Semiclassical Theory of Atom–Radiation Interaction

The semiclassical theory is not a historical stepping stone — it is the current working theory wherever fields are strong and coherent. Attosecond science and high-harmonic generation model the laser pulse classically all the way into the non-perturbative regime; the workhorse "strong-field approximation" is minimal coupling with the field kept classical, exactly equation (1). Laser engineering, from gain media to frequency combs, runs on semiclassical rate and Maxwell–Bloch equations. And the humble oscillator strength remains big science: stellar chemical abundances are only as good as the laboratory $f$ values behind each absorption line. When the field itself must be quantum — single photons, vacuum decay, entangled light — the baton passes to quantum electrodynamics, whose first landing in this course is Einstein's $A$ coefficient two articles ahead.

Hold the two length scales in your head at once. A hydrogen atom is about one ångström across. The green light from a laser pointer has a wavelength of 5320 ångström. If the atom were a person, the light wave would be a gentle tide with crests five kilometres apart: at any instant the entire electron cloud feels one common field value, pushed uniformly back and forth $5\times10^{14}$ times per second. The atom responds like a tiny driven antenna — and when the drive frequency matches a Bohr frequency $\omega_{fi}=(E_f-E_i)/\hbar$, the pushes arrive in phase with the atom's internal beat and population transfers resonantly between two stationary states. Why is it legitimate to leave the field classical while the atom gets the full quantum treatment? Count photons. A milliwatt of visible light is $\sim3\times10^{15}$ photons per second; the modes of a laser beam hold enormous mean photon numbers, so the field's granularity — one photon more or less — is invisible against the coherent amplitude. The approximation fails exactly where photon counting matters: single-photon states, vacuum fluctuations, and the spontaneous decay they drive.

The wave does two things and refuses to do a third. Written as a sum of rotating exponentials, the drive contains an $e^{-i\omega t}$ half that hands the atom one quantum of energy $\hbar\omega$ — absorption — and an $e^{+i\omega t}$ half that coaxes an excited atom down, adding its energy to the wave in phase — stimulated emission. Both doors swing on the same hinge: one matrix element, squared. But turn the field off and both doors lock. An isolated excited atom, in this theory, sits in a stationary state forever. Real atoms decay in nanoseconds. The missing third door — spontaneous emission — simply has no classical key, and the ledger below keeps that boundary honest.

The atom alone is $\hat{H}_0=\hat{\mathbf{p}}^2/2m+V(\mathbf{r})$, with stationary states $|n\rangle$ and Bohr frequencies $\omega_{fi}=(E_f-E_i)/\hbar$. How does an external electromagnetic field enter? Not by guesswork: the classical Hamiltonian of a charge $q$ in a field described by potentials $(\mathbf{A},\phi)$ is fixed by the requirement that Hamilton's equations reproduce the Lorentz force, and quantization promotes it wholesale:

where: $\hat{\mathbf{p}}$ is the canonical momentum operator, $q=-e$ for the electron, and $V$ is the binding Coulomb potential. The substitution $\hat{\mathbf{p}}\to\hat{\mathbf{p}}-q\mathbf{A}$ is minimal coupling — the same gauge principle that runs through all of electrodynamics; the full gauge-invariance story of the coupled Schrödinger equation deserves (and will get) an article of its own, but everything this band needs is the weak-field reduction derived next.

The factor $e^{i\mathbf{k}\cdot\mathbf{r}}$ still rides inside $\mathbf{A}(\hat{\mathbf{r}},t)$ in matrix elements. This article keeps it; the demonstration that it may be replaced by $1$ — and exactly when that fails — is the whole subject of The Dipole Approximation, the next article in the band.

Here is a question that has quietly bothered generations of students. The interaction just derived couples through $\hat{\mathbf{p}}$. But every textbook formula for transition rates — and the entire language of "dipole transitions" — uses $\hat{\mathbf{r}}$ through the dipole operator $\hat{\mathbf{d}}=q\hat{\mathbf{r}}$. Are these the same theory? They are, and the bridge is one commutator. Since $\hat{H}_0=\hat{\mathbf{p}}^2/2m+V(\hat{\mathbf{r}})$ and $V$ commutes with $\hat{\mathbf{r}}$,

where: the left identity is the operator form of "velocity is the rate of change of position." Sandwiched between eigenstates the commutator becomes a plain energy difference — and that is the entire proof, executed next on a real atom.

Spectroscopists of the 1920s tabulated line strengths long before anyone could compute a wavefunction, and their bookkeeping variable survives because it works. Define, for a transition $i\to f$ (summing over the degenerate final states),

where: the prefactor is engineered to make $f$ dimensionless and to make a perfect classical Lorentz oscillator have $f=1$. An $f$ value is "what fraction of one classical electron does this line act like." For hydrogen's resonance line the machinery above gives $\sum_{m}|\langle 2p\,m|\hat{\mathbf{r}}|1s\rangle|^2=3|\langle 2p_0|\hat z|1s\rangle|^2$ and

where: all quantities are in atomic units. Lyman-α behaves like $42\%$ of a classical oscillating electron. The Thomas–Reiche–Kuhn sum rule — stated here, proved with the rate machinery in Absorption and Stimulated Emission — says the missing $58\%$ is real and findable: $\sum_f f_{i\to f}=1$ exactly, the sum running over all discrete and continuum final states. One electron's worth of response, conserved and merely distributed. That is why $f$ values are the currency of stellar-abundance work: measure an absorption line's equivalent width, divide by $f$, and you are counting atoms.

Everything the radiation band computes downstream traces to the coupling built above, so it pays to state exactly what the semiclassical theory delivers and what it structurally cannot.

The first-order amplitude that powers rows one to three is the standard integral $c_f^{(1)}(t)=-\tfrac{i}{\hbar}\int_0^t\langle f|\hat{H}'(t')|i\rangle e^{i\omega_{fi}t'}dt'$ — built in First-Order Transition Amplitudes and Probabilities, which this article deliberately does not re-derive.

Example 1 — the resonance line of hydrogen, computed both ways. Take the $1s\to2p$ transition: enumerate what is needed — the Bohr frequency, the matrix element in both forms, and the oscillator strength.

Step 1. Bohr frequency. $E_n=-\tfrac{1}{2n^2}$ a.u., so $\omega_{fi}=E_2-E_1=\tfrac{3}{8}$ a.u. $=10.20\,\mathrm{eV}/\hbar$ — the ultraviolet Lyman-α line.

Step 2. Dipole-form integral. With $\psi_{1s}=2e^{-r}Y_0^0$ and $\psi_{2p_0}=\tfrac{1}{\sqrt{24}}re^{-r/2}Y_1^0$, the angular factor gives $1/\sqrt3$ and the radial integral $\int_0^\infty R_{21}R_{10}\,r^3dr$ evaluates to give $\langle 2p_0|\hat z|1s\rangle=\tfrac{128\sqrt2}{243}\,a_0=0.74494\,a_0$.

Step 3. Momentum-form integral, independently: $\langle 2p_0|\hat p_z|1s\rangle=\tfrac{16\sqrt2}{81}i=0.27935\,i$ a.u. Check the theorem: $im\omega_{fi}\times0.74494=i\cdot0.375\cdot0.74494=0.27935\,i$. Identical.

Step 4. Oscillator strength. Summing the three final $m$ states triples the squared element: $f=\tfrac{2}{3}\cdot\tfrac{3}{8}\cdot3\cdot(0.74494)^2=\mathbf{0.4162}$ — the tabulated Lyman-α $f$ value used in astrophysics.

Example 2 — where "weak field" ends: the field-strength ladder. The internal yardstick is the field one Bohr radius from a proton.

Step 1. Atomic field: $\mathcal{E}_{\text{atomic}}=e/(4\pi\varepsilon_0a_0^2)=\mathbf{5.14\times10^{11}\,V/m}$ — the Bohr Radius and Characteristic Atomic Scales article builds this same entry in its atomic-units table.

Step 2. Laser pointer: $P=1\,\mathrm{mW}$ into a $0.5\,\mathrm{mm}$-radius spot gives intensity $I=P/\pi r^2=1.27\times10^{3}\,\mathrm{W/m^2}$ and field $E_0=\sqrt{2I/\varepsilon_0c}=\mathbf{980\,V/m}$ — a fraction $1.9\times10^{-9}$ of atomic. Perturbation theory is magnificently safe.

Step 3. Femtosecond amplifier at $10^{14}\,\mathrm{W/cm^2}=10^{18}\,\mathrm{W/m^2}$: $E_0=\mathbf{2.7\times10^{10}\,V/m}$, now $5.3\%$ of atomic — the shore of strong-field physics, where the dropped $A^2$ term and non-perturbative ionization return.

Step 4. Photon-number sanity check on the whole framework: the pointer delivers $P/\hbar\omega=3.2\times10^{15}$ photons per second — the classical-field idealization is the appropriate one whenever a beam is bright enough to matter.

References. Griffiths, Introduction to Quantum Mechanics, 3rd ed., Ch. 11 (Time-Dependent Perturbation Theory; Emission and Absorption of Radiation) · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 5 (Time-Dependent Potentials; Interaction with the Classical Radiation Field) · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 18 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Ch. XIII and Complements · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 14 · Einstein, A., 'Zur Quantentheorie der Strahlung', Phys. Z. 18, 121 (1917)

Provenance. All equations are standard textbook results. A verifier should: (1) reproduce the minimal-coupling expansion and dipole reduction (Griffiths 3rd ed. §11.2–11.3, Sakurai §5.7); (2) re-derive the first-order $\mathrm{sinc}^2$ probability and its $2\pi t\,\delta(\omega-\omega_{fi})$ long-time limit; (3) confirm the $1/3$ angular average and the $B=\pi|\langle f|\mathbf{d}|i\rangle|^2/3\epsilon_0\hbar^2$ coefficient; (4) check the Einstein relations $B_{12}=B_{21}$ and $A/B=\hbar\omega^3/\pi^2c^3$ against Planck's law (Einstein 1917); and (5) verify the spontaneous rate $A=\omega^3|\langle f|\mathbf{d}|i\rangle|^2/3\pi\epsilon_0\hbar c^3$ and the hydrogen $2p\to1s$ lifetime $\tau\approx1.6\,\mathrm{ns}$. Cross-check: Griffiths Ch. 11, Sakurai Ch. 5, Cohen-Tannoudji Vol. II Ch. XIII, Shankar Ch. 18. No non-standard numbers are introduced. (Lane note: items (2)–(5) are executed downstream by the owner articles — the first-order machinery by First-Order Transition Amplitudes and Probabilities, the rate pair and 1/3 average by Absorption and Stimulated Emission, the Einstein relations and the 1.6 ns lifetime by Spontaneous Emission and the Einstein A Coefficient — per the band's ownership map; this article's script verifies item (1) plus the equivalence theorem and every number shipped here.)

Builds on: The Interaction Picture and Time-Dependent Hamiltonians · Time-Dependent Perturbation Theory: General Formalism · Harmonic (Sinusoidal) Perturbations and Resonance · The Dipole Approximation

Leads to: Fermi's Golden Rule · Absorption and Stimulated Emission · Spontaneous Emission and the Einstein A Coefficient · Transition Probability for a Periodic Two-Level System