Formula sheet · PU-306

Atomic & Molecular Physics — formula sheet

Every boxed result from the 15 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.

D-298

Relativistic Kinetic-Energy Correction

\[E^{(1)}_{\mathrm{rel}}=-\frac{E_n^{2}}{2mc^{2}}\left(\frac{4n}{\ell+\tfrac12}-3\right)=\frac{E_n\,\alpha^{2}}{n^{2}}\left(\frac{n}{\ell+\tfrac12}-\frac34\right)\]

The shift is negative for every allowed \((n,\ell)\) — relativity always lowers the levels, because a genuinely relativistic electron has slightly less kinetic energy than the Newtonian \(p^2/2m\) predicts for the same momentum. Its size relative to \(|E_n|\) is of order \((Z\alpha)^2/n^2\). Crucially the shift depends on \(\ell\): it partially lifts the accidental Coulomb degeneracy, states of smaller \(\ell\) (more penetrating, faster near the nucleus) being pushed down most.

D-299

Spin-Orbit Coupling from the Rest Frame

\[ \hat H_{\text{SO}} = \frac{1}{2m^2c^2}\,\frac{1}{r}\frac{dU}{dr}\,\vec S\cdot\vec L,\qquad \big\langle\vec S\cdot\vec L\big\rangle = \frac{\hbar^2}{2}\big[j(j{+}1)-\ell(\ell{+}1)-s(s{+}1)\big]. \]

The electron feels an internal magnetic field \(\vec B'\propto\vec L\) because in its own frame the nucleus circulates around it (Biot–Savart). Its spin moment couples to that field; the Thomas precession of the accelerating frame halves the result. States of different total angular momentum \(j\) are split, with \(j=\ell+\tfrac12\) raised and \(j=\ell-\tfrac12\) lowered.

D-301

Hyperfine Splitting and the 21 cm Line

\[ \Delta E_{\mathrm{hf}} = \frac{4}{3}\,g_p\,\frac{m_e}{m_p}\,\alpha^4\, m_e c^2 \approx 5.88\ \mu\mathrm{eV},\qquad \nu = \frac{\Delta E_{\mathrm{hf}}}{h}\approx 1420\ \mathrm{MHz},\qquad \lambda=\frac{c}{\nu}\approx 21\ \mathrm{cm} \]

The magnetic energy of the electron sitting in the proton's contact field depends on whether the two spins are parallel (triplet, higher) or antiparallel (singlet, lower). The gap is suppressed relative to the gross structure \(\alpha^2 m_e c^2\) by a further \(\alpha^2 (m_e/m_p)\) — one factor \(\alpha^2\) from the magnetic (rather than Coulomb) coupling and \(m_e/m_p\) from the small nuclear moment. A spin flip from triplet to singlet emits one 21 cm photon.

D-302

Anomalous Zeeman Effect and the Lande g-Factor

\[ \boxed{\;\Delta E_{\text{weak}} = g_J\,\mu_B B\,m_J,\qquad g_J = 1 + \frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}\;} \]

D-303

Linear Stark Effect in Hydrogen

\[ \Delta E^{(1)}_{n=2} = \{\,+3ea_0\mathcal{E},\ \ 0,\ \ 0,\ \ -3ea_0\mathcal{E}\,\} \]

The applied field splits the four-fold degenerate \(n=2\) level into three: two shifted states (the \(sp\)-hybrids, one raised and one lowered by \(3ea_0\mathcal{E}\)) and a doubly-degenerate unshifted pair (the \(2p_{\pm1}\) states). The splitting grows linearly with field strength — the signature of the effect — and its scale \(3ea_0\) is an induced-independent permanent dipole moment of the hybrid states, magnitude three Bohr radii times the elementary charge.

D-304

Exchange Symmetry and the Ortho/Para Splitting

\[ \boxed{\;E_\pm=\varepsilon_{1s}+\varepsilon_{2s}+J\pm K,\qquad K=\frac{e^2}{4\pi\varepsilon_0}\!\iint\!\frac{\varphi_{1s}^{*}(\mathbf r_1)\varphi_{2s}^{*}(\mathbf r_2)\varphi_{1s}(\mathbf r_2)\varphi_{2s}(\mathbf r_1)}{r_{12}}\,d^3r_1 d^3r_2,\qquad \Delta E_{\text{para}-\text{ortho}}=2K\;}\]

Antisymmetrizing the two-electron wavefunction produces two energies differing only in the sign of the exchange integral. The spin-blind Coulomb repulsion nonetheless separates the spin singlet (para) and spin triplet (ortho) by \(2K\), because antisymmetry forces the triplet into a spatial state \(\psi_-\) that vanishes when the electrons coincide (the Fermi/exchange hole), lowering their mutual repulsion. Orthohelium is the lower, more stable term; the two ladders scarcely intercombine because photons carry no spin flip.

D-305

Variational Ground-State Energy of Helium

\[ \boxed{\;\zeta^\star=Z-\frac{5}{16}=\frac{27}{16},\qquad E_{\min}=-\left(Z-\frac{5}{16}\right)^{2}=-\left(\frac{27}{16}\right)^{2}E_h\approx-2.848\,E_h\approx-77.5\ \text{eV}\;} \]

Each helium electron does not feel the full \(+2\) nuclear charge; the other electron shields \(5/16\) of a unit, leaving an effective \(+27/16\approx1.69\). Using this self-consistently softened orbital gives a ground-state energy of \(-77.5\) eV, an upper bound lying \(1.5\) eV (about \(1.9\%\)) above the experimental \(-79.0\) eV. Letting \(\zeta\) relax from the naive value \(2\) recovers roughly two-thirds of that gap.

D-306

The Hartree-Fock Self-Consistent-Field Equations

\[ \hat{f}\,\chi_a=\varepsilon_a\,\chi_a,\qquad \hat{f}=\hat{h}+\sum_{b=1}^{N}\big(\hat{J}_b-\hat{K}_b\big),\qquad E=\sum_a\langle a|\hat{h}|a\rangle+\frac12\sum_{a,b}\big(J_{ab}-K_{ab}\big) \]

Each electron obeys a one-body Schrödinger-like equation in an effective potential built from all the others: a classical Coulomb (Hartree) repulsion \( \hat{J}_b \) from the smeared-out charge of every occupied orbital, minus a quantum exchange \( \hat{K}_b \) that acts only between parallel spins and lowers the energy by keeping like-spin electrons apart. The equations are coupled and nonlinear — the potential depends on the very orbitals it determines — so they are solved by iteration: guess orbitals, build \( \hat{f} \), diagonalise, rebuild, repeat until the field stops changing. That loop is the self-consistent field (SCF). The canonical orbital energy \( \varepsilon_a \) equals \( \langle a|\hat{h}|a\rangle+\sum_b(J_{ab}-K_{ab}) \), so \( \sum_a\varepsilon_a=E+\tfrac12\sum_{ab}(J_{ab}-K_{ab}) \) double-counts the electron–electron energy: the total energy is not the sum of orbital energies.

D-307

LS Coupling, Term Symbols, and Hund's Rules

\[ E\bigl(^{2S+1}L_J\bigr)=\underbrace{E_0+E_{\text{term}}(L,S)}_{\text{electrostatic}}+\underbrace{\frac{A}{2}\bigl[J(J+1)-L(L+1)-S(S+1)\bigr]}_{\text{spin–orbit}} \]

D-309

Einstein Coefficients and Spontaneous Emission

\[ g_1 B_{12}=g_2 B_{21},\qquad \frac{A_{21}}{B_{21}}=\frac{8\pi h\nu^3}{c^3},\qquad A_{21}=\frac{\omega^3\,|\mathbf d_{21}|^2}{3\pi\varepsilon_0\hbar c^3} \]

Absorption and stimulated emission are the same intrinsic strength (up to degeneracy), and the "extra" process Einstein was forced to add — spontaneous emission — is not free: its rate is locked to the stimulated rate by \(8\pi h\nu^3/c^3\), the density of electromagnetic modes times \(h\nu\). The last equality shows that this rate is nothing but the transition dipole squared, driven by the \(\omega^3\) mode density; a strong dipole and a high frequency both make the excited state short-lived. Spontaneous emission is stimulated emission driven by the vacuum's zero-point modes.

D-310

The Born-Oppenheimer Approximation

\[ \boxed{\;\Big[-\sum_A\frac{\hbar^2}{2M_A}\nabla_A^2+E_k(\mathbf R)\Big]\chi_k(\mathbf R)=E\,\chi_k(\mathbf R),\qquad \hat H_e\phi_k=E_k(\mathbf R)\phi_k,\qquad \Psi\approx\chi_k(\mathbf R)\,\phi_k(\mathbf r;\mathbf R)\;} \]

The electrons solve their own Schrödinger equation at every frozen nuclear geometry, producing an energy \(E_k(\mathbf R)\) that acts back on the nuclei as a potential-energy surface. The nuclei then move quantum-mechanically on that single surface, blind to the other electronic states. Molecular structure, force constants, and reaction paths are all features of \(E_k(\mathbf R)\); the approximation is controlled by \(m_e/M_A\ll1\).

D-311

The Hydrogen Molecular Ion and the Chemical Bond

\[ \boxed{\,E_\pm(R) = E_{1s} + \frac{1}{R} - \frac{J(R) \pm K(R)}{1 \pm S(R)}\,},\qquad E_{1s}=-\tfrac12\ \text{hartree} \]

The bonding curve \(E_+(R)\) uses \(+\) throughout: the resonance integral \(K>0\) enters as \(-(J+K)/(1+S)\), a large lowering, and produces a minimum at \(R_e\approx 2.5\,a_0\) that lies \(D_e\approx1.76\,\mathrm{eV}\) below the separated \(\mathrm{H}+\mathrm{H^+}\) limit. The antibonding curve \(E_-(R)\) uses \(-\); the resonance term now raises the energy and the curve is purely repulsive, with a nodal plane midway between the nuclei where \(\psi_-=0\). Bonding piles electron amplitude between the nuclei; antibonding evacuates it.

D-312

Molecular Rotational and Vibrational Spectra

\[ F(J)=\tilde{B}\,J(J+1),\qquad G(v)=\tilde{\omega}_e\!\left(v+\tfrac12\right)-\tilde{\omega}_e x_e\!\left(v+\tfrac12\right)^2 \]

D-313

The Franck-Condon Principle

\[ \boxed{\; q_{v''v'} = \big|\langle \chi_{v''}|\chi_{v'}\rangle\big|^{2} = \left| \int \chi_{v''}^{*}(R)\,\chi_{v'}(R)\,dR \right|^{2}, \qquad I_{v''v'} \propto \omega^{3}\,|\mathbf{M}(R_0)|^{2}\,q_{v''v'} \;} \]

The intensity of a vibronic line is the electronic transition strength times the squared overlap of the two vibrational wavefunctions. Crucially, \( \chi_{v'} \) and \( \chi_{v''} \) live on different electronic potential-energy surfaces — generally with different equilibrium geometries and force constants — so they are not orthogonal, and the overlap can be large for \( v'' \neq v' \). The most intense line corresponds to the upper vibrational state whose wavefunction has greatest amplitude directly above the lower state's equilibrium geometry: the "vertical" transition.

D-435

Lasers: Stimulated Emission, Population Inversion and Gain

\[ g_1 B_{12} = g_2 B_{21}, \quad \frac{A_{21}}{B_{21}} = \frac{8 \pi h f^3}{c^3}, \quad \text{Gain} \propto N_2 - \frac{g_2}{g_1} N_1 \]

The Einstein relations establish the fundamental constants governing atomic transitions. The gain coefficient is positive only if the upper-level population exceeds the lower-level population by a factor of the degeneracy ratio. In thermal equilibrium, \(N_2/N_1 = (g_2/g_1) e^{-h f / k T} < g_2/g_1\), so the gain is always negative (absorption dominates). Population inversion requires non-thermal pumping.

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