152 Total Angular Momentum
An angular momentum equal to the sum of orbital and spin parts that is used to describe the full rotational properties of a quantum system.
\(\mathbf{J}=\mathbf{L}+\mathbf{S}\). The total angular momentum is the vector sum of orbital and spin angular momentum. This principle is used to couple \(\mathbf{L}\) and \(\mathbf{S}\) in atoms.
The total angular momentum is
\[ \mathbf{J} = \mathbf{L} + \mathbf{S} \]
where
- \(\mathbf{J}\) is the total angular momentum.
- \(\mathbf{L}\) is the orbital angular momentum.
- \(\mathbf{S}\) is the spin angular momentum.
The \(\mathbf{J}\) algebra. The components of \(\mathbf{J}\) obey the same commutation relations as \(\mathbf{L}\). This principle is used to label states by \(j\) and \(m_{j}\).
The total-angular-momentum commutation relations are
\[ [J_{i}, J_{j}] = i\hbar\sum_{k}\epsilon_{ijk}J_{k} \]
where
- \(J_{i}\) are the components of \(\mathbf{J}\).
- \(\hbar\) is the reduced Planck constant.
The \(j=\ell\pm 1/2\) values. For one electron with orbital quantum number \(\ell\) and spin \(1/2\), the allowed \(j\) values are \(\ell\pm 1/2\). This principle is used to write fine-structure doublets.
The allowed \(j\) values are
\[ j = \ell \pm \dfrac{1}{2} \]
where
- \(j\) labels eigenvalues of \(J^{2}\).
- \(\ell\) is the orbital quantum number.
Conservation of \(\mathbf{J}\). In a rotationally invariant system, \(\mathbf{J}\) is often the conserved angular momentum. This principle is used to replace separate conservation of \(\mathbf{L}\) and \(\mathbf{S}\) when they are coupled.
Note: Also denoted \(\mathbf{J}\).
152.1 References
- Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — \(\mathbf{J}=\mathbf{L}+\mathbf{S}\).
- Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \([J_{i},J_{j}]=i\hbar\sum_{k}\epsilon_{ijk}J_{k}\).
- Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — total angular momentum \(\mathbf{J}=\mathbf{L}+\mathbf{S}\).
- 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