4.4 Angular Momentum and Hydrogen Atom

Angular momentum in quantum mechanics is quantized, unlike its classical counterpart. This fundamental property shapes the behavior of particles at the atomic level, influencing electronic structure and spectral characteristics.

The hydrogen atom serves as a crucial model in quantum mechanics. Its Schrödinger equation solution reveals quantized energy levels and wave functions, providing insights into atomic structure and spectroscopic observations.

Quantum Mechanical Angular Momentum

Angular Momentum in Quantum Mechanics

Top images from around the web for Angular Momentum in Quantum Mechanics

• 

  Angular momentum in quantum mechanics | Introduction to the physics of atoms, molecules and photons View original

  Is this image relevant?

• 

  30.8 Quantum Numbers and Rules – College Physics View original

  Is this image relevant?

• 

  Angular Momentum and Its Conservation | Physics View original

  Is this image relevant?

• 

  Angular momentum in quantum mechanics | Introduction to the physics of atoms, molecules and photons View original

  Is this image relevant?

• 

  30.8 Quantum Numbers and Rules – College Physics View original

  Is this image relevant?

1 of 3

Top images from around the web for Angular Momentum in Quantum Mechanics

• 

  Angular momentum in quantum mechanics | Introduction to the physics of atoms, molecules and photons View original

  Is this image relevant?

• 

  30.8 Quantum Numbers and Rules – College Physics View original

  Is this image relevant?

• 

  Angular Momentum and Its Conservation | Physics View original

  Is this image relevant?

• 

  Angular momentum in quantum mechanics | Introduction to the physics of atoms, molecules and photons View original

  Is this image relevant?

• 

  30.8 Quantum Numbers and Rules – College Physics View original

  Is this image relevant?

1 of 3

• Angular momentum is a fundamental property of particles in quantum mechanics, analogous to classical angular momentum but with distinct quantized values
• The angular momentum operators (Lx, Ly, Lz) are derived from the position and momentum operators in quantum mechanics
• The total angular momentum operator (L2) is the sum of the squares of the individual angular momentum operators (Lx2 + Ly2 + Lz2)
• The eigenvalues of the total angular momentum operator (L2) are given by $l(l+1)ħ^2$, where l is the angular momentum quantum number ($l = 0, 1, 2, ...$)
• The eigenvalues of the z-component of the angular momentum operator (Lz) are given by $mħ$, where m is the magnetic quantum number ($m = -l, -l+1, ..., l-1, l$)

Spherical Harmonics as Eigenfunctions

• The eigenfunctions of the angular momentum operators are the spherical harmonics (Yl,m), which are solutions to the angular part of the Schrödinger equation
• Spherical harmonics describe the angular distribution of the wave function in quantum mechanics
• Examples of spherical harmonics include $Y_{0,0}$ (s orbital), $Y_{1,0}$ (pz orbital), $Y_{1,±1}$ (px and py orbitals)

Angular Momentum Operators and Eigenfunctions

Applying Angular Momentum Operators

• The angular momentum operators (Lx, Ly, Lz) and the total angular momentum operator (L2) act on the spherical harmonics (Yl,m) to yield eigenvalues and eigenfunctions
• The spherical harmonics are orthonormal functions that form a complete basis set for describing the angular distribution of particles in quantum mechanics
• The commutation relations between the angular momentum operators ($[Lx, Ly] = iħLz, [Ly, Lz] = iħLx, [Lz, Lx] = iħLy$) lead to the uncertainty principle for angular momentum components

Generating and Coupling Angular Momenta

• The raising and lowering operators (L+ and L-) can be used to generate spherical harmonics with different magnetic quantum numbers (m) for a given angular momentum quantum number (l)
• The raising operator increases m by 1, while the lowering operator decreases m by 1
• The Clebsch-Gordan coefficients describe the coupling of angular momenta in quantum mechanics, allowing the addition of angular momentum quantum numbers for composite systems
• Example: coupling two spin-1/2 particles leads to a singlet state (total spin 0) and a triplet state (total spin 1)

Hydrogen Atom Schrödinger Equation

Hydrogen Atom Model

• The hydrogen atom consists of a single electron bound to a proton, serving as a fundamental model system in quantum mechanics
• The Schrödinger equation for the hydrogen atom involves the kinetic energy of the electron, the potential energy due to the Coulomb interaction between the electron and proton, and the energy eigenvalue
• The solution to the Schrödinger equation for the hydrogen atom yields the energy eigenvalues and the corresponding wave functions (orbitals)

Solving the Schrödinger Equation

• The energy eigenvalues of the hydrogen atom are given by $E_n = -13.6 eV / n^2$, where n is the principal quantum number ($n = 1, 2, 3, ...$)
• The wave functions of the hydrogen atom are characterized by three quantum numbers: the principal quantum number (n), the angular momentum quantum number (l), and the magnetic quantum number (m)
• The radial part of the hydrogen atom wave functions is described by the associated Laguerre polynomials, while the angular part is described by the spherical harmonics
• Example: the ground state (1s orbital) wave function is $ψ_{1,0,0} = (1/√π)(1/a_0^{3/2})e^{-r/a_0}$, where $a_0$ is the Bohr radius

Quantum Numbers and Electronic Structure of Hydrogen

Quantum Numbers and Orbital Shapes

• The principal quantum number (n) determines the energy level and the size of the orbital, with larger n corresponding to higher energy and larger orbital size
• The angular momentum quantum number (l) determines the shape of the orbital, with l = 0 (s orbitals), l = 1 (p orbitals), l = 2 (d orbitals), and so on
• The magnetic quantum number (m) determines the orientation of the orbital in space relative to an external magnetic field, with m ranging from -l to +l
• The electron spin quantum number (ms) describes the intrinsic angular momentum of the electron, with values of +1/2 and -1/2

Electronic Configuration and Spectral Lines

• The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers (n, l, m, ms), leading to the electronic configuration of the hydrogen atom
• The ground state of the hydrogen atom has the electron in the 1s orbital (n = 1, l = 0, m = 0), while excited states involve the electron occupying higher energy orbitals (2s, 2p, 3s, 3p, 3d, etc.)
• The electronic transitions between different energy levels in the hydrogen atom give rise to the characteristic spectral lines observed in the hydrogen spectrum, such as the Lyman, Balmer, and Paschen series
• Example: the Balmer series corresponds to transitions from higher energy levels to the n = 2 level, with the Hα line (656.3 nm) resulting from the transition from n = 3 to n = 2