Atoms & Nuclei
MIT‑style treatment – first principles, rigorous reasoning, deeper theory, edge cases
Audience – Advanced undergraduate / beginning graduate physics students preparing for the MHT CET and interested in a deep, theoretical understanding of atomic and nuclear physics.
Prerequisites – Classical mechanics, electromagnetism, basic quantum mechanics (wavefunctions, operators, perturbation theory), special relativity, and introductory nuclear physics.
1. Atomic Structure
| Concept | Key Equation | Significance |
|---|---|---|
| Bohr Radius | (a_0=\frac{4\pi\varepsilon_0\hbar^2}{m_e e^2}) | Scale of the hydrogen atom. |
| Rydberg Formula | (E_n=-R_\infty \frac{Z_{\text{eff}}^2}{n^2}) | Energy levels of hydrogen‑like ions. |
| Fine‑structure constant | (\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}\approx\frac{1}{137}) | Small, dimensionless coupling constant. |
| Dirac Equation | ((\boldsymbol{\alpha}\cdot\mathbf{p}+\beta m c^2+\mathbb{V})\psi=E\psi) | Relativistic treatment of electrons. |
1.1 The Schrödinger Equation for Hydrogen‑Like Systems
In the non‑relativistic limit the time‑independent Schrödinger equation reads
[ \left[-\frac{\hbar^2}{2m_e}\nabla^2 - \frac{Z e^2}{4\pi\varepsilon_0 r}\right]\psi_{n\ell m}(\mathbf{r}) = E_n \psi_{n\ell m}(\mathbf{r}). ]
Solving by separation of variables in spherical coordinates yields the well‑known eigenfunctions
[ \psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r),Y_{\ell m}(\theta,\phi), ] with radial parts
[ R_{n\ell}(r) = N_{n\ell} , \rho^\ell e^{-\rho/2} L_{n-\ell-1}^{2\ell+1}(\rho),\qquad \rho=\frac{2Zr}{na_0}, ] and energies
[ E_n = -\frac{Z^2 m_e e^4}{2(4\pi\varepsilon_0)^2 \hbar^2}\frac{1}{n^2} = -R_\infty \frac{Z^2}{n^2}. ]
Rigorous derivation
- The Coulomb potential is central → (L_z) and (L^2) commute with the Hamiltonian.
- Angular part solved by spherical harmonics (Y_{\ell m}).
- Radial equation transformed to a confluent hypergeometric differential equation → quantization condition (n=\ell+1+m).
Edge cases
- Z→∞: The non‑relativistic equation breaks down; need Dirac or even QED corrections.
- Rydberg states ((n\gg1)): Wavefunctions extend far beyond the nucleus → become sensitive to external fields (Stark, Zeeman).
1.2 Relativistic Corrections – Dirac Equation
The Dirac equation for an electron in a Coulomb field is
[ \left[c\boldsymbol{\alpha}\cdot\mathbf{p}+\beta m_e c^2 - \frac{Z e^2}{4\pi\varepsilon_0 r}\right]\psi = E\psi, ] with (\boldsymbol{\alpha}) and (\beta) being the Dirac matrices. Its eigenvalues are
[ E_{n\kappa} = m_e c^2 \left[1+\left(\frac{Z\alpha}{n-\delta_\kappa}\right)^2\right]^{-1/2}, ] where (\delta_\kappa = |\kappa| - \sqrt{\kappa^2 - (Z\alpha)^2}) and (\kappa=\pm(\ell+1)).
Fine structure
Expanding (E_{n\kappa}) to order ((Z\alpha)^4) gives
[ \Delta E_{\text{fs}} = \frac{(Z\alpha)^4 m_e c^2}{2n^3}\left[\frac{1}{j+1/2} - \frac{3}{4n}\right]. ]
Lamb shift – QED correction (self‑energy, vacuum polarization) adds a small shift (\sim 10^{-6}) eV to the (2S_{1/2}) state.