Angular momentum operator explained

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.^[1]

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

L = r x p . The quantum-mechanical counterparts of these objects share the same relationship:$$\backslash mathbf\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf$$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. L = \left(L _{x,} L _{y,} L _{z\right)} 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:$$\backslash mathbf\; =\; -i\backslash hbar(\backslash mathbf\; \backslash times\; \backslash nabla)$$where is the vector differential operator, del.

Spin angular momentum

See main article: Spin (physics). There is another type of angular momentum, called spin angular momentum (more often shortened to spin), represented by the spin operator

S = \left(S _{x,} S _{y,} S _{z\right)} . Spin is often depicted as a particle literally spinning around an axis, but this is only a metaphor: the closest classical analog is based on wave circulation.^[2] All elementary particles have a characteristic spin (scalar bosons have zero spin). For example, electrons always have "spin 1/2" while photons always have "spin 1" (details below).

Total angular momentum

J = \left(J _{x,} J _{y,} J _{z\right)} , which combines both the spin and orbital angular momentum of a particle or system:$$\backslash mathbf\; =\; \backslash mathbf\; +\; \backslash mathbf.$$

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

L = \left(L _{x,} L _{y,} L _{z\right)} . The components have the following commutation relations with each other:^[3] $$\backslash left[L\_x,\; L\_y\backslash right]\; =\; i\backslash hbar\; L\_z,\; \backslash ;\backslash ;\; \backslash left[L\_y,\; L\_z\backslash right]\; =\; i\backslash hbar\; L\_x,\; \backslash ;\backslash ;\; \backslash left[L\_z,\; L\_x\backslash right]\; =\; i\backslash hbar\; L\_y,$$

where denotes the commutator$$[X,\; Y]\; \backslash equiv\; XY\; -\; YX.$$

This can be written generally as$$\backslash left[L\_l,\; L\_m\backslash right]\; =\; i\; \backslash hbar\; \backslash sum\_^\; \backslash varepsilon\_\; L\_n,$$where l, m, n are the component indices (1 for x, 2 for y, 3 for z), and denotes the Levi-Civita symbol.

A compact expression as one vector equation is also possible:^[4] $$\backslash mathbf\; \backslash times\; \backslash mathbf\; =\; i\backslash hbar\; \backslash mathbf$$

The commutation relations can be proved as a direct consequence of the canonical commutation relations

[x _{l,p} _{m]} = i \hbar \delta _{lm} , where is the Kronecker delta.

There is an analogous relationship in classical physics:^[5] $$\backslash left\backslash \; =\; \backslash varepsilon\_\; L\_k$$where L_n is a component of the classical angular momentum operator, and

\{ ,\} is the Poisson bracket.

The same commutation relations apply for the other angular momentum operators (spin and total angular momentum):$$\backslash left[S\_l,\; S\_m\backslash right]\; =\; i\; \backslash hbar\; \backslash sum\_^\; \backslash varepsilon\_\; S\_n,\; \backslash quad\; \backslash left[J\_l,\; J\_m\backslash right]\; =\; i\; \backslash hbar\; \backslash sum\_^\; \backslash varepsilon\_\; J\_n.$$

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 are its structure constants. In this case, the Lie algebra is SU(2) or SO(3) in physics notation (

\operatorname{su}( 2 ) or \operatorname{so}( 3 ) 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,^[6] 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,$$L^2\; \backslash equiv\; L\_x^2\; +\; L\_y^2\; +\; L\_z^2.$$

L ^{2} is another quantum operator. It commutes with the components of L ,$$\backslash left[L^2,\; L\_x\backslash right]\; =\; \backslash left[L^2,\; L\_y\backslash right]\; =\; \backslash left[L^2,\; L\_z\backslash right]\; =\; 0\; .$$

One way to prove that these operators commute is to start from the [''L''_''ℓ'', ''L''_''m''] commutation relations in the previous section:

Mathematically,

L ^{2} is a Casimir invariant of the Lie algebra SO(3) spanned by L .

As above, there is an analogous relationship in classical physics:$$\backslash left\backslash \; =\; \backslash left\backslash \; =\; \backslash left\backslash \; =\; 0$$where

L _{i} is a component of the classical angular momentum operator, and \{ ,\} is the Poisson bracket.^[7]

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

See main article: Uncertainty principle and Uncertainty principle derivations. 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:$$\backslash sigma\_\; \backslash sigma\_\; \backslash geq\; \backslash frac\; \backslash left|\; \backslash langle\; L\_z\; \backslash rangle\; \backslash right|.$$where

\sigma _{X} is the standard deviation in the measured values of X and \langle X \rangle 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 L_x and L_y) are complementary and cannot be simultaneously known or measured, except in special cases such as

L _{x} = L _{y} = L _{z} = 0 .

It is, however, possible to simultaneously measure or specify L² and any one component of L; for example, L² 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 L² 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

See also: Azimuthal quantum number and Magnetic quantum number. 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

\hbar is reduced Planck constant:^[8]

If you measure...	...the result can be...	Notes
L ^{2}	\hbar ^{2} \ell (\ell + 1 ) ,    where \ell = 0 , 1 , 2 , \ldots	\ell is sometimes called azimuthal quantum number or orbital quantum number.
L _{z}	\hbar m _{\ell} ,    where m _{\ell} = - \ell, ( - \ell + 1 ), \ldots, (\ell - 1 ), \ell	m _{\ell} is sometimes called magnetic quantum number. This same quantization rule holds for any component of L ; e.g., L _{x} or L _{y} . This rule is sometimes called spatial quantization.^[9]
S ^{2}	\hbar ^{2} s(s + 1 ) ,    where s = 0 , \tfrac{ 1 }{ 2 }, 1 , \tfrac{ 3 }{ 2 }, \ldots	s is called spin quantum number or just spin. For example, a spin- particle is a particle where s = .
S _{z}	\hbar m _{s} ,    where m _{s} = - s, ( - s + 1 ), \ldots, (s - 1 ), s	m _{s} is sometimes called spin projection quantum number. This same quantization rule holds for any component of S ; e.g., S _{x} or S _{y} .
J ^{2}	\hbar ^{2} j(j + 1 ) ,    where j = 0 , \tfrac{ 1 }{ 2 }, 1 , \tfrac{ 3 }{ 2 }, \ldots	j is sometimes called total angular momentum quantum number.
J _{z}	\hbar m _{j} ,    where m _{j} = - j, ( - j + 1 ), \ldots, (j - 1 ), j	m _{j} is sometimes called total angular momentum projection quantum number. This same quantization rule holds for any component of J ; e.g., J _{x} or J _{y} .

Derivation using ladder operators

A common way to derive the quantization rules above is the method of ladder operators.^[10] The ladder operators for the total angular momentum

J = \left(J _{x,} J _{y,} J _{z\right)} are defined as:

Suppose

|\psi\rangle is a simultaneous eigenstate of J ^{2} and J _{z} (i.e., a state with a definite value for J ^{2} and a definite value for J _{z} ). Then using the commutation relations for the components of J , one can prove that each of the states J _{+} |\psi\rangle and J _{-|\psi\rangle} is either zero or a simultaneous eigenstate of J ^{2} and J _{z} , with the same value as |\psi\rangle for J ^{2} but with values for J _{z} that are increased or decreased by \hbar respectively. The result is zero when the use of a ladder operator would otherwise result in a state with a value for J _{z} that is outside the allowable range. Using the ladder operators in this way, the possible values and quantum numbers for J ^{2} and J _{z} can be found.

Since

S and L have the same commutation relations as J , the same ladder analysis can be applied to them, except that for L there is a further restriction on the quantum numbers that they must be integers.

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

\ell = 2 , and m _{\ell} = -2 , -1 , 0 , 1 , 2 for the five cones from bottom to top. Since |L| = \sqrt{L ^{2}} = \hbar \sqrt{ 6 } , the vectors are all shown with length \hbar \sqrt{ 6 } . The rings represent the fact that L _{z} is known with certainty, but L _{x} and L _{y} are unknown; therefore every classical vector with the appropriate length and z-component is drawn, forming a cone. The expected value of the angular momentum for a given ensemble of systems in the quantum state characterized by \ell and m _{\ell} could be somewhere on this cone while it cannot be defined for a single system (since the components of L do not commute with each other).

Quantization in macroscopic systems

The quantization rules are widely thought to be true even for macroscopic systems, like the angular momentum L of a spinning tire. However they have no observable effect so this has not been tested. For example, if

L _{z/\hbar} is roughly 100000000, it makes essentially no difference whether the precise value is an integer like 100000000 or 100000001, or a non-integer like 100000000.2—the discrete steps are currently too small to measure.^[11]

Angular momentum as the generator of rotations

See also: Total angular momentum quantum number. The most general and fundamental definition of angular momentum is as the generator of rotations.^[12] More specifically, let

R(\hat{n},\phi) be a rotation operator, which rotates any quantum state about axis \hat{n} by angle \phi . As \phi → 0 , the operator R(\hat{n},\phi) approaches the identity operator, because a rotation of 0° maps all states to themselves. Then the angular momentum operator J _{\hat{n} } about axis \hat{n} is defined as:^[12] $$J\_\backslash hat\; \backslash equiv\; i\backslash hbar\; \backslash lim\_\; \backslash frac\; =\; \backslash left.\; i\backslash hbar\; \backslash frac\; \backslash right|\_$$

where 1 is the identity operator. Also notice that R is an additive morphism :

R\left(\hat{n}, \phi _{1} + \phi _{2\right)} = R\left(\hat{n}, \phi _{1\right)R\left(\hat{n},} \phi _{2\right)} ; as a consequence^[12] $$R\backslash left(\backslash hat,\; \backslash phi\backslash right)\; =\; \backslash exp\backslash left(-\backslash frac\backslash right)$$where exp is matrix exponential. The existence of the generator is guaranteed by the Stone's theorem on one-parameter unitary groups.

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$$R\_\backslash text\backslash left(\backslash hat,\; \backslash phi\backslash right)\; =\; \backslash exp\backslash left(-\backslash frac\backslash right),$$rotates the position (in space) of all particles and fields, without rotating the internal (spin) state of any particle. Likewise, the operator$$R\_\backslash text\backslash left(\backslash hat,\; \backslash phi\backslash right)\; =\; \backslash exp\backslash left(-\backslash frac\backslash right),$$rotates the internal (spin) state of all particles, without moving any particles or fields in space. The relation J = L + S comes from:$$R\backslash left(\backslash hat,\; \backslash phi\backslash right)\; =\; R\_\backslash text\backslash left(\backslash hat,\; \backslash phi\backslash right)\; R\_\backslash text\backslash left(\backslash hat,\; \backslash phi\backslash right)$$

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

See main article: Spin (physics). Although one might expect

R\left(\hat{n}, 360 ^{\circ\right)} = 1 (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.), R\left(\hat{n}, 360 ^{\circ\right)} = -1 , and when it is an integer, R\left(\hat{n}, 360 ^{\circ\right)} = +1 .^[12] 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°.)^[12]

On the other hand,

R _{spatial\left(\hat{n},} 360 ^{\circ\right)} = +1 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 R _{spatial} operators carry the structure of SO(3), while R and R _{internal} carry the structure of SU(2).

From the equation

+1 = R _{spatial\left(\hat{z},} 360 ^{\circ\right)} = \exp\left( -2 \pi i L _{z} / \hbar\right) , one picks an eigenstate L _{z} |\psi\rangle = m\hbar |\psi\rangle and draws$$e^\; =\; 1$$which is to say that the orbital angular momentum quantum numbers can only be integers, not half-integers.

Connection to representation theory

See main article: Particle physics and representation theory and Representation theory of SU(2). Starting with a certain quantum state

|\psi _{0\rangle} , consider the set of states R\left(\hat{n}, \phi\right) \left|\psi _{0\right\rangle} for all possible \hat{n} and \phi , 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 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.^[12]

(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:$$RHR^\; =\; H$$where R is a rotation operator. As a consequence,

[H, R] = 0 , and then [H, J ] = 0 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

\left| r \right| ). 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

See main article: Angular momentum coupling and Clebsch–Gordan coefficients.

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

\left(J _{1\right)} _{z,} \left(J ^{2 ,} _{1 \right)} \left(J _{2\right)} _{z,} \left(J ^{2} _{2 \right)} all have definite values, and on the other hand, states where \left(J ^{2 ,} _{1 \right)} \left(J ^{2 ,} _{2 \right)} J ^{2,} J _{z} 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

\left(J ^{2 ,} _{1 \right)} \left(J ^{2 ,} _{2 \right)} J ^{2} :$$j\; \backslash in\; \backslash left\backslash \; .$$

For an atom or molecule with J = L + S, the term symbol gives the quantum numbers associated with the operators

L ^{2,} S ^{2,} J ^{2} .

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^{[13] [14]}

In spherical coordinates the angular part of the Laplace operator can be expressed by the angular momentum. This leads to the relation$$\backslash Delta\; =\; \backslash frac\; \backslash frac\; \backslash left(r^2\backslash ,\; \backslash frac\backslash right)\; -\; \backslash frac.$$

L ^{2} , we obtain the followingwhere$$\backslash left\backslash langle\; \backslash theta,\; \backslash phi\; |\; \backslash ell,\; m\; \backslash right\backslash rangle\; =\; Y\_(\backslash theta,\; \backslash phi)$$are the spherical harmonics.^[15]

See also

• Runge–Lenz vector (used to describe the shape and orientation of bodies in orbit)
• Holstein–Primakoff transformation
• Jordan map (Schwinger's bosonic model of angular momentum^[16])
• Pauli–Lubanski pseudovector
• Angular momentum diagrams (quantum mechanics)
• Spherical basis
• Tensor operator
• Orbital magnetization
• Orbital angular momentum of free electrons
• Orbital angular momentum of light

References

1. Introductory Quantum Mechanics, Richard L. Liboff, 2nd Edition,
2. Ohanian . Hans C. . 1986-06-01 . What is spin? . American Journal of Physics . en . 54 . 6 . 500–505 . 10.1119/1.14580 . 1986AmJPh..54..500O . 0002-9505.
3. Book: Aruldhas, G. . https://books.google.com/books?id=dRsvmTFpB3wC&pg=PA171. Quantum Mechanics. 171. formula (8.8) . 978-81-203-1962-2 . 2004-02-01. Prentice Hall India.
4. Book: Shankar. R. . Principles of quantum mechanics . limited . 1994. Kluwer Academic / Plenum. New York . 9780306447907 . 319. 2nd.
5. H. Goldstein, C. P. Poole and J. Safko, Classical Mechanics, 3rd Edition, Addison-Wesley 2002, pp. 388 ff.
6. J. H. Van Vleck. The Coupling of Angular Momentum Vectors in Molecules. Reviews of Modern Physics. 23 . 213 . 1951 . 3. 10.1103/RevModPhys.23.213. 1951RvMP...23..213V.
7. Goldstein et al, p. 410
8. Book: Condon . E. U. . Edward Condon . Shortley . G. H. . Quantum Theory of Atomic Spectra . Cambridge University Press . 1935 . Chapter III: Angular Momentum . https://books.google.com/books?id=hPyD-Nc_YmgC&pg=PA45 . 9780521092098.
9. Introduction to quantum mechanics: with applications to chemistry, by Linus Pauling, Edgar Bright Wilson, page 45, google books link
10. Book: Griffiths, David J. . Introduction to Quantum Mechanics . limited . . 1995 . 147–149.
11. Web site: Downes . Sean . Spin Angular Momentum . Physics! . 29 July 2022.
12. Web site: Lecture notes on rotations in quantum mechanics. Robert. Littlejohn. Robert Grayson Littlejohn. 13 Jan 2012. Physics 221B Spring 2011. 2011. 26 August 2014. https://web.archive.org/web/20140826003155/http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf. dead.
13. Book: Bes , Daniel R. . Springer Berlin Heidelberg . 978-3-540-46215-6 . Quantum Mechanics . Advanced Texts in Physics . Berlin, Heidelberg . 2007 . 70 . 10.1007/978-3-540-46216-3 . 2007qume.book.....B .
14. Compare and contrast with the contragredient classical L.
15. Sakurai, JJ & Napolitano, J (2010), Modern Quantum Mechanics (2nd edition) (Pearson)
16. Book: Schwinger, Julian . 1952 . On Angular Momentum . U.S. Atomic Energy Commission.

Further reading

• Book: Abers, E. . Quantum Mechanics. Addison Wesley, Prentice Hall Inc. 2004. 978-0-13-146100-0.
• Book: Biedenharn . L. C. . Angular Momentum in Quantum Physics: Theory and Application . Louck . James D. . 1984 . Cambridge University Press . 978-0-521-30228-9 . Encyclopedia of Mathematics and its Applications . Cambridge . 10.1017/cbo9780511759888. 1984amqp.book.....B . Lawrence Biedenharn.
• Book: Physics of Atoms and Molecules. B.H.. Bransden. C.J.. Joachain. Longman. 1983. 0-582-44401-2.
• Book: https://www.feynmanlectures.caltech.edu/III_18.html. The Feynman Lectures on Physics Vol. III. The New Millennium. Ch. 18: Angular Momentum. Richard P.. Feynman. Robert B.. Leighton. Matthew. Sands.
• Book: McMahon, D.. Quantum Mechanics Demystified. Mc Graw Hill (USA). 2006. 0-07-145546 9.
• Book: Zare, R.N.. Angular Momentum. Understanding Spatial Aspects in Chemistry and Physics. Wiley-Interscience. 1991. 978-0-47-1858928.