Angular momentum is a crucial concept in quantum mechanics, combining orbital and spin components. This topic explores how these components interact and add together, forming the total angular momentum. It's essential for understanding atomic structure and spectra.

The addition of angular momenta follows specific rules and leads to new quantum states. This process explains fine structure in atomic spectra and is key to predicting how atoms behave in magnetic fields. It's a fundamental tool for analyzing complex quantum systems.

Total Angular Momentum

Vector Sum and Quantum Numbers

• Total angular momentum (J) combines orbital angular momentum (L) and spin angular momentum (S) as vector sum $\mathbf{J} = \mathbf{L} + \mathbf{S}$
• Magnitude of total angular momentum calculated using $|\mathbf{J}| = \sqrt{j(j+1)}\hbar$
  • j represents total angular momentum quantum number
• Z-component of total angular momentum quantized as $J_z = m_j\hbar$
  • $m_j$ denotes magnetic quantum number for total angular momentum
• Allowed values for j range from $|l - s|$ to $|l + s|$ in integer steps
  • l and s signify orbital and spin angular momentum quantum numbers

Coupling Mechanisms and Vector Model

• LS coupling (Russell-Saunders coupling) applies to lighter atoms with weak spin-orbit interaction
• Vector model depicts L and S precessing around J
  • Constant angle maintained between L, S, and J during precession
• Coupling strength influences energy level splitting and spectral features (fine structure)
• Examples of LS coupling:
  • Hydrogen atom fine structure
  • Alkali metal spectra (lithium, sodium)

Angular Momentum Coupling Rules

Triangle Inequality and Magnetic Quantum Numbers

• Triangle inequality rule constrains total angular momentum quantum number J
  • For angular momenta $j_1$ and $j_2$, J must satisfy $|j_1 - j_2| \leq J \leq j_1 + j_2$
• Magnetic quantum number $m_J$ for total angular momentum sums individual values
  • $m_J = m_1 + m_2$
• Allowed $m_J$ values range from -J to +J in integer steps
• Degeneracy of coupled state determined by (2J + 1) possible $m_J$ values
• Examples:
  • Coupling two spin-1/2 particles yields singlet (J=0) and triplet (J=1) states
  • p-orbital electron (l=1) coupled with spin-1/2 results in j=1/2 and j=3/2 states

Multiple Angular Momenta and Parity

• Coupling more than two angular momenta requires sequential application of addition rules
  • Couple two angular momenta at a time
  • Order of coupling can affect intermediate states but not final outcome
• Parity of coupled state determined by product of individual state parities
  • Even parity: (-1)^l = +1
  • Odd parity: (-1)^l = -1
• Examples:
  • Coupling three spin-1/2 particles can result in total J = 1/2 or J = 3/2
  • Two-electron configuration 1s2s has even parity (product of two even-parity states)

Clebsch-Gordan Coefficients

Definition and Properties

• Clebsch-Gordan coefficients ⟨j1, j2, m1, m2|J, M⟩ describe probability amplitude for coupling angular momenta states
• Coefficients satisfy normalization and orthogonality conditions
  • Ensures conservation of probability in quantum mechanics
• Symmetry properties include invariance under certain permutations and sign changes of quantum numbers
• Square of coefficient represents probability of finding individual angular momentum states within coupled state
• Examples:
  • ⟨1/2, 1/2, 1/2, -1/2|0, 0⟩ = -1/√2 for singlet state of two spin-1/2 particles
  • ⟨1, 1/2, 1, 1/2|3/2, 3/2⟩ = 1 for maximum aligned state of l=1 and s=1/2

Calculation Methods and Related Symbols

• Coefficients calculated using recursive formulas or obtained from tables for common couplings
• Wigner 3-j symbols related to Clebsch-Gordan coefficients
  • Used in advanced calculations due to symmetry properties
• Racah W-coefficients and 6-j symbols useful for coupling three angular momenta
• Examples:
  • Recursive formula for Clebsch-Gordan coefficients involves square roots of factorial terms
  • Wigner 3-j symbol (j1 j2 J; m1 m2 M) related to Clebsch-Gordan coefficient by phase factor and normalization

Energy Levels from Coupling

Fine Structure and Degeneracy

• Angular momentum coupling lifts degeneracies in uncoupled systems
  • Leads to fine structure in atomic spectra
• Total degeneracy of coupled state given by (2J + 1)
  • J represents total angular momentum quantum number
• Spin-orbit coupling splits energy levels according to total angular momentum J
  • Splitting proportional to spin-orbit coupling strength
• Examples:
  • Hydrogen atom fine structure splits n=2 level into 2S1/2 and 2P1/2, 2P3/2 states
  • Sodium D-line splitting results from spin-orbit coupling in 3p state

Zeeman Effect and Selection Rules

• Landé g-factor determines magnitude of Zeeman splitting in magnetic field
  • Depends on coupled angular momenta L, S, and J
• Selection rules for transitions between coupled states based on allowed changes in total angular momentum and z-component
  • ΔJ = 0, ±1 (except J=0 to J=0)
  • ΔmJ = 0, ±1
• Many-electron atoms may require different coupling schemes
  • LS coupling for lighter atoms
  • jj coupling for heavier atoms with strong spin-orbit interaction
• Examples:
  • Anomalous Zeeman effect in sodium D-lines due to different g-factors
  • LS coupling in carbon atom ground state configuration leads to 3P0, 3P1, and 3P2 terms