Physics:Quantum Angular momentum operator



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Angular momentum operator in quantum mechanics, the angular momentum operator is the operator associated with rotational motion and rotational symmetry, and is the quantum analogue of angular momentum in classical mechanics. Geometrically, this corresponds to a situation in which the magnitude of the angular momentum vector is well-defined while only one of its directional components is sharp, so the vector cannot be assigned a definite direction in three-dimensional space. Instead, it is often visualized as lying on a sphere of fixed radius (set by L^2) with uncertainty in its orientation, forming a “cone” of possible directions. As an observable, angular momentum is represented by operators whose eigenstates correspond to states with definite angular momentum, and whose eigenvalues give the possible results of measurement; it plays a central role in atomic physics, molecular physics, spectroscopy, and quantum theory more generally. Angular momentum is one of the fundamental conserved quantities of motion, together with linear momentum and energy. Depending on context, the expression angular momentum operator may refer either to orbital angular momentum or to total angular momentum. For a closed system, the total angular momentum is conserved, in accordance with rotational symmetry and Noether's theorem. In quantum mechanics, angular momentum appears in three closely related forms: orbital angular momentum, spin angular momentum, and total angular momentum.

Quantum Angular momentum operator.

Overview

Vector cones of total angular momentum $\mathbf{𝐉}$ (green), orbital angular momentum $\mathbf{𝐋}$ (blue), and spin angular momentum $\mathbf{𝐒}$ (red). The cone structure reflects the fact that not all components can be known simultaneously; see Visual interpretation.

In quantum mechanics, angular momentum appears in three closely related forms: orbital angular momentum, spin angular momentum, and total angular momentum.

Orbital angular momentum

The classical angular momentum of a particle is $$\mathbf{𝐋}=\mathbf{𝐫}\times \mathbf{𝐩}.$$ The same formal expression holds in quantum mechanics, except that $\mathbf{𝐫}$ and $\mathbf{𝐩}$ are now the position operator and momentum operator: $$\mathbf{𝐋}=\mathbf{𝐫}\times \mathbf{𝐩}.$$

Thus $\mathbf{𝐋}$ is a vector operator, with components $$\mathbf{𝐋}=(L_x,L_y,L_z).$$

For a single spinless, uncharged particle in the position representation, $$\mathbf{𝐋}=-i\hslash (\mathbf{𝐫}\times \mathrm{\nabla }),$$ where $\mathrm{\nabla }$ is the gradient operator.

Spin angular momentum

Related topic: Spin

In addition to orbital angular momentum, quantum systems may possess an intrinsic form of angular momentum called spin, represented by $$\mathbf{𝐒}=(S_x,S_y,S_z).$$

Spin has no exact classical counterpart. It is often illustrated as if a particle were spinning about an axis, but this picture is only heuristic; spin is an intrinsic quantum property. Elementary particles have fixed intrinsic spin: for example, electrons have spin $\frac{1}{2}$, while photons have spin 1.

Total angular momentum

The total angular momentum operator combines orbital and spin contributions: $$\mathbf{𝐉}=\mathbf{𝐋}+\mathbf{𝐒}.$$

For a closed system, the total angular momentum is conserved. By contrast, $\mathbf{𝐋}$ and $\mathbf{𝐒}$ need not be conserved separately. For example, in spin–orbit interaction, angular momentum may be exchanged between orbital and spin parts while the total $\mathbf{𝐉}$ remains constant.

Commutation relations

Commutation relations between components

The components of the orbital angular momentum operator satisfy the commutation relations $$[L_x,L_y]=i\hslash L_z,[L_y,L_z]=i\hslash L_x,[L_z,L_x]=i\hslash L_y,$$ where the commutator is defined by $$[X,Y]\equiv XY-YX.$$

In index notation, $$[L_l,L_m]=i\hslash \sum _{n=1}^3{\epsilon }_{lmn}{L}_n,$$ or, using Einstein summation convention, $$[L_l,L_m]=i\hslash {\epsilon }_{lmn}{L}_n.$$

These relations can also be written compactly as $$\mathbf{𝐋}\times \mathbf{𝐋}=i\hslash \mathbf{𝐋}.$$

They follow from the canonical commutation relations $$[x_l,p_m]=i\hslash {\delta }_{lm}.$$

The same algebra holds for spin and total angular momentum: $$[S_l,S_m]=i\hslash \sum _{n=1}^3{\epsilon }_{lmn}{S}_n,[J_l,J_m]=i\hslash \sum _{n=1}^3{\epsilon }_{lmn}{J}_n.$$

These commutation relations show that angular momentum operators generate the Lie algebra associated with three-dimensional rotations, usually written in physics as the algebra of SU(2) or SO(3).

Commutation relations involving the magnitude

The square of the orbital angular momentum operator is defined by $$L^2\equiv L_x^2+L_y^2+L_z^2.$$

This operator commutes with each component of $\mathbf{𝐋}$: $$[L^2,L_x]=[L^2,L_y]=[L^2,L_z]=0.$$

Thus one may simultaneously specify the values of $L^2$ and one chosen component, usually $L_z$. The same property holds for spin and total angular momentum: $$[S^2,S_i]=0,[J^2,J_i]=0.$$

Mathematically, $L^2$ is a Casimir invariant of the rotation algebra.

Uncertainty principle

Related topic: Uncertainty principle

Because different components of angular momentum do not commute, they cannot in general be measured simultaneously with arbitrary precision. For example, the Robertson–Schrödinger relation gives $${\sigma }_{L_x}{\sigma }_{L_y}\ge \frac{\hslash }{2}|⟨L_z⟩|,$$ where ${\sigma }_X$ is the standard deviation of measurements of $X$, and $⟨X⟩$ is the expectation value.

Thus two orthogonal components, such as $L_x$ and $L_y$, are complementary observables. By contrast, $L^2$ and one component such as $L_z$ can be measured simultaneously.

Quantization

Related topic: Azimuthal quantum number, Magnetic quantum number

In quantum mechanics, angular momentum is quantized: only certain discrete measurement results are allowed. For orbital angular momentum,

Quantity	Allowed values	Notes
$L^2$	${\hslash }^2\ell (\ell +1)$, where $\ell =0,1,2,\dots$	$\ell$ is the orbital or azimuthal quantum number.
$L_z$	$\hslash m_{\ell }$, where $m_{\ell }=-\ell ,-\ell +1,\dots ,\ell$	$m_{\ell }$ is the magnetic quantum number.
$S^2$	${\hslash }^{2}s(s+1)$, where $s=0,\frac{1}{2},1,\frac{3}{2},\dots$	$s$ is the spin quantum number.
$S_z$	$\hslash m_s$, where $m_s=-s,-s+1,\dots ,s$	$m_s$ is the spin projection quantum number.
$J^2$	${\hslash }^{2}j(j+1)$, where $j=0,\frac{1}{2},1,\frac{3}{2},\dots$	$j$ is the total angular momentum quantum number.
$J_z$	$\hslash m_j$, where $m_j=-j,-j+1,\dots ,j$	$m_j$ is the total angular momentum projection quantum number.

For orbital angular momentum, both $\ell$ and $m_{\ell }$ are always integers. For spin and total angular momentum, half-integer values are also possible.

Derivation using ladder operators

Related topic: Angular_momentum

A standard derivation of the allowed values uses the ladder operators $$J_{+}\equiv J_x+i{J}_y,J_{-}\equiv J_x-i{J}_y.$$

If $|\psi ⟩$ is a simultaneous eigenstate of $J^2$ and $J_z$, then $J_{+}|\psi ⟩$ and $J_{-}|\psi ⟩$ are either zero or new simultaneous eigenstates with the same value of $J^2$ but with the $J_z$ eigenvalue shifted by $\pm \hslash$. Repeated application of these operators leads to the quantization rules above.

Since $\mathbf{𝐋}$ obeys the same algebra as $\mathbf{𝐉}$, the same ladder-operator argument applies to orbital angular momentum. In the case of $\mathbf{𝐋}$, single-valuedness of the wavefunction in the azimuthal angle $\varphi$ imposes the further restriction that $\ell$ and $m_{\ell }$ must be integers.

Visual interpretation

Heuristic vector model of orbital angular momentum.

Although angular momentum in quantum mechanics is represented by operators rather than classical vectors, it is often illustrated heuristically by vectors of fixed length whose tip can lie only on a cone. For a state with given $\ell$ and $m_{\ell }$, the magnitude is $$|L|=\sqrt{L^2}=\hslash \sqrt{\ell (\ell +1)},$$ while the component $L_z$ has the definite value $$L_z=\hslash m_{\ell }.$$

The uncertainty in the transverse components $L_x$ and $L_y$ is represented by the circular spread around the cone.

Quantization in macroscopic systems

The same quantum rules apply in principle to macroscopic bodies. In practice, however, the allowed steps in angular momentum are so small compared with the total angular momentum of ordinary objects that the spectrum appears continuous for all observable purposes.

Angular momentum as the generator of rotations

Related topic: Total angular momentum quantum number

The most fundamental characterization of angular momentum is that it generates rotations. If $R(\hat{n},\varphi )$ denotes the operator that rotates a system by an angle $\varphi$ about the axis $\hat{n}$, then the angular momentum component along that axis is defined by $$J_{\hat{n}}\equiv i\hslash \underset{\varphi \to 0}{\lim }\frac{R(\hat{n},\varphi )-1}{\varphi }={i\hslash \frac{\partial R(\hat{n},\varphi )}{\partial \varphi }|}_{\varphi =0}.$$

Equivalently, $$R(\hat{n},\varphi )=\exp (-\frac{i\varphi J_{\hat{n}}}{\hslash }).$$

Thus angular momentum governs how quantum states transform under rotations.

Different kinds of rotation operators

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The orbital and spin operators similarly generate spatial and internal rotations: $$R_{\text{spatial}}(\hat{n},\varphi )=\exp (-\frac{i\varphi L_{\hat{n}}}{\hslash }),$$ $$R_{\text{internal}}(\hat{n},\varphi )=\exp (-\frac{i\varphi S_{\hat{n}}}{\hslash }).$$

The relation $$\mathbf{𝐉}=\mathbf{𝐋}+\mathbf{𝐒}$$ reflects the corresponding decomposition of a full rotation into spatial and internal parts.

SU(2), SO(3), and 360° rotations

Related topic: Spin

In classical mechanics, a rotation by $360^{\circ }$ is identical to doing nothing. In quantum mechanics, however, a state with half-integer total angular momentum may change sign under a full $360^{\circ }$ rotation: $$R(\hat{n},360^{\circ })=-1$$ for half-integer $j$, whereas $$R(\hat{n},360^{\circ })=+1$$ for integer $j$.

This reflects the fact that quantum rotations are described by SU(2), which is the double cover of SO(3). Orbital angular momentum, by contrast, corresponds to ordinary spatial rotations and therefore only allows integer quantum numbers.

Connection to representation theory

Related topic: Particle physics and representation theory, Representation theory of SU(2), A note on Lie algebras

When rotation operators act on quantum states, they define a representation of the rotation group. Correspondingly, angular momentum operators define a representation of the associated Lie algebra. The classification of possible angular momentum quantum numbers is therefore a representation-theoretic problem for SU(2) and SO(3).

Connection to commutation relations

Rotations about different axes do not commute. This noncommutativity is reflected at the operator level in the angular momentum commutation relations. Thus the algebra of angular momentum is a direct expression of the geometry of rotations in three-dimensional space.

Conservation of angular momentum

If the Hamiltonian $H$ is rotationally invariant, then angular momentum is conserved. Rotational invariance means $$RH{R}^{-1}=H,$$ where $R$ is a rotation operator. This implies $$[H,R]=0,$$ and therefore $$[H,\mathbf{𝐉}]=\mathbf{𝟎}.$$

By the Ehrenfest theorem, the total angular momentum is then conserved. For a spinless particle in a central potential, this reduces to conservation of orbital angular momentum. When spin is present, spin–orbit coupling may transfer angular momentum between $\mathbf{𝐋}$ and $\mathbf{𝐒}$, while the total $\mathbf{𝐉}$ remains conserved.

Angular momentum coupling

Related topic: Angular momentum coupling, Clebsch–Gordan coefficients

When a system contains multiple sources of angular momentum, the individual contributions may combine to form a conserved total angular momentum. For example, for two angular momenta ${\mathbf{𝐉}}_1$ and ${\mathbf{𝐉}}_2$, $$\mathbf{𝐉}={\mathbf{𝐉}}_1+{\mathbf{𝐉}}_2.$$

The corresponding total quantum number satisfies $$j\in \{|j_1-j_2|,|j_1-j_2|+1,\dots ,j_1+j_2\}.$$

Transformations between uncoupled and coupled angular momentum bases are described by Clebsch–Gordan coefficients. In atomic and molecular physics, this structure underlies term symbols and the classification of energy levels.

Orbital angular momentum in spherical coordinates

Angular momentum operators naturally arise in problems with spherical symmetry. In the position representation and spherical coordinates, the orbital angular momentum operator is $$\begin{array}{rl}\mathbf{𝐋}& =i\hslash (\frac{\hat{\mathit{\theta }}}{\sin (\theta )}\frac{\partial }{\partial \varphi }-\hat{\mathit{\varphi }}\frac{\partial }{\partial \theta })\\ & =i\hslash (\widehat{\mathbf{𝐱}}(\sin (\varphi )\frac{\partial }{\partial \theta }+\cot (\theta )\cos (\varphi )\frac{\partial }{\partial \varphi })+\widehat{\mathbf{𝐲}}(-\cos (\varphi )\frac{\partial }{\partial \theta }+\cot (\theta )\sin (\varphi )\frac{\partial }{\partial \varphi })-\widehat{\mathbf{𝐳}}\frac{\partial }{\partial \varphi }),\\ L_{+}& =\hslash e^{i\varphi }(\frac{\partial }{\partial \theta }+i\cot (\theta )\frac{\partial }{\partial \varphi }),\\ L_{-}& =\hslash e^{-i\varphi }(-\frac{\partial }{\partial \theta }+i\cot (\theta )\frac{\partial }{\partial \varphi }),\\ L^2& =-{\hslash }^2(\frac{1}{\sin (\theta )}\frac{\partial }{\partial \theta }(\sin (\theta )\frac{\partial }{\partial \theta })+\frac{1}{{\sin }^2(\theta )}\frac{{\partial }^2}{\partial {\varphi }^2}),\\ L_z& =-i\hslash \frac{\partial }{\partial \varphi }.\end{array}$$

The angular part of the Laplace operator can be written in terms of $L^2$: $$\mathrm{\Delta }=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)-\frac{L^2}{{\hslash }^2{r}^2}.$$

The simultaneous eigenstates of $L^2$ and $L_z$ satisfy $$\begin{array}{rl}{L}^2|\ell ,m⟩& ={\hslash }^2\ell (\ell +1)|\ell ,m⟩,\\ L_z|\ell ,m⟩& =\hslash m|\ell ,m⟩,\end{array}$$ with wavefunctions $$⟨\theta ,\varphi |\ell ,m⟩=Y_{\ell ,m}(\theta ,\varphi ),$$ where $Y_{\ell ,m}$ are the spherical harmonics.

Index

Core theory

1. Foundations

2. Conceptual and interpretations

3. Mathematical structure and systems

4. Atomic and spectroscopy

5. Wavefunctions and modes

6. Quantum dynamics and evolution

7. Measurement and information

8. Quantum information and computing

Applications and extensions

9. Quantum optics and experiments

10. Open quantum systems

11. Quantum field theory

12. Statistical mechanics and kinetic theory

13. Condensed matter and solid-state physics

14. Plasma and fusion physics

15. Timeline

16. Advanced and frontier topics

Book II

• Matter by scale

Book III

• Methods and tools

Book IV

• Data Analysis Techniques

Full contents

1. Foundations (14) Back to index

1. Physics:Quantum basics

2. Physics:Quantum photoelectric effect

3. Physics:Quantum black-body radiation

4. Physics:Quantum Planck constant

5. Physics:Quantum Postulates

6. Physics:Quantum Hilbert space

7. Physics:Quantum Observables and operators

8. Physics:Quantum mechanics

9. Physics:Quantum mechanics measurements

10. Physics:Quantum state

11. Physics:Quantum system

12. Physics:Quantum superposition

13. Physics:Quantum probability

14. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (14) Back to index

15. Physics:Quantum Interpretations of quantum mechanics

16. Physics:Quantum Wave–particle duality

17. Physics:Quantum Complementarity principle

18. Physics:Quantum Uncertainty principle

19. Physics:Quantum Measurement problem

20. Physics:Quantum Bell's theorem

21. Physics:Quantum Hidden variable theory

22. Physics:Quantum nonlocality

23. Physics:Quantum contextuality

24. Physics:Quantum Darwinism

25. Physics:Quantum A Spooky Action at a Distance

26. Physics:Quantum A Walk Through the Universe

27. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

28. Physics:Quantum measurement problem

3. Mathematical structure and systems (15) Back to index

29. Physics:Quantum Density matrix

30. Physics:Quantum Exactly solvable quantum systems

31. Physics:Quantum many-body problem

32. Physics:Quantum Formulas Collection

33. Physics:Quantum A Matter Of Size

34. Physics:Quantum Symmetry in quantum mechanics

35. Physics:Quantum Noether theorem

36. Physics:Quantum Angular momentum operator

37. Physics:Quantum Runge–Lenz vector

38. Physics:Quantum Approximation Methods

39. Physics:Quantum Matter Elements and Particles

40. Physics:Quantum Dirac equation

41. Physics:Quantum Klein–Gordon equation

42. Physics:Quantum pendulum

43. Physics:Quantum configuration space

4. Atomic and spectroscopy (14) Back to index

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

44. Physics:Quantum Atomic structure and spectroscopy

45. Physics:Quantum Hydrogen atom

46. Physics:Quantum number

47. Physics:Quantum Multi-electron atoms

48. Physics:Quantum Fine structure

49. Physics:Quantum Hyperfine structure

50. Physics:Quantum Isotopic shift

51. Physics:Quantum defect

52. Physics:Quantum Zeeman effect

53. Physics:Quantum Stark effect

54. Physics:Quantum Spectral lines and series

55. Physics:Quantum Selection rules

56. Physics:Quantum Fermi's golden rule

57. Physics:Quantum beats

5. Wavefunctions and modes (9) Back to index

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

58. Physics:Quantum Wavefunction

59. Physics:Quantum Superposition principle

60. Physics:Quantum Eigenstates and eigenvalues

61. Physics:Quantum Boundary conditions and quantization

62. Physics:Quantum Standing waves and modes

63. Physics:Quantum Normal modes and field quantization

64. Physics:Number of independent spatial modes in a spherical volume

65. Physics:Quantum Density of states

66. Physics:Quantum carpet

6. Quantum dynamics and evolution (20) Back to index

67. Physics:Quantum Time evolution

68. Physics:Quantum Schrödinger equation

69. Physics:Quantum Time-dependent Schrödinger equation

70. Physics:Quantum Stationary states

71. Physics:Quantum Perturbation theory

72. Physics:Quantum Time-dependent perturbation theory

73. Physics:Quantum Adiabatic theorem

74. Physics:Quantum Berry phase

75. Physics:Quantum Aharonov-Bohm effect

76. Physics:Quantum Scattering theory

77. Physics:Quantum Scattering matrix

78. Physics:Quantum S-matrix

79. Physics:Quantum tunnelling

80. Physics:Quantum speed limit

81. Physics:Quantum revival

82. Physics:Quantum reflection

83. Physics:Quantum oscillations

84. Physics:Quantum jump

85. Physics:Quantum boomerang effect

86. Physics:Quantum chaos

7. Measurement and information (9) Back to index

87. Physics:Quantum Measurement theory

88. Physics:Quantum Measurement operators

89. Physics:Quantum Projective measurement

90. Physics:Quantum POVM

91. Physics:Quantum Weak measurement

92. Physics:Quantum Measurement collapse

93. Physics:Quantum entanglement

94. Physics:Quantum Zeno effect

95. Physics:Quantum limit

8. Quantum information and computing (10) Back to index

96. Physics:Quantum information theory

97. Physics:Quantum Qubit

98. Physics:Quantum Entanglement

99. Physics:Quantum Gates and circuits

100. Physics:Quantum Computing Algorithms in the NISQ Era

101. Physics:Quantum Noisy Qubits

102. Physics:Quantum random access code

103. Physics:Quantum pseudo-telepathy

104. Physics:Quantum network

105. Physics:Quantum money

9. Quantum optics and experiments (5) Back to index

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

106. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

107. Physics:Quantum optics beam splitter experiments

108. Physics:Quantum Ultra fast lasers

109. Physics:Quantum Experimental quantum physics

110. Physics:Quantum optics

111. Template:Quantum optics operators

10. Open quantum systems (9) Back to index

112. Physics:Quantum Open systems

113. Physics:Quantum Master equation

114. Physics:Quantum Lindblad equation

115. Physics:Quantum Decoherence

116. Physics:Quantum dissipation

117. Physics:Quantum Markov semigroup

118. Physics:Quantum Markovian dynamics

119. Physics:Quantum Non-Markovian dynamics

120. Physics:Quantum Trajectories

11. Quantum field theory (23) Back to index

Structural dependency map of quantum field theory.

121. Physics:Quantum field theory (QFT) basics

122. Physics:Quantum field theory (QFT) core

123. Physics:Quantum Fields and Particles

124. Physics:Quantum Second quantization

125. Physics:Quantum Fock space

126. Physics:Quantum Harmonic Oscillator field modes

127. Physics:Quantum Creation and annihilation operators

128. Physics:Quantum vacuum fluctuations

129. Physics:Quantum Casimir effect

130. Physics:Quantum Propagators in quantum field theory

131. Physics:Quantum Feynman diagrams

132. Physics:Quantum Path integral formulation

133. Physics:Quantum Renormalization in field theory

134. Physics:Quantum Renormalization group

135. Physics:Quantum Field Theory Gauge symmetry

136. Physics:Quantum Spontaneous symmetry breaking

137. Physics:Quantum Non-Abelian gauge theory

138. Physics:Quantum Electrodynamics (QED)

139. Physics:Quantum chromodynamics (QCD)

140. Physics:Quantum Electroweak theory

141. Physics:Quantum Standard Model

142. Physics:Quantum triviality

143. Physics:Quantum confinement problem

12. Statistical mechanics and kinetic theory (9) Back to index

144. Physics:Quantum Statistical mechanics

145. Physics:Quantum Partition function

146. Physics:Quantum Distribution functions

147. Physics:Quantum Liouville equation

148. Physics:Quantum Kinetic theory

149. Physics:Quantum Boltzmann equation

150. Physics:Quantum BBGKY hierarchy

151. Physics:Quantum Relaxation and thermalization

152. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (15) Back to index

153. Physics:Quantum Band structure

154. Physics:Quantum Fermi surfaces

155. Physics:Quantum Landau levels

156. Physics:Quantum Semiconductor physics

157. Physics:Quantum Phonons

158. Physics:Quantum Electron-phonon interaction

159. Physics:Quantum Exchange interaction

160. Physics:Quantum Superconductivity

161. Physics:Quantum Topological phases of matter

162. Physics:Quantum well

163. Physics:Quantum spin liquid

164. Physics:Quantum spin Hall effect

165. Physics:Quantum phase transition

166. Physics:Quantum critical point

167. Physics:Quantum dot

14. Plasma and fusion physics (8) Back to index

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

168. Physics:Quantum Fusion reactions and Lawson criterion

169. Physics:Quantum Plasma (fusion context)

170. Physics:Quantum Magnetic confinement fusion

171. Physics:Quantum Inertial confinement fusion

172. Physics:Quantum Plasma instabilities and turbulence

173. Physics:Quantum Tokamak core plasma

174. Physics:Quantum Tokamak edge physics and recycling asymmetries

175. Physics:Quantum Stellarator

15. Timeline (8) Back to index

176. Physics:Quantum mechanics/Timeline

177. Physics:Quantum mechanics/Timeline/Pre-quantum era

178. Physics:Quantum mechanics/Timeline/Old quantum theory

179. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

180. Physics:Quantum mechanics/Timeline/Quantum field theory era

181. Physics:Quantum mechanics/Timeline/Quantum information era

182. Physics:Quantum mechanics/Timeline/Quantum technology era

183. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (16) Back to index

184. Physics:Quantum topology

185. Physics:Quantum battery

186. Physics:Quantum Supersymmetry

187. Physics:Quantum Black hole thermodynamics

188. Physics:Quantum Holographic principle

189. Physics:Quantum gravity

190. Physics:Quantum De Sitter invariant special relativity

191. Physics:Quantum Doubly special relativity

192. Physics:Quantum arithmetic geometry

193. Physics:Quantum unsolved problems

194. Physics:Quantum Yang-Mills mass gap

195. Physics:Quantum gravity problem

196. Physics:Quantum black hole information paradox

197. Physics:Quantum dark matter problem

198. Physics:Quantum neutrino mass problem

199. Physics:Quantum matter-antimatter asymmetry problem

• Runge–Lenz vector
• Holstein–Primakoff transformation
• Jordan map
• 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

Further reading

• Abers, E. (2004). Quantum Mechanics. Addison Wesley, Prentice Hall Inc. ISBN 978-0-13-146100-0. 
• Biedenharn, L. C.; Louck, James D. (1984). Angular Momentum in Quantum Physics: Theory and Application. Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511759888. ISBN 978-0-521-30228-9. Bibcode: 1984amqp.book.....B. https://www.cambridge.org/core/books/angular-momentum-in-quantum-physics/53AFDEE1D64D0256AD874534F084C402. 
• Bransden, B.H.; Joachain, C.J. (1983). Physics of Atoms and Molecules. Longman. ISBN 0-582-44401-2. 
• Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew. "Ch. 18: Angular Momentum". The Feynman Lectures on Physics Vol. III (The New Millennium ed.). https://www.feynmanlectures.caltech.edu/III_18.html. 
• McMahon, D. (2006). Quantum Mechanics Demystified. McGraw-Hill. ISBN 0-07-145546-9. 
• Zare, R.N. (1991). Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics. Wiley-Interscience. ISBN 978-0-471-85892-8. 

Author: Harold Foppele

Source attribution: Physics:Quantum Angular momentum operator