Glossary

3.4 Angular momentum in quantum mechanics

4 min read•august 7, 2024

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 , and . It also covers the , , and the coupling of different types of angular momenta in complex systems.

Angular Momentum Operators

Defining Angular Momentum in Quantum Mechanics

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  • L^\hat{L} represents the angular momentum of a particle in quantum mechanics
  • L^\hat{L} describes the angular momentum associated with the motion of a particle in space
    • Depends on the particle's position r^\hat{r} and linear momentum p^\hat{p}
    • Components of orbital angular momentum: L^x\hat{L}_x, L^y\hat{L}_y, and L^z\hat{L}_z
  • S^\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: S^x\hat{S}_x, S^y\hat{S}_y, and S^z\hat{S}_z
  • J^\hat{J} is the sum of orbital and spin angular momenta: J^=L^+S^\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
    • [L^i,L^j]=iℏϵijkL^k[\hat{L}_i, \hat{L}_j] = i\hbar\epsilon_{ijk}\hat{L}_k and [S^i,S^j]=iℏϵijkS^k[\hat{S}_i, \hat{S}_j] = i\hbar\epsilon_{ijk}\hat{S}_k, where ϵijk\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)ℏ2l(l+1)\hbar^2, where ll is a non-negative integer
    • Spin angular momentum eigenvalues: s(s+1)ℏ2s(s+1)\hbar^2, where ss is a half-integer (1/2, 3/2, ...)
    • mlm_l for orbital angular momentum: −l≤ml≤l-l \leq m_l \leq l
    • Magnetic quantum number msm_s for spin angular momentum: −s≤ms≤s-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: ∣L^∣=l(l+1)ℏ|\hat{L}| = \sqrt{l(l+1)}\hbar, where ll is the orbital quantum number
    • Magnitude of spin angular momentum: ∣S^∣=s(s+1)ℏ|\hat{S}| = \sqrt{s(s+1)}\hbar, where ss is the
  • Projection of angular momentum along a specific axis (usually z-axis) is also quantized
    • L^z\hat{L}_z eigenvalues: mlℏm_l\hbar, where mlm_l is the magnetic quantum number for orbital angular momentum
    • S^z\hat{S}_z eigenvalues: msℏm_s\hbar, where msm_s is the magnetic quantum number for spin angular momentum

Spherical Harmonics and Angular Wave Functions

  • Spherical harmonics Ylml(θ,ϕ)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 ll and the magnetic quantum number mlm_l
    • Describe the angular distribution of the particle's probability density
  • Properties of spherical harmonics:
    • Orthonormal: ∫Ylml∗(θ,ϕ)Yl′ml′(θ,ϕ)dΩ=δll′δmlml′\int Y_l^{m_l*}(\theta, \phi)Y_{l'}^{m_l'}(\theta, \phi) d\Omega = \delta_{ll'}\delta_{m_lm_l'}
    • Parity: Ylml(π−θ,ϕ+π)=(−1)lYlml(θ,ϕ)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: Δl=±1\Delta l = \pm 1 and Δml=0,±1\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:
    • (Russell-Saunders coupling): Orbital angular momenta of individual particles are added to form a total orbital angular momentum L^\hat{L}, and spin angular momenta are added to form a total spin angular momentum S^\hat{S}. Then, L^\hat{L} and S^\hat{S} are coupled to form the total angular momentum J^\hat{J}.
    • : 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.
  • ⟨j1m1;j2m2∣JM⟩\langle j_1 m_1; j_2 m_2 | J M \rangle describe the coupling of two angular momenta j^1\hat{j}_1 and j^2\hat{j}_2 to form a total angular momentum J^\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
  • 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

Key Terms to Review (23)

Angular momentum operator: The angular momentum operator is a key concept in quantum mechanics, represented by the operator that corresponds to the physical observable of angular momentum in a quantum system. It plays a crucial role in understanding rotational motion and is defined in terms of position and momentum operators, reflecting the quantization of angular momentum. This operator helps describe the behavior of particles with rotational symmetry, connecting deeply with both wavefunctions and eigenvalues in quantum mechanics.
Azimuthal Quantum Number: The azimuthal quantum number, often denoted as 'l', is a quantum number that describes the angular momentum of an electron in an atom and its subshell designation. It determines the shape of the electron's orbital and can take on integer values from 0 to (n-1), where 'n' is the principal quantum number. This number is crucial in understanding the distribution of electrons within an atom and their energy levels, linking directly to the angular momentum in quantum mechanics.
Clebsch-Gordan Coefficients: Clebsch-Gordan coefficients are mathematical factors that arise in the addition of angular momentum in quantum mechanics, helping to combine different angular momentum states. They provide a way to express the total angular momentum of a system in terms of its individual components, which is essential for understanding phenomena like atomic and molecular interactions. The coefficients also play a crucial role in simplifying calculations related to quantum states when multiple angular momenta are involved.
Commutation Relations: Commutation relations are mathematical expressions that describe how different operators act on quantum states in a way that reflects the fundamental uncertainties inherent in quantum mechanics. These relations are crucial in understanding the properties of observable quantities like position and momentum, as well as angular momentum, by establishing whether two operators can be measured simultaneously without affecting each other.
Conservation of Angular Momentum: Conservation of angular momentum is a principle stating that the total angular momentum of a closed system remains constant over time, provided that no external torque acts on it. This concept is crucial in understanding the behavior of rotating systems, as it illustrates how changes in the distribution of mass and rotational speed can affect overall motion without altering the total angular momentum.
Eigenvalues: Eigenvalues are special scalar values that arise in linear transformations and play a crucial role in various mathematical and physical applications, particularly in quantum mechanics. They are associated with eigenvectors, representing the factors by which these eigenvectors are stretched or compressed during a transformation. This concept connects deeply with matrices, vectors, and the fundamental principles governing quantum systems.
Hydrogen Atom: The hydrogen atom is the simplest and most abundant element in the universe, consisting of one proton and one electron. Its simplicity allows it to serve as a fundamental model for understanding more complex atomic systems and key concepts in quantum mechanics, including angular momentum, the Schrödinger equation, and wave functions. The hydrogen atom's behavior illustrates essential principles that govern atomic structure and interactions.
Jj coupling: jj coupling is a model used in quantum mechanics to describe the coupling of angular momentum between particles, particularly in multi-electron systems. It focuses on how individual angular momenta of particles combine, where each particle's total angular momentum is considered separately before being combined with other particles. This approach is especially relevant for understanding the energy levels and spectral properties of atoms with multiple electrons, where the interactions between their spins and orbital angular momenta play a critical role.
Ls coupling: ls coupling, or Russell-Saunders coupling, is a method used to describe the interactions between the angular momentum of electrons in atoms. In this framework, the total orbital angular momentum ($$L$$) and total spin angular momentum ($$S$$) of an atom are coupled together to form the total angular momentum ($$J$$). This approach is particularly useful for understanding the energy levels and spectral characteristics of light elements with weak interactions among their electrons.
Magnetic quantum number: The magnetic quantum number, denoted as $$m_l$$, describes the orientation of an atomic orbital in space relative to an external magnetic field. It is one of the four quantum numbers that specify the state of an electron in an atom, connecting to angular momentum, the arrangement of electrons in multi-electron atoms, and the solutions to eigenvalue equations that represent quantum systems.
Niels Bohr: Niels Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, notably through his model of the hydrogen atom. His work laid the groundwork for explaining the behavior of electrons in atoms, which connects directly to the formulation of quantum mechanical operators and observables, as well as principles like the uncertainty principle and angular momentum in quantum mechanics.
Orbital Angular Momentum: Orbital angular momentum refers to the momentum associated with an object moving in a circular path around a central point, specifically in the context of quantum mechanics, it describes the angular momentum of electrons in atoms. This concept is crucial for understanding electron configurations, atomic shapes, and how these properties influence chemical behavior. In quantum mechanics, orbital angular momentum is quantized and is characterized by quantum numbers, which determine the allowed states of an electron.
Pauli Exclusion Principle: The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of quantum numbers. This fundamental principle helps explain the arrangement of electrons in atoms, which in turn influences their angular momentum and plays a critical role in determining the electron configurations of multi-electron atoms, molecular orbitals, and the behavior of fermions in quantum statistics.
Quantization of Angular Momentum: Quantization of angular momentum refers to the principle that angular momentum in quantum mechanics can only take on discrete values, rather than a continuous range. This concept is fundamental in understanding how particles, such as electrons, behave in atomic and molecular systems, leading to the idea that certain states are permitted while others are forbidden, shaping the structure of atoms and their energy levels.
Raman Scattering: Raman scattering is a phenomenon in which light interacts with molecular vibrations, leading to a shift in the energy of the scattered light. This effect provides insights into the molecular structure and dynamics, making it a powerful tool for characterization. The interaction is crucial in understanding angular momentum, as the change in energy levels relates to vibrational transitions and their associated quantum states.
Spectroscopy: Spectroscopy is the study of the interaction between matter and electromagnetic radiation, which helps to identify and analyze the properties of substances based on their spectral characteristics. By examining how matter absorbs, emits, or scatters light at different wavelengths, spectroscopy provides insights into molecular structure, energy levels, and dynamics. This technique is crucial in understanding various phenomena in quantum mechanics, perturbation theory, and the role of symmetry in molecular systems.
Spherical Harmonics: Spherical harmonics are mathematical functions that arise in the solution of problems involving angular momentum in quantum mechanics, particularly when dealing with systems that exhibit spherical symmetry. These functions are defined on the surface of a sphere and are crucial for describing the angular part of wave functions in quantum mechanics, allowing for the quantization of angular momentum. They play a key role in the analysis of atomic orbitals, helping to explain the shapes and orientations of these orbitals.
Spin Angular Momentum: Spin angular momentum is a fundamental property of quantum particles that describes their intrinsic angular momentum, independent of any external motion. This form of angular momentum arises from the quantum mechanical spin property, which is not related to literal spinning but rather to a particle's inherent quantum state. Understanding spin angular momentum is crucial in quantum mechanics, as it plays a significant role in the behavior of particles and their interactions.
Spin Quantum Number: The spin quantum number is a fundamental quantum number that describes the intrinsic angular momentum, or 'spin,' of a particle, typically an electron. It takes on values of +1/2 or -1/2, indicating the two possible orientations of the spin. This property is crucial in understanding various phenomena in quantum mechanics, including the behavior of particles in a magnetic field and the application of the Pauli exclusion principle to fermions.
Total Angular Momentum: Total angular momentum is a quantum mechanical property that quantifies the rotational motion of a system. It is the vector sum of both the orbital angular momentum and the intrinsic spin angular momentum of particles, playing a crucial role in determining the behavior of quantum systems under rotational transformations.
Uncertainty Principle: The uncertainty principle is a fundamental concept in quantum mechanics that states it is impossible to simultaneously know both the exact position and exact momentum of a particle. This principle highlights the inherent limitations of measurement at the quantum level, emphasizing that observing one property more precisely results in increased uncertainty in the other property. It is a crucial aspect that influences various concepts in quantum mechanics, including energy levels, angular momentum, and the behavior of electrons in atoms.
Werner Heisenberg: Werner Heisenberg was a German physicist known for formulating the Heisenberg Uncertainty Principle, which fundamentally altered our understanding of quantum mechanics. His work established that certain pairs of physical properties, like position and momentum, cannot be simultaneously known with arbitrary precision. This principle has deep implications in the study of commutation relations and angular momentum, highlighting the inherent limitations in measuring quantum systems.
Zeeman Effect: The Zeeman Effect refers to the splitting of a spectral line into multiple components in the presence of a magnetic field. This phenomenon occurs due to the interaction between the magnetic field and the magnetic dipole moment associated with angular momentum in quantum systems, revealing critical insights into atomic structure and electron configurations.
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