Glossary

9.1 Orbital angular momentum in quantum mechanics

3 min read•august 9, 2024

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 , 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

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  • Angular momentum operator in quantum mechanics represents rotational motion of particles
  • Defined as L^=r^×p^\hat{L} = \hat{r} \times \hat{p}, where r^\hat{r} is the position operator and p^\hat{p} is the momentum operator
  • Components of angular momentum operator in Cartesian coordinates
    • L^x=yp^z−zp^y\hat{L}_x = y\hat{p}_z - z\hat{p}_y
    • L^y=zp^x−xp^z\hat{L}_y = z\hat{p}_x - x\hat{p}_z
    • L^z=xp^y−yp^x\hat{L}_z = x\hat{p}_y - y\hat{p}_x
  • Commutation relations for angular momentum components
    • [L^x,L^y]=iℏL^z[\hat{L}_x, \hat{L}_y] = i\hbar\hat{L}_z
    • [L^y,L^z]=iℏL^x[\hat{L}_y, \hat{L}_z] = i\hbar\hat{L}_x
    • [L^z,L^x]=iℏL^y[\hat{L}_z, \hat{L}_x] = i\hbar\hat{L}_y
  • Angular momentum components do not commute with each other
  • Total angular momentum operator L^2=L^x2+L^y2+L^z2\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 L^2∣ℓ,m⟩=ℏ2ℓ(ℓ+1)∣ℓ,m⟩\hat{L}^2|\ell,m\rangle = \hbar^2\ell(\ell+1)|\ell,m\rangle
  • Eigenvalue equation for z-component of angular momentum L^z∣ℓ,m⟩=ℏm∣ℓ,m⟩\hat{L}_z|\ell,m\rangle = \hbar m|\ell,m\rangle
  • arises from these eigenvalue equations
  • Azimuthal ℓ\ell determines the magnitude of angular momentum
    • Takes integer values ℓ=0,1,2,...\ell = 0, 1, 2, ...
  • Magnetic quantum number mm determines the z-component of angular momentum
    • Takes integer values from −ℓ-\ell to +ℓ+\ell
  • Angular momentum eigenfunctions represented by spherical harmonics Yℓm(θ,ϕ)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 ℓ\ell determines the shape of atomic orbitals
  • Corresponds to subshells in atomic structure (s, p, d, f orbitals)
    • ℓ=0\ell = 0 (s orbital): spherical shape
    • ℓ=1\ell = 1 (p orbital): dumbbell shape
    • ℓ=2\ell = 2 (d orbital): more complex shapes (cloverleaf, doughnut)
  • Determines the total orbital angular momentum through L=ℓ(ℓ+1)ℏL = \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 mm specifies the orientation of angular momentum in space
  • Takes integer values from −ℓ-\ell to +ℓ+\ell, including zero
  • Determines the z-component of angular momentum through Lz=mℏL_z = m\hbar
  • Represents different spatial orientations of orbitals within a subshell
    • (p orbitals: pxp_x, pyp_y, pzp_z)
  • Splits energy levels in the presence of external magnetic fields ()
  • Plays a crucial role in magnetic resonance techniques (NMR, EPR)

Mathematical Tools

Levi-Civita Symbol and Angular Momentum Algebra

  • Levi-Civita symbol ϵijk\epsilon_{ijk} used to express angular momentum commutation relations compactly
  • Defined as:
    • ϵijk=+1\epsilon_{ijk} = +1 for even permutations of (i,j,k)(i,j,k)
    • ϵijk=−1\epsilon_{ijk} = -1 for odd permutations of (i,j,k)(i,j,k)
    • ϵijk=0\epsilon_{ijk} = 0 if any two indices are equal
  • Allows concise expression of cross products (A⃗×B⃗)i=ϵijkAjBk(\vec{A} \times \vec{B})_i = \epsilon_{ijk}A_jB_k
  • Simplifies angular momentum commutation relations to [L^i,L^j]=iℏϵijkL^k[\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

  • L^+\hat{L}_+ and L^−\hat{L}_- raise or lower the magnetic quantum number mm
  • Defined as linear combinations of L^x\hat{L}_x and L^y\hat{L}_y:
    • L^+=L^x+iL^y\hat{L}_+ = \hat{L}_x + i\hat{L}_y
    • L^−=L^x−iL^y\hat{L}_- = \hat{L}_x - i\hat{L}_y
  • Action on angular momentum states:
    • L^+∣ℓ,m⟩=ℏℓ(ℓ+1)−m(m+1)∣ℓ,m+1⟩\hat{L}_+|\ell,m\rangle = \hbar\sqrt{\ell(\ell+1)-m(m+1)}|\ell,m+1\rangle
    • L^−∣ℓ,m⟩=ℏℓ(ℓ+1)−m(m−1)∣ℓ,m−1⟩\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)

Key Terms to Review (18)

Angular Momentum Operator: The angular momentum operator in quantum mechanics represents the quantized version of angular momentum, which is a fundamental concept for understanding the behavior of particles in rotational motion. It plays a crucial role in various physical phenomena, influencing the properties of systems like atoms and molecules through quantization rules. The operator is defined mathematically and relates to the eigenvalues and eigenstates that determine measurable quantities associated with angular momentum.
Degeneracy: Degeneracy refers to the phenomenon where two or more distinct quantum states share the same energy level. This concept is essential in understanding how systems behave under various conditions and helps explain the structure of atomic and molecular states. In quantum mechanics, degeneracy often arises in the context of symmetries and conservation laws, leading to multiple eigenstates corresponding to a single eigenvalue.
Electron spin: Electron spin is a fundamental property of electrons, representing an intrinsic form of angular momentum. This quantum mechanical property is not related to the actual spinning motion of the electron, but rather describes its magnetic moment and how it behaves in a magnetic field. Spin plays a crucial role in determining the electron's state and its interactions with other particles, significantly influencing atomic structure and behavior.
Erwin Schrödinger: Erwin Schrödinger was an Austrian physicist who made foundational contributions to quantum mechanics, most notably through the development of the Schrödinger equation. His work established a wave-based perspective of particles, linking the concepts of eigenstates and eigenvalues to the behavior of quantum systems and helping to describe the nature of atomic and subatomic particles.
ℓ: In quantum mechanics, ℓ represents the orbital angular momentum quantum number, which quantifies the angular momentum of an electron in an atom. This value can take on non-negative integer values (0, 1, 2, ...), corresponding to the different orbital shapes and orientations of electrons. The value of ℓ is crucial for understanding electron configurations and the overall behavior of atoms in various states.
L = r × p: The equation $$l = r \times p$$ defines the orbital angular momentum in quantum mechanics, where 'l' represents the angular momentum vector, 'r' is the position vector of the particle from a chosen origin, and 'p' is the linear momentum vector of the particle. This relationship highlights how rotational motion can be characterized by the positioning and motion of a particle, bridging concepts from classical mechanics into quantum mechanics.
L² = l(l + 1)ħ²: The equation $$l^{2} = l(l + 1) ext{ħ}^{2}$$ represents the quantization of orbital angular momentum in quantum mechanics. In this equation, 'l' is the orbital angular momentum quantum number, and 'ħ' is the reduced Planck's constant, which links the quantized nature of angular momentum to specific values rather than a continuum. This relationship indicates that angular momentum can only take on certain discrete values, influencing the behavior and properties of quantum systems such as electrons in atoms.
Ladder operators: Ladder operators are mathematical tools used in quantum mechanics to raise or lower the energy levels of quantum states. These operators, often referred to as creation and annihilation operators, play a crucial role in simplifying the calculations of quantum systems, particularly in contexts involving harmonic oscillators and angular momentum. By acting on a state, they help transition between different energy eigenstates, making it easier to understand the behavior of quantum systems.
Mℓ: The quantum number mℓ, known as the magnetic quantum number, specifies the orientation of an electron's orbital in a magnetic field. This value can take on integer values ranging from -l to +l, where l is the azimuthal quantum number that defines the shape of the orbital. The different values of mℓ indicate how many orientations are possible for a given type of orbital, impacting the electron's behavior and distribution in space.
Niels Bohr: Niels Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, particularly with his model of the hydrogen atom. His work established crucial principles that paved the way for modern quantum mechanics, influencing various topics related to wave-particle duality, measurement, and the behavior of particles in potential wells.
Orbital angular momentum: Orbital angular momentum is a fundamental concept in quantum mechanics that describes the rotational motion of a particle about a point, typically the nucleus for electrons in an atom. It quantifies how the electron's motion contributes to its overall angular momentum and is characterized by quantized values, which means it can only take on certain discrete values. This concept is crucial for understanding the behavior of electrons in atomic structures and plays a key role in various physical phenomena, including atomic spectra and the interaction of magnetic fields with matter.
Orbital magnetic moment: The orbital magnetic moment is a property of an electron in an atom that arises from its motion around the nucleus, essentially acting like a tiny current loop. This moment is connected to the electron's orbital angular momentum and contributes to the overall magnetic behavior of the atom, influencing phenomena such as electron interactions and atomic spectra.
P-orbital: A p-orbital is a type of atomic orbital that has a dumbbell shape and can hold a maximum of two electrons. These orbitals are associated with angular momentum and are crucial for understanding how atoms bond with one another. In terms of energy levels, p-orbitals start appearing in the second energy level and come in three different orientations, which are aligned along the x, y, and z axes, allowing for a variety of chemical bonding arrangements.
Pauli Exclusion Principle: The Pauli Exclusion Principle states that no two fermions, such as electrons, can occupy the same quantum state within a quantum system simultaneously. This principle is crucial in understanding the behavior of particles at the microscopic level and has significant implications for the arrangement of electrons in atoms and the structure of matter.
Quantization of angular momentum: Quantization of angular momentum refers to the concept that angular momentum, which describes the rotational motion of a particle, can only take on discrete values rather than any continuous value. This idea is crucial in quantum mechanics, where it leads to the understanding of various physical systems, particularly in defining orbital and spin angular momentum and connecting them to specific mathematical functions.
Quantum Number: A quantum number is a value that describes the quantized states of a quantum system, providing essential information about the energy levels and angular momentum of particles, particularly electrons in atoms. Each quantum number corresponds to specific properties, such as energy, shape, orientation, and spin of an electron, helping to explain the structure of atomic orbitals and their behavior in various potential fields.
S-orbital: An s-orbital is a spherical region in an atom where the probability of finding an electron is highest. Unlike other types of orbitals, s-orbitals have no angular momentum and are characterized by their simple shape, which allows them to hold a maximum of two electrons with opposite spins. This unique feature connects to the concept of orbital angular momentum, highlighting that s-orbitals have an angular momentum quantum number (l) of zero, leading to specific implications in electron configuration and chemical bonding.
Zeeman Effect: The Zeeman Effect is the phenomenon where spectral lines are split into multiple components in the presence of a magnetic field. This effect reveals important insights about atomic structure and the behavior of electrons, demonstrating how external forces can influence energy levels and transitions within atoms, highlighting the inadequacies of classical physics in explaining such microscopic phenomena. Additionally, it connects to concepts of angular momentum and the intrinsic properties of particles, which are central to understanding quantum mechanics.
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