⚛️Quantum Mechanics Unit 5 – Angular Momentum and Spin

Key Concepts

• Angular momentum plays a crucial role in quantum mechanics, describing the rotational motion and intrinsic spin of particles
• Classical angular momentum is a vector quantity, while quantum angular momentum is quantized and follows specific rules
• Orbital angular momentum arises from the motion of a particle around a central point, such as an electron orbiting an atomic nucleus
• Spin angular momentum is an intrinsic property of particles, not related to spatial motion, and is characterized by the spin quantum number
• The total angular momentum of a system is the vector sum of the orbital and spin angular momenta of its constituent particles
• Angular momentum is conserved in closed systems, leading to the selection rules that govern transitions between quantum states
• The commutation relations between angular momentum operators give rise to the uncertainty principle for angular momentum components

Classical Angular Momentum Review

• In classical mechanics, angular momentum is defined as the cross product of the position vector and the linear momentum vector: $\vec{L} = \vec{r} \times \vec{p}$
• The magnitude of the angular momentum is given by $L = |\vec{r}| |\vec{p}| \sin \theta$, where $\theta$ is the angle between the position and momentum vectors
• Angular momentum is a conserved quantity in the absence of external torques, as stated by the law of conservation of angular momentum
• The rotational kinetic energy of a rigid body is related to its angular momentum and moment of inertia: $K = \frac{1}{2} I \omega^2 = \frac{L^2}{2I}$
• The torque on an object is equal to the time rate of change of its angular momentum: $\vec{\tau} = \frac{d\vec{L}}{dt}$
• In a central force field, such as the gravitational field, the angular momentum of a particle is conserved, resulting in stable orbits (planets orbiting the Sun)

Quantum Angular Momentum

• In quantum mechanics, angular momentum is quantized and can only take on discrete values
• The angular momentum operators $\hat{L}_x$, $\hat{L}_y$, and $\hat{L}_z$ do not commute with each other, leading to the uncertainty principle for angular momentum components
• The eigenvalues of the square of the angular momentum operator $\hat{L}^2$ are given by $l(l+1)\hbar^2$, where $l$ is the angular momentum quantum number
• The eigenvalues of the $z$-component of the angular momentum operator $\hat{L}_z$ are given by $m_l \hbar$, where $m_l$ is the magnetic quantum number, ranging from $-l$ to $l$ in integer steps
• The commutation relations between angular momentum operators are: $[\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z$, $[\hat{L}_y, \hat{L}_z] = i\hbar \hat{L}_x$, and $[\hat{L}_z, \hat{L}_x] = i\hbar \hat{L}_y$
• The raising and lowering operators, $\hat{L}_+$ and $\hat{L}_-$, are used to change the magnetic quantum number by $\pm 1$, respectively

Orbital Angular Momentum

• Orbital angular momentum describes the angular momentum of a particle due to its motion around a central point, such as an electron orbiting an atomic nucleus
• The orbital angular momentum quantum number $l$ determines the shape of the atomic orbital and the magnitude of the orbital angular momentum
• The magnetic quantum number $m_l$ determines the orientation of the orbital angular momentum vector in space relative to a chosen axis (usually the $z$-axis)
• The shape of the atomic orbitals is related to the orbital angular momentum quantum number:
  • $l = 0$ corresponds to s-orbitals (spherical)
  • $l = 1$ corresponds to p-orbitals (dumbbell-shaped)
  • $l = 2$ corresponds to d-orbitals (various shapes)
• The number of orbitals with a given $l$ value is equal to $2l + 1$, corresponding to the possible values of $m_l$
• The energy levels of an atom are partially determined by the orbital angular momentum, with higher $l$ values generally corresponding to higher energies

Spin Angular Momentum

• Spin angular momentum is an intrinsic property of particles, not related to their spatial motion
• The spin quantum number $s$ determines the magnitude of the spin angular momentum, with the eigenvalues of $\hat{S}^2$ given by $s(s+1)\hbar^2$
• Fermions (particles with half-integer spin, such as electrons and protons) obey the Pauli exclusion principle, while bosons (particles with integer spin, such as photons) do not
• The $z$-component of the spin angular momentum has eigenvalues $m_s \hbar$, where $m_s$ ranges from $-s$ to $s$ in integer steps
• Electrons have a spin quantum number of $s = 1/2$, with two possible values for $m_s$: $+1/2$ (spin up) and $-1/2$ (spin down)
• The magnetic moment of a particle is related to its spin angular momentum, leading to the splitting of energy levels in the presence of a magnetic field (Zeeman effect)

Addition of Angular Momenta

• When two or more angular momenta are combined, the resulting total angular momentum is found by vector addition
• The total angular momentum quantum number $j$ ranges from $|j_1 - j_2|$ to $j_1 + j_2$ in integer steps, where $j_1$ and $j_2$ are the individual angular momentum quantum numbers
• The $z$-component of the total angular momentum has eigenvalues $m_j \hbar$, where $m_j$ ranges from $-j$ to $j$ in integer steps
• The Clebsch-Gordan coefficients determine the relative contributions of the individual angular momenta to the total angular momentum eigenstates
• The addition of orbital and spin angular momenta leads to the fine structure of atomic energy levels, such as the splitting of the p-orbitals in hydrogen
• The addition of angular momenta is essential for understanding the coupling schemes in multi-electron atoms (LS coupling, jj coupling)

Applications in Atomic Physics

• Angular momentum plays a crucial role in determining the energy levels and spectra of atoms
• The selection rules for electric dipole transitions between atomic states are based on the conservation of angular momentum:
  • $\Delta l = \pm 1$
  • $\Delta m_l = 0, \pm 1$
  • $\Delta s = 0$
• The fine structure of atomic energy levels arises from the coupling between the orbital and spin angular momenta of electrons
• The hyperfine structure results from the interaction between the electron's angular momentum and the nuclear spin angular momentum
• The Zeeman effect describes the splitting of atomic energy levels in the presence of an external magnetic field, which interacts with the magnetic moments associated with orbital and spin angular momenta
• The Stern-Gerlach experiment demonstrated the quantization of spin angular momentum by observing the deflection of silver atoms in a non-uniform magnetic field

Problem-Solving Strategies

• Identify the type of angular momentum involved in the problem (orbital, spin, or total) and the relevant quantum numbers
• Use the eigenvalue equations for $\hat{L}^2$, $\hat{L}_z$, $\hat{S}^2$, and $\hat{S}_z$ to determine the possible values of the angular momentum and its components
• Apply the selection rules for angular momentum when considering transitions between quantum states
• Use the Clebsch-Gordan coefficients or the vector addition rules to determine the total angular momentum when combining multiple angular momenta
• Consider the symmetry of the system and any conserved quantities, such as total angular momentum, when simplifying the problem
• Apply perturbation theory to calculate the energy level shifts and splittings due to interactions such as spin-orbit coupling or the Zeeman effect
• Use the commutation relations between angular momentum operators to derive any necessary equations or to determine the compatibility of simultaneous eigenstates