Chapter 3: Electric Dipole Transitions in Real Atoms

Source: Anthony E. Siegman, Lasers (1986), Chapter 3. Use each section/problem identifier with the book; the original prompts are not reproduced here. Each entry gives the governing model, the decisive solution route, and a physical verification.

Section 3.1: Decay Rates And Transition Strengths In Real Atoms

Problem 3.1.1 — Quantum calculation: Hydrogen-atom oscillator strengths

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Section 3.2: Line-Broadening Mechanisms In Real Atoms

Problem 3.2.1 — Derivative spectroscopy on a variable-pressure gas sample

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Section 3.3: Polarization Properties Of Atomic Transitions

Problem 3.3.1 — Two-dimensional Zeeman-split classical oscillator model

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Problem 3.3.2 — Computer plots of oscillating atomic charge distributions (research problem)

Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits. Relate microscopic strength to decay with \(A_{21}=\omega_{21}^3|\boldsymbol\mu_{21}|^2/(3\pi\epsilon_0\hbar c^3)\) and include degeneracy and polarization projections before summing sublevels. Check selection rules, normalization over polarization/orientation, and the cubic frequency scaling of spontaneous decay.

Section 3.4: Tensor Susceptibilities

Problem 3.4.1 — Negative circular polarization response of a gyrotropic tensor

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Problem 3.4.2 — Tensor response of an anisotropic two-dimensional classical oscillator

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Problem 3.4.3 — Tensor response of a three-dimensional Zeeman-split classical oscillator

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Problem 3.4.4 — Field patterns in a “twisted-mode” laser cavity

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Problem 3.4.5 — More on the twisted-mode cavity

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Section 3.5: The “Factor Of Three”

Problem 3.5.1 — Averaging cos2 0 over 47r steradians

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Relate microscopic strength to decay with \(A_{21}=\omega_{21}^3|\boldsymbol\mu_{21}|^2/(3\pi\epsilon_0\hbar c^3)\) and include degeneracy and polarization projections before summing sublevels. Check selection rules, normalization over polarization/orientation, and the cubic frequency scaling of spontaneous decay.

Section 3.7: Inhomogeneous Line Broadening

Problem 3.7.1 — Inhomogeneous broadening with a Lorentzian (rather than Gaussian) inhomogeneous distribution

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Problem 3.7.2 — Inhomogeneous broadening with a uniform inhomogeneous distribution

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Problem 3.7.3 — Ditto with a triangular distribution

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Problem 3.7.4 — Midband absorption versus pressure in a gas

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Problem 3.7.5 — Chemical lasers, and absorption versus pressure in a deuterium fluoride gas cell

Evaluate both cases from the same symbolic expression before taking their ratio; this keeps normalization and sign conventions from obscuring the comparison. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).

Problem 3.7.6 — Inhomogeneous Voight profiles far out in the wings

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Use the driven-oscillator response \(\chi(\omega)\propto[\omega_0^2-\omega^2-j\gamma\omega]^{-1}\); its real part gives dispersion and its imaginary part gives absorption. Normalize the line profile to unit area and check its value and symmetry at \(\omega=\omega_0\).