⚛️Atomic Physics Unit 3 – One–Electron Atoms

Key Concepts and Definitions

• Atomic structure consists of a positively charged nucleus surrounded by negatively charged electrons
• Electrons occupy discrete energy levels or shells around the nucleus
• Quantum numbers describe the state and properties of an electron in an atom
  • Principal quantum number (n) represents the energy level or shell
  • Angular momentum quantum number (l) describes the subshell or orbital shape
  • Magnetic quantum number (m) specifies the orientation of the orbital in space
  • Spin quantum number (s) indicates the intrinsic angular momentum of the electron
• Spectral lines result from electrons transitioning between energy levels, emitting or absorbing photons
• Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers
• Hund's rule determines the order in which electrons fill orbitals within a subshell
• Aufbau principle guides the electron configuration of an atom, filling orbitals from lowest to highest energy

Historical Background

• Bohr model introduced the concept of stationary states and quantized energy levels in atoms
  • Explained the discrete spectral lines observed in the hydrogen atom
  • Postulated that electrons orbit the nucleus in circular paths with fixed radii
• Sommerfeld extended the Bohr model to include elliptical orbits and introduced additional quantum numbers
• Wave-particle duality, proposed by de Broglie, suggested that particles can exhibit wave-like properties
  • Wavelength is inversely proportional to momentum ($\lambda = h/p$)
• Heisenberg's uncertainty principle states that the position and momentum of a particle cannot be simultaneously determined with arbitrary precision
• Schrödinger equation, a fundamental equation in quantum mechanics, describes the behavior of a quantum system
  • Solutions to the Schrödinger equation for the hydrogen atom yield the electron's wavefunction and energy levels

Quantum Mechanical Model

• Electrons are described by wavefunctions ($\Psi$) that represent the probability distribution of finding an electron in a given region of space
• Born interpretation relates the square of the wavefunction's absolute value ($|\Psi|^2$) to the probability density of an electron's location
• Schrödinger equation for the hydrogen atom is solved using separation of variables, leading to the electron's wavefunction and energy levels
  • Wavefunction is a product of radial and angular components ($\Psi(r,\theta,\phi) = R(r)Y(\theta,\phi)$)
• Quantum numbers emerge as a consequence of solving the Schrödinger equation
  • Principal quantum number (n) arises from the radial component of the wavefunction
  • Angular momentum quantum numbers (l and m) result from the angular component of the wavefunction
• Probability density plots visually represent the likelihood of finding an electron in various regions around the nucleus

Energy Levels and Spectra

• Energy levels in a one-electron atom are given by the Bohr formula: $E_n = -13.6 \text{ eV}/n^2$
  • n is the principal quantum number, and the negative sign indicates a bound state
• Transitions between energy levels result in the emission or absorption of photons with specific frequencies
  • Frequency of the photon is related to the energy difference between levels: $\Delta E = hf$
• Selection rules govern the allowed transitions between energy levels based on changes in quantum numbers
  • Electric dipole transitions require $\Delta l = \pm 1$ and $\Delta m = 0, \pm 1$
• Spectral series (Lyman, Balmer, Paschen) correspond to transitions ending in specific energy levels
  • Lyman series involves transitions to the ground state (n = 1)
  • Balmer series involves transitions to the first excited state (n = 2)
• Fine structure and hyperfine structure result from additional interactions within the atom
  • Fine structure arises from the coupling of the electron's orbital angular momentum and spin
  • Hyperfine structure originates from the interaction between the electron and the nuclear spin

Angular Momentum and Spin

• Orbital angular momentum (L) is associated with the electron's motion around the nucleus
  • Magnitude of orbital angular momentum is quantized: $|L| = \sqrt{l(l+1)}\hbar$
  • Projection of orbital angular momentum along a chosen axis is quantized: $L_z = m_l\hbar$
• Spin angular momentum (S) is an intrinsic property of the electron
  • Magnitude of spin angular momentum is fixed: $|S| = \sqrt{3/4}\hbar$
  • Projection of spin angular momentum along a chosen axis is quantized: $S_z = m_s\hbar$, where $m_s = \pm 1/2$
• Total angular momentum (J) is the vector sum of orbital and spin angular momenta: $J = L + S$
  • Magnitude of total angular momentum is quantized: $|J| = \sqrt{j(j+1)}\hbar$
  • Projection of total angular momentum along a chosen axis is quantized: $J_z = m_j\hbar$
• Spin-orbit coupling is the interaction between the electron's orbital angular momentum and spin
  • Leads to the fine structure splitting of energy levels
  • Strength of the coupling depends on the atomic number and the electron's proximity to the nucleus

Electron Orbitals

• Orbitals are three-dimensional regions of space where an electron is likely to be found
  • Characterized by the quantum numbers n, l, and m
  • Shapes of orbitals are determined by the angular component of the wavefunction
• s orbitals (l = 0) are spherically symmetric and have no angular dependence
  • Probability density is highest near the nucleus and decreases with increasing distance
• p orbitals (l = 1) have a dumbbell shape and are oriented along the x, y, or z axes
  • Characterized by a node at the nucleus and two lobes with opposite phases
• d orbitals (l = 2) have more complex shapes, including cloverleaf and double-dumbbell configurations
  • Possess additional angular nodes compared to s and p orbitals
• f orbitals (l = 3) have even more intricate shapes and a greater number of angular nodes
• Hybrid orbitals are linear combinations of atomic orbitals that better describe the bonding in molecules
  • sp, sp2, and sp3 hybridization are common in organic compounds

Applications and Experiments

• Atomic spectroscopy techniques, such as absorption and emission spectroscopy, rely on the discrete energy levels in atoms
  • Used to identify elements and determine their concentrations in a sample
  • Provide information about the electronic structure and transitions in atoms
• Laser cooling and trapping techniques exploit the interaction between atoms and light
  • Doppler cooling uses the Doppler effect to slow down atoms by selectively absorbing photons
  • Magneto-optical traps (MOTs) combine laser cooling with magnetic fields to confine atoms in a small region of space
• Precision measurements, such as the determination of fundamental constants, benefit from the understanding of one-electron atoms
  • Fine structure constant ($\alpha$) can be measured using the energy levels of hydrogen-like atoms
  • Rydberg constant ($R_\infty$) is determined from the spectral lines of hydrogen and other one-electron systems
• Quantum computing and information processing harness the properties of one-electron atoms
  • Qubits can be realized using the spin states or energy levels of trapped ions or neutral atoms
  • Entanglement and superposition, fundamental concepts in quantum mechanics, are explored using one-electron systems

Advanced Topics and Current Research

• Relativistic effects become significant in heavy one-electron atoms, such as high-Z hydrogen-like ions
  • Dirac equation, which incorporates special relativity, is used to describe the electron's behavior
  • Fine structure and hyperfine structure are modified by relativistic corrections
• Quantum electrodynamics (QED) provides a more accurate description of one-electron atoms
  • Accounts for the interaction between the electron and the quantized electromagnetic field
  • Lamb shift, a small energy difference between the 2S1/2 and 2P1/2 states in hydrogen, is explained by QED
• Many-body effects, such as electron correlation and screening, become important in multi-electron atoms
  • Configuration interaction and coupled-cluster methods are used to include these effects in atomic structure calculations
• Attosecond science explores the ultrafast dynamics of electrons in atoms and molecules
  • Attosecond laser pulses can probe the real-time motion of electrons and the formation of chemical bonds
• Fundamental symmetries, such as parity and time-reversal symmetry, can be tested using one-electron atoms
  • Searches for electric dipole moments (EDMs) in atoms can provide insights into physics beyond the Standard Model
• Antimatter studies benefit from the understanding of one-electron atoms
  • Antihydrogen, the bound state of an antiproton and a positron, is being investigated to test the symmetry between matter and antimatter