93 Angular Momentum
An operator that is used to represent rotational motion of a quantum system, equal to the cross product of position and momentum in the orbital case.
\(\mathbf{L}=\mathbf{r}\times\mathbf{p}\). Orbital angular momentum is the operator \(\mathbf{r}\times\mathbf{p}\). This principle is used to assign a rotational observable to spatial motion.
The orbital angular momentum is
\[ \mathbf{L} = \mathbf{r}\times\mathbf{p} \]
where
- \(\mathbf{L}\) is the angular momentum operator.
- \(\mathbf{r}\) is the position operator.
- \(\mathbf{p}\) is the momentum operator.
The commutation relations. The Cartesian components do not commute. This principle is used to explain why \(L_{x}\), \(L_{y}\), and \(L_{z}\) cannot all have sharp values at once.
The angular-momentum commutation relations are
\[ [L_{i}, L_{j}] = i\hbar\sum_{k}\epsilon_{ijk}L_{k} \]
where
- \(L_{i}\) are the Cartesian components of \(\mathbf{L}\).
- \(\epsilon_{ijk}\) is the Levi-Civita symbol.
- \(\hbar\) is the reduced Planck constant.
Simultaneous eigenstates of \(L^{2}\) and \(L_{z}\). \(L^{2}\) commutes with each component, so \(L^{2}\) and \(L_{z}\) can be sharp together. This principle is used to label states by \(\ell\) and \(m\).
The simultaneous eigenvalue equations are
\[ L^{2}|\ell,m\rangle = \hbar^{2}\ell(\ell+1)|\ell,m\rangle \]
\[ L_{z}|\ell,m\rangle = m\hbar|\ell,m\rangle \]
where
- \(\ell\) is the angular-momentum quantum number.
- \(m\) is the projection quantum number.
The general \(\mathbf{J}\) algebra. The same commutation relations hold for any angular momentum \(\mathbf{J}\), including spin. This principle is used to treat \(\mathbf{L}\), \(\mathbf{S}\), and \(\mathbf{J}\) with one algebra.
Note: Also denoted \(\mathbf{L}\). Also denoted \(\hat{\mathbf{L}}\). Spin is an intrinsic angular momentum not built from \(\mathbf{r}\) and \(\mathbf{p}\).
93.1 References
- Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — generators and commutation relations.
- Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — commutation relations.
- Shankar, R. Fundamentals of Physics. Yale University Press. — operator \(\mathbf{L}=\mathbf{r}\times\mathbf{p}\).
- Absorption
- Angular Momentum
- Atomic Orbitals
- Aufbau Principle
- Bohr Radius
- Bra and Ket
- Commutators
- Conjugate Variable
- Conservation Laws
- Conservation of Angular Momentum
- Conservation of Charge
- Conservation of Energy
- Conservation of Energy Transition Law
- Conservation of Momentum
- de Broglie Wavelength
- Derivation of Hamiltonian
- Derivation of Lagrangian
- Dipole Selection Rules
- Eigenvalue
- Eigenvector
- Einstein Coefficients
- Electromagnetic Interaction
- Electromagnetic Interaction
- Electromagnetic Radiation
- Electron Configurations
- Energy Quantization
- Expectation Values
- Fermi’s Golden Rule
- Hamiltonian
- Hund’s Rule
- Hydrogen Energy Levels
- Lagrangian
- Magnetic Moment
- Measurement
- Momentum Operator
- Normalization
- Operators
- Orbital Angular Momentum
- Pauli Exclusion Principle
- Photon Momentum
- Planck Relation
- Position Operator
- Potential Wells
- Probability Current
- Probability Density
- Quantum Harmonic Oscillator
- Quantum States
- Quantum Tunneling
- Rydberg Formula
- Scattering Theory
- Schrodinger Equation Time-Independent
- Schrodinger Equations
- Selection Rules
- Spin
- Spin-Orbit Coupling
- Spontaneous Emission
- Stimulated Emission
- Superposition
- Time Dependent Schrodinger Equation 1-Dimensional
- Time Dependent Schrodinger Equation Generalized
- Total Angular Momentum
- Uncertainty Principle
- Wave-Particle Duality
- Wavefunctions