation between J and rotation operators,
When angular momentum operators act on quantum states, it forms a representation of the Lie algebra SU(2) or SO(3).(The Lie algebras of SU(2) and SO(3) are identical.)
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.
(This same calculational procedure is one way to answer the mathematical question "What is the Lie algebra of the Lie groups SO(3) or SU(2)?")
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,
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
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
One important result in this field is that a relationship between the quantum numbers for
For an atom or molecule with J = L + S, the term symbol gives the quantum numbers associated with the operators
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
where
are the spherical harmonics.