Spontaneous Emission and Radiative Decay from the Quantised Field

The emission and absorption matrix elements are the first-order amplitudes of the A⋅p coupling that take an atom with n photons in one mode to its other level with n + 1 or n − 1 photons: √(n + 1) and √n times atomic factors that are complex conjugates of each other. Use the supplied topic model Y_out = 47*sqrt(n+1)*cosd(th), where Y_out is EMISSION RABI 2g√(n+1)/2π (kHz). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; sind/cosd/tand use degrees. th is Angle θ between ε and d_ba, measured in °. Keep n (Photons in the mode n) = 1 fixed. For this exercise use only 9.5625 ° ≤ th ≤ 80.4375 ° 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 = 40.189982 kHz, with a bounded reading uncertainty ±3.8032 kHz. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown th by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0070875 °, 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 A coefficient is the rate at which an excited atom alone in the photon vacuum emits, obtained here as the golden-rule rate into every empty mode of the field: for an electric-dipole line, A = ω₀³|d|²/(3πε₀ħc³). Use the supplied topic model Y_out = (n0-n1)*sind(th)^2/max((n0*sind(th)^2+n1*(1+cosd(th)^2)),0.000001), where Y_out is POLARISATION, + IN z–k PLANE (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value; sind/cosd/tand use degrees. th is Viewing angle θ from z, measured in °. Keep n0 (Atoms in m = 0, n₀) = 6; n1 (Atoms in each of m = ±1, n₁) = 2 fixed. For this exercise use only 19.125 ° ≤ th ≤ 85.275 ° 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 = 0.42596235, with a bounded reading uncertainty ±0.01085. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown th by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.006615 °, 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 Einstein relations, A/B₂₁ = ħω³/(π²c³) and g₁B₁₂ = g₂B₂₁, fix the absorption and stimulated-emission coefficients B from the spontaneous rate A; with the quantised field they follow from the photon factors n̄ and n̄ + 1, and Planck's law follows from them. Use the supplied topic model Y_out = nf/(nf+s), where Y_out is STIMULATED SHARE OF EMISSION (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. nf is Field photons per mode n̄, dimensionless. Keep hw (Level spacing ħω/kT) = 0.7; s (Vacuum term (1 quantised, 0 classical)) = 1 fixed. For this exercise use only 0.469687 ≤ nf ≤ 3.58031 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 = 0.70301459, with a bounded reading uncertainty ±0.011799. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown nf by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000311062, 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.

Magnetic-dipole emission is spontaneous decay driven by the coupling −μ⋅B of an atom's magnetic moment to the vacuum fluctuations of the quantised magnetic field, at the rate μ₀ω³|⟨f|μ|i⟩|²/(3πħc³). Use the supplied topic model Y_out = 75115/mu^2, where Y_out is E1 / M1 RATE AT EQUAL ω (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. mu is Transition moment |⟨f|μ|i⟩| / μ_B, dimensionless. Keep dec (Frequency log₁₀(ν / 1420 MHz)) = 0; tb (Background temperature T) = 2.725 K fixed. For this exercise use only 0.301875 ≤ mu ≤ 1.79812 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 = 50917.733, with a bounded reading uncertainty ±4152.9. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown mu by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000149625, 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.