💫Intro to Quantum Mechanics II Unit 3 – Angular Momentum and Spin in Quantum Mechanics

Key Concepts

• Angular momentum quantization leads to discrete energy levels in atoms and molecules
• Orbital angular momentum $\vec{L}$ arises from the motion of a particle in a central potential
• Spin angular momentum $\vec{S}$ is an intrinsic property of particles like electrons and quarks
• Total angular momentum $\vec{J} = \vec{L} + \vec{S}$ combines orbital and spin contributions
• Commutation relations between angular momentum components $[L_x, L_y] = i\hbar L_z$ and cyclic permutations
• Clebsch-Gordan coefficients describe the coupling of two angular momenta into a total angular momentum
• Wigner-Eckart theorem relates matrix elements of tensor operators to reduced matrix elements

Angular Momentum Basics

• Angular momentum is a vector quantity $\vec{L}$ that describes the rotational motion of a system
• In quantum mechanics, angular momentum is quantized in units of the reduced Planck constant $\hbar$
• The magnitude of angular momentum is given by $|\vec{L}| = \sqrt{l(l+1)}\hbar$, where $l$ is the angular momentum quantum number
• The z-component of angular momentum is quantized as $L_z = m_l\hbar$, where $m_l$ is the magnetic quantum number
  • $m_l$ can take values from $-l$ to $+l$ in integer steps
• Angular momentum operators satisfy the commutation relations $[L_i, L_j] = i\hbar\epsilon_{ijk}L_k$, where $\epsilon_{ijk}$ is the Levi-Civita symbol
• The raising and lowering operators $L_{\pm} = L_x \pm iL_y$ change the magnetic quantum number by $\pm 1$

Orbital Angular Momentum

• Orbital angular momentum describes the angular momentum of a particle moving in a central potential $V(r)$
• The orbital angular momentum operator is $\vec{L} = \vec{r} \times \vec{p}$, where $\vec{r}$ is the position and $\vec{p}$ is the momentum
• Eigenfunctions of $L^2$ and $L_z$ are the spherical harmonics $Y_l^{m_l}(\theta, \phi)$
  • $l$ is the orbital angular momentum quantum number and $m_l$ is the magnetic quantum number
• The radial part of the wavefunction is described by the radial Schrödinger equation, which depends on the potential $V(r)$
• The hydrogen atom is a prime example of a system with orbital angular momentum
  • The energy levels depend on the principal quantum number $n$ and the orbital angular momentum quantum number $l$

Spin Angular Momentum

• Spin is an intrinsic form of angular momentum possessed by particles like electrons, protons, and quarks
• Spin is not related to the spatial motion of the particle but is a fundamental property
• The spin angular momentum operator $\vec{S}$ satisfies the same commutation relations as the orbital angular momentum operator
• For a spin-1/2 particle, the spin quantum number is $s = 1/2$, and the magnetic quantum number $m_s$ can be either $+1/2$ or $-1/2$
• The Pauli matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$ are 2x2 matrices that represent the spin operators for a spin-1/2 particle
• The Stern-Gerlach experiment demonstrated the quantization of spin angular momentum by splitting a beam of silver atoms in an inhomogeneous magnetic field

Addition of Angular Momenta

• When two angular momenta $\vec{J_1}$ and $\vec{J_2}$ are combined, the total angular momentum is given by $\vec{J} = \vec{J_1} + \vec{J_2}$
• The quantum numbers of the total angular momentum $J$ satisfy the triangle inequality: $|J_1 - J_2| \leq J \leq J_1 + J_2$
• The z-component of the total angular momentum is the sum of the individual z-components: $M_J = M_{J_1} + M_{J_2}$
• Clebsch-Gordan coefficients $\langle J_1 M_{J_1} J_2 M_{J_2} | J M_J \rangle$ describe the coupling of two angular momenta
  • They are used to construct the eigenstates of the total angular momentum from the eigenstates of the individual angular momenta
• The addition of orbital and spin angular momenta leads to the total angular momentum $\vec{J} = \vec{L} + \vec{S}$
  • This coupling gives rise to the fine structure in atomic spectra

Applications in Atomic Physics

• Angular momentum plays a crucial role in understanding the structure and properties of atoms
• The electronic configuration of an atom is determined by the orbital and spin angular momenta of its electrons
  • The Pauli exclusion principle states that no two electrons can have the same set of quantum numbers $(n, l, m_l, m_s)$
• The total angular momentum $\vec{J} = \vec{L} + \vec{S}$ determines the fine structure of atomic energy levels
  • The coupling between $\vec{L}$ and $\vec{S}$ is described by the LS (Russell-Saunders) coupling scheme for lighter atoms and the jj coupling scheme for heavier atoms
• The interaction of the atom's total angular momentum with an external magnetic field leads to the Zeeman effect
  • The magnetic field splits the energy levels into multiple sublevels, depending on the magnetic quantum number $M_J$
• Selection rules for atomic transitions are based on the conservation of angular momentum
  • Electric dipole transitions require $\Delta l = \pm 1$ and $\Delta m_l = 0, \pm 1$, while magnetic dipole transitions have different selection rules

Mathematical Tools

• The theory of angular momentum heavily relies on the use of spherical harmonics $Y_l^{m_l}(\theta, \phi)$
  • Spherical harmonics are eigenfunctions of the orbital angular momentum operators $L^2$ and $L_z$
• The Wigner-Eckart theorem simplifies the calculation of matrix elements of tensor operators $T_q^{(k)}$
  • It states that the matrix elements can be expressed as a product of a Clebsch-Gordan coefficient and a reduced matrix element: $\langle j' m' | T_q^{(k)} | j m \rangle = \langle j' || T^{(k)} || j \rangle \langle j' m' | j k ; m q \rangle$
• The Wigner 3j symbols are related to the Clebsch-Gordan coefficients and are used in the calculation of angular momentum coupling and matrix elements
• The Wigner rotation matrices $D_{m'm}^{(j)}(\alpha, \beta, \gamma)$ describe the transformation of angular momentum eigenstates under rotations parametrized by the Euler angles $(\alpha, \beta, \gamma)$
• The Racah coefficients, or Wigner 6j symbols, are used in the calculation of matrix elements involving the coupling of three angular momenta

Common Misconceptions

• Angular momentum is not always conserved in quantum systems, especially when the Hamiltonian is time-dependent or the system is open
• The spin of a particle is not related to its actual rotation in space, but is an intrinsic property
• The orbital and spin angular momenta are not always independent, as they can interact through spin-orbit coupling
• The Stern-Gerlach experiment does not directly measure the spin of a particle, but rather the projection of the spin along a specific axis
• The Wigner-Eckart theorem does not imply that all matrix elements of a tensor operator are zero, but rather that they are proportional to a reduced matrix element
• The addition of angular momenta is not commutative, as $\vec{J_1} + \vec{J_2} \neq \vec{J_2} + \vec{J_1}$ in general
• The magnetic quantum number $m$ is not the same as the spin projection quantum number $m_s$, although they have similar roles in describing the z-component of angular momentum