Atoms & Nuclei – Advanced Study Notes
Physics, Engineering Services (ESE) – MIT‑level
Audience: Graduate‑level physics students or advanced undergraduates with a solid background in quantum mechanics, electromagnetism, and special relativity.
Goal: Provide a rigorous, first‑principles treatment of atomic and nuclear structure, with deep theoretical insights, edge cases, and an advanced problem for practice.
1. Overview
| Topic | Key Questions | Typical Tools |
|---|---|---|
| Atomic structure | How are electrons arranged? What determines energy levels? | Schrödinger/Dirac equations, perturbation theory |
| Spectroscopy | How do transitions obey selection rules? | Angular‑momentum algebra, Wigner–Eckart theorem |
| Nuclear structure | What binds protons & neutrons? How does the nucleus behave collectively? | Liquid‑drop model, shell model, mean‑field theory |
| Decay & reactions | How does a nucleus transform? What energy is released? | Quantum tunneling, Fermi’s golden rule, cross‑section formalism |
| Exotic systems | How do muonic atoms, hypernuclei, and quark‑gluon plasmas differ? | Effective field theories, lattice QCD |
2. Atomic Structure from First Principles
2.1 The Non‑Relativistic Schrödinger Equation
For an electron of mass (m_e) in the Coulomb field of a nucleus of charge (+Ze):
[ \hat H \psi(\mathbf r)=E\psi(\mathbf r),\quad \hat H = \frac{\hat{\mathbf p}^2}{2m_e} - \frac{Ze^2}{4\pi\varepsilon_0 r},. ]
- Assumptions: nucleus fixed (Born–Oppenheimer approximation), electron spin neglected.
- Solution: Separation into radial and angular parts; spherical harmonics (Y_{lm}(\theta,\phi)) for angular dependence; radial Schrödinger equation for (R_{nl}(r)).
2.2 Hydrogen Atom
- Energy eigenvalues:
[ E_{n} = -\frac{Z^2 m_e e^4}{8\varepsilon_0^2 h^2}\frac{1}{n^2} = -13.6,Z^2;\text{eV};\frac{1}{n^2};. ]
- Wavefunctions:
[ \psi_{nlm}(\mathbf r)=R_{nl}(r)Y_{lm}(\theta,\phi),. ]
- Degeneracy: For hydrogen, energy depends only on (n), not on (l) or (m). This degeneracy follows from the hidden SU(2) symmetry generated by the Runge–Lenz vector.
2.3 Multi‑Electron Atoms
2.3.1 Hartree–Fock Approximation
- Construct Slater determinant (|\Phi\rangle) of single‑particle spinors (\phi_i(\mathbf r,\sigma)).
- Energy functional:
[ E[\Phi] = \sum_i \langle \phi_i | h_0 | \phi_i \rangle + \frac{1}{2}\sum_{i,j} \left( \langle \phi_i\phi_j | v | \phi_i\phi_j\rangle - \langle \phi_i\phi_j | v | \phi_j\phi_i\rangle \right). ]
- Self‑consistent field (SCF) equations yield effective potentials incorporating exchange.
2.3.2 Configuration Interaction & Correlation
- Beyond HF: expand wavefunction in a basis of configuration state functions (CSFs).
- Correlation energy recovered by including excitations (single, double, etc.).
2.3.3 Fine & Hyperfine Structure
- Relativistic corrections: Dirac equation, Breit interaction.
- Spin–orbit coupling, Darwin term, Lamb shift (QED).
3. Quantum Numbers & Periodic Table
| Symbol | Value | Physical Meaning |
|---|---|---|
| (n) | Principal | Energy level |
| (\ell) | Orbital | Angular momentum |
| (m_\ell) | Magnetic | Projection of (\ell) |
| (s) | Spin | Intrinsic spin |
| (m_s) | Spin projection | |
| (j) | Total angular momentum | (j = \ell \pm s) |
| (m_j) | Projection of (j) |
- Pauli Exclusion Principle: no two electrons may occupy identical set of quantum numbers.
- Aufbau Principle: electrons fill orbitals in order of increasing (n + \ell); ties resolved by increasing (n).
- Hund’s Rule: maximize total spin (S) within degenerate subshells.
4. Spectroscopic Transitions
4.1 Selection Rules
- Electric dipole ((E1)) transitions: (\Delta l = \pm 1), (\Delta m = 0, \pm 1), (\Delta s = 0).
- Magnetic dipole ((M1)) and higher multipoles: more restrictive.
4.2 Wigner–Eckart Theorem
[ \langle \alpha'