Angular momentum in quantum mechanics is a crucial concept that builds on classical physics ideas. It describes the rotational motion of particles and systems, introducing quantization and uncertainty principles that challenge our everyday intuitions about rotation.

This topic explores angular momentum operators, their commutation relations, and eigenvalues. It also covers the quantization of angular momentum, spherical harmonics, and the coupling of different types of angular momenta in complex systems.

Angular Momentum Operators

Defining Angular Momentum in Quantum Mechanics

• Angular momentum operator $\hat{L}$ represents the angular momentum of a particle in quantum mechanics
• Orbital angular momentum $\hat{L}$ describes the angular momentum associated with the motion of a particle in space
  • Depends on the particle's position $\hat{r}$ and linear momentum $\hat{p}$
  • Components of orbital angular momentum: $\hat{L}_x$, $\hat{L}_y$, and $\hat{L}_z$
• Spin angular momentum $\hat{S}$ represents the intrinsic angular momentum of a particle
  • Not related to the particle's spatial motion
  • Fundamental property of particles like electrons, protons, and neutrons
  • Components of spin angular momentum: $\hat{S}_x$, $\hat{S}_y$, and $\hat{S}_z$
• Total angular momentum $\hat{J}$ is the sum of orbital and spin angular momenta: $\hat{J} = \hat{L} + \hat{S}$
  • Describes the overall angular momentum of a particle or system
  • Relevant in systems where both orbital and spin angular momenta are important (atoms, molecules)

Commutation Relations and Eigenvalues

• Angular momentum operators obey specific commutation relations
  • $[\hat{L}_i, \hat{L}j] = i\hbar\epsilon{ijk}\hat{L}_k$ and $[\hat{S}_i, \hat{S}j] = i\hbar\epsilon{ijk}\hat{S}k$, where $\epsilon{ijk}$ is the Levi-Civita symbol
  • These relations lead to the quantization of angular momentum
• Eigenvalues of angular momentum operators are quantized
  • Orbital angular momentum eigenvalues: $l(l+1)\hbar^2$, where $l$ is a non-negative integer
  • Spin angular momentum eigenvalues: $s(s+1)\hbar^2$, where $s$ is a half-integer (1/2, 3/2, ...)
  • Magnetic quantum number $m_l$ for orbital angular momentum: $-l \leq m_l \leq l$
  • Magnetic quantum number $m_s$ for spin angular momentum: $-s \leq m_s \leq s$

Quantization and Spherical Harmonics

Quantization of Angular Momentum

• Angular momentum is quantized in quantum mechanics
  • Magnitude of orbital angular momentum: $|\hat{L}| = \sqrt{l(l+1)}\hbar$, where $l$ is the orbital quantum number
  • Magnitude of spin angular momentum: $|\hat{S}| = \sqrt{s(s+1)}\hbar$, where $s$ is the spin quantum number
• Projection of angular momentum along a specific axis (usually z-axis) is also quantized
  • $\hat{L}_z$ eigenvalues: $m_l\hbar$, where $m_l$ is the magnetic quantum number for orbital angular momentum
  • $\hat{S}_z$ eigenvalues: $m_s\hbar$, where $m_s$ is the magnetic quantum number for spin angular momentum

Spherical Harmonics and Angular Wave Functions

• Spherical harmonics $Y_l^{m_l}(\theta, \phi)$ are the angular part of the wave function for a particle in a central potential
  • Depend on the orbital quantum number $l$ and the magnetic quantum number $m_l$
  • Describe the angular distribution of the particle's probability density
• Properties of spherical harmonics:
  • Orthonormal: $\int Y_l^{m_l}(\theta, \phi)Y_{l'}^{m_l'}(\theta, \phi) d\Omega = \delta_{ll'}\delta_{m_lm_l'}$
  • Parity: $Y_l^{m_l}(\pi - \theta, \phi + \pi) = (-1)^l Y_l^{m_l}(\theta, \phi)$
• Selection rules for transitions between states with different angular momenta
  • Determined by the overlap integral of the initial and final state wave functions
  • Electric dipole transitions: $\Delta l = \pm 1$ and $\Delta m_l = 0, \pm 1$
  • Transitions that do not satisfy these rules are forbidden or less likely to occur

Coupling of Angular Momenta

Addition of Angular Momenta

• Angular momenta can be coupled or added together to form a total angular momentum
• Coupling schemes:
  • LS coupling (Russell-Saunders coupling): Orbital angular momenta of individual particles are added to form a total orbital angular momentum $\hat{L}$, and spin angular momenta are added to form a total spin angular momentum $\hat{S}$. Then, $\hat{L}$ and $\hat{S}$ are coupled to form the total angular momentum $\hat{J}$.
  • jj coupling: Individual orbital and spin angular momenta are coupled to form a total angular momentum for each particle, then these total angular momenta are coupled to form the overall total angular momentum.
• Clebsch-Gordan coefficients $\langle j_1 m_1; j_2 m_2 | J M \rangle$ describe the coupling of two angular momenta $\hat{j}_1$ and $\hat{j}_2$ to form a total angular momentum $\hat{J}$
  • Determine the probability amplitudes for the different possible combinations of individual angular momentum states that can form a given total angular momentum state

Conservation of Angular Momentum

• Total angular momentum is conserved in a closed system
  • Follows from the rotational symmetry of the system's Hamiltonian
• Conservation of angular momentum has important consequences:
  • Selection rules for transitions between states with different angular momenta
  • Determines the allowed transitions and decay processes in atoms and molecules
  • Plays a crucial role in the conservation of angular momentum in particle interactions and decays (particle physics)
• Examples of conservation of angular momentum:
  • Electron transitions in atoms: The change in the electron's orbital angular momentum must be compensated by the emission or absorption of a photon with the appropriate angular momentum
  • Molecular rotations: The total angular momentum of a molecule is conserved during rotational transitions, leading to specific selection rules for rotational spectra