9.1 Orbital angular momentum in quantum mechanics

Orbital angular momentum in quantum mechanics describes how particles rotate in space. It's quantized, meaning it can only take specific values determined by quantum numbers. This concept is crucial for understanding atomic structure and spectroscopy.

The angular momentum operator, its components, and commutation relations form the mathematical foundation. Eigenvalues and quantum numbers define the allowed states, shaping atomic orbitals and influencing chemical properties. This topic sets the stage for exploring spin and total angular momentum.

Angular Momentum Operator and Quantization

Angular Momentum Operator and Commutation Relations

• Angular momentum operator in quantum mechanics represents rotational motion of particles
• Defined as $\hat{L} = \hat{r} \times \hat{p}$, where $\hat{r}$ is the position operator and $\hat{p}$ is the momentum operator
• Components of angular momentum operator in Cartesian coordinates
  • $\hat{L}_x = y\hat{p}_z - z\hat{p}_y$
  • $\hat{L}_y = z\hat{p}_x - x\hat{p}_z$
  • $\hat{L}_z = x\hat{p}_y - y\hat{p}_x$
• Commutation relations for angular momentum components
  • $[\hat{L}_x, \hat{L}_y] = i\hbar\hat{L}_z$
  • $[\hat{L}_y, \hat{L}_z] = i\hbar\hat{L}_x$
  • $[\hat{L}_z, \hat{L}_x] = i\hbar\hat{L}_y$
• Angular momentum components do not commute with each other
• Total angular momentum operator $\hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2$ commutes with all components

Eigenvalues and Quantization of Angular Momentum

• Eigenvalue equation for total angular momentum $\hat{L}^2|\ell,m\rangle = \hbar^2\ell(\ell+1)|\ell,m\rangle$
• Eigenvalue equation for z-component of angular momentum $\hat{L}_z|\ell,m\rangle = \hbar m|\ell,m\rangle$
• Quantization of angular momentum arises from these eigenvalue equations
• Azimuthal quantum number $\ell$ determines the magnitude of angular momentum
  • Takes integer values $\ell = 0, 1, 2, ...$
• Magnetic quantum number $m$ determines the z-component of angular momentum
  • Takes integer values from $-\ell$ to $+\ell$
• Angular momentum eigenfunctions represented by spherical harmonics $Y_{\ell m}(\theta, \phi)$
• Quantization leads to discrete energy levels in atoms and molecules

Quantum Numbers

Azimuthal Quantum Number and Atomic Orbitals

• Azimuthal quantum number $\ell$ determines the shape of atomic orbitals
• Corresponds to subshells in atomic structure (s, p, d, f orbitals)
  • $\ell = 0$ (s orbital): spherical shape
  • $\ell = 1$ (p orbital): dumbbell shape
  • $\ell = 2$ (d orbital): more complex shapes (cloverleaf, doughnut)
• Determines the total orbital angular momentum through $L = \sqrt{\ell(\ell+1)}\hbar$
• Affects electron distribution and chemical bonding in atoms and molecules
• Influences spectroscopic selection rules in atomic transitions

Magnetic Quantum Number and Spatial Orientation

• Magnetic quantum number $m$ specifies the orientation of angular momentum in space
• Takes integer values from $-\ell$ to $+\ell$, including zero
• Determines the z-component of angular momentum through $L_z = m\hbar$
• Represents different spatial orientations of orbitals within a subshell
  • (p orbitals: $p_x$, $p_y$, $p_z$)
• Splits energy levels in the presence of external magnetic fields (Zeeman effect)
• Plays a crucial role in magnetic resonance techniques (NMR, EPR)

Mathematical Tools

Levi-Civita Symbol and Angular Momentum Algebra

• Levi-Civita symbol $\epsilon_{ijk}$ used to express angular momentum commutation relations compactly
• Defined as:
  • $\epsilon_{ijk} = +1$ for even permutations of $(i,j,k)$
  • $\epsilon_{ijk} = -1$ for odd permutations of $(i,j,k)$
  • $\epsilon_{ijk} = 0$ if any two indices are equal
• Allows concise expression of cross products $(\vec{A} \times \vec{B})_i = \epsilon_{ijk}A_jB_k$
• Simplifies angular momentum commutation relations to $[\hat{L}_i, \hat{L}_j] = i\hbar\epsilon_{ijk}\hat{L}_k$
• Facilitates calculations involving rotations and angular momentum in quantum mechanics
• Appears in many areas of physics (electromagnetism, fluid dynamics)

Ladder Operators and Angular Momentum States

• Ladder operators $\hat{L}_+$ and $\hat{L}_-$ raise or lower the magnetic quantum number $m$
• Defined as linear combinations of $\hat{L}_x$ and $\hat{L}_y$:
  • $\hat{L}_+ = \hat{L}_x + i\hat{L}_y$
  • $\hat{L}_- = \hat{L}_x - i\hat{L}_y$
• Action on angular momentum states:
  • $\hat{L}_+|\ell,m\rangle = \hbar\sqrt{\ell(\ell+1)-m(m+1)}|\ell,m+1\rangle$
  • $\hat{L}_-|\ell,m\rangle = \hbar\sqrt{\ell(\ell+1)-m(m-1)}|\ell,m-1\rangle$
• Simplify calculations of matrix elements and transition probabilities
• Used to construct angular momentum eigenstates
• Facilitate the study of angular momentum coupling in multi-particle systems
• Appear in other areas of quantum mechanics (harmonic oscillator, spin)