Emission and Absorption of Radiation

The electric-dipole Hamiltonian H′ = −d⋅E(t), with d = qr, is the coupling of an atom to a light wave whose field is treated as uniform across the atom: the leading term of an expansion in k a, the atom's size times the wave number, which is of order α. Use the supplied topic model Y_out = 10^lg, where Y_out is WAVELENGTH λ (nm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. lg is Wavelength, log₁₀(λ / 1 nm), dimensionless. Keep wt (Phase of the wave ωt) = 60 ° fixed. For this exercise use only -0.575 ≤ lg ≤ 2.575 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 22.207517 nm, with a bounded reading uncertainty ±3.2912 nm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown lg by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

In incoherent light, absorption and stimulated emission are constant rates, each proportional to the squared transition dipole and to the light's spectral energy density at the transition frequency, and equal to each other for non-degenerate levels. Use the supplied topic model Y_out = 1/(2*w), where Y_out is INCOHERENT RATE R = ω₁²/(2γ) (ω₁). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. w is Band half-width γ, measured in ω₁. Keep tm (Time marker ω₁t) = 3 fixed. For this exercise use only 1.50937 ω₁ ≤ w ≤ 8.99062 ω₁ on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.082332479 ω₁, with a bounded reading uncertainty ±0.0040078 ω₁. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown w by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000748125 ω₁, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Einstein's A coefficient is the rate at which an excited atom decays with no light present; the B coefficients turn the spectral energy density at the transition frequency into absorption and stimulated-emission rates. Use the supplied topic model Y_out = 1/(exp(0.0143878*10^(-lg-lt))-1), where Y_out is PHOTONS PER MODE n̄ = B_baρ/A (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. lg is Wavelength, log₁₀(λ / 1 m), dimensionless. Keep lt (Temperature, log₁₀(T / 1 K)) = 3.8; gr (Degeneracy ratio g_b/g_a) = 2 fixed. For this exercise use only -5.8625 ≤ lg ≤ -1.1375 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 458.43986, with a bounded reading uncertainty ±68.731. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown lg by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0004725, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The 2p lifetime is τ = 1/A, the inverse of the spontaneous-emission rate to 1s, which is fixed by the cube of the Lyman-α frequency and by the squared dipole matrix element e²|⟨100|r|21m⟩|², the same for all three m. Use the supplied topic model Y_out = 1.29027*(1-exp(-1.5*z*r)*(1+z*r*(1.5+z*r*(1.125+z*r*(0.5625+0.2109375*z*r)))))/z, where Y_out is RADIAL INTEGRAL FROM 0 TO R (a₀). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. r is Upper limit R of the radial integral, measured in a₀. Keep z (Nuclear charge Z (hydrogen-like ion)) = 1 fixed. For this exercise use only 1.72188 a₀ ≤ r ≤ 10.7781 a₀ on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 1.2689892 a₀, with a bounded reading uncertainty ±0.003124 a₀. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown r by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000905625 a₀, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.