Time-Dependent Perturbation Theory

The coefficient equations are the Schrödinger equation for H₀ + H′(t) rewritten, with no approximation, as coupled first-order equations for the amplitudes cₙ(t) of the unperturbed stationary states, each state keeping its own phase e^(−iEₙt/ħ). Use the supplied topic model Y_out = (1-bz/sqrt(bz^2+bx^2))/2, where Y_out is UPPER STATE OF H₀ + H′ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. bz is Static field B₀ along z, measured in μT. Keep bx (Transverse field B₁ along x) = 30 μT; tau (Switch-off time τ) = 0.3 ms fixed. For this exercise use only 19.5625 μT ≤ bz ≤ 90.4375 μT 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 = 0.048840679, with a bounded reading uncertainty ±0.0035117. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown bz 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 μT, 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 interaction picture is the frame that turns with the solved Hamiltonian H₀, so a state changes only under the dressed perturbation H′_I(t); the Dyson series expands that change in the number of times H′_I acts, each later action written to the left. Use the supplied topic model Y_out = 1/(1+d^2), where Y_out is CEILING ω₁²/Ω² (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is Detuning Δ/ω₁, dimensionless. Keep s (Elapsed time ω₁t) = 6.25 rad fixed. For this exercise use only 0.31875 ≤ d ≤ 2.68125 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 = 0.2440715, with a bounded reading uncertainty ±0.018294. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown d by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00023625, 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 first-order amplitude cₘ(t) is the amplitude for reaching |m⟩ when the perturbation may act only once on the initial state |n⟩, at any moment t′ between 0 and t, each moment weighted by the phase e^(iωₘₙt′). Use the supplied topic model Y_out = 1000/(t-ramp/2), where Y_out is FIRST ZEROS ±1/(t − T/2) (MHz). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. t is Time since switch-on t, measured in ns. Keep ramp (Linear ramp duration T) = 0 ns fixed. For this exercise use only 46.375 ns ≤ t ≤ 93.625 ns 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 = 13.298314 MHz, with a bounded reading uncertainty ±0.39261 MHz. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.004725 ns, 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.

A harmonic perturbation H′ = V cos ωt drives a pair of levels with a first-order transition probability that peaks where ħω matches their spacing, and is the same whether the system starts in the lower level (absorption) or the upper one (stimulated emission). Use the supplied topic model Y_out = (4*(sin((1+u)*pi*n)^2/(1+u)+sin((1-u+1e-7)*pi*n)^2/(1-u+1e-7))^2+(sin((1+u)*2*pi*n)/(1+u)+sin((1-u+1e-7)*2*pi*n)/(1-u+1e-7))^2)/(2*pi*n)^2, where Y_out is BOTH TERMS, a → b (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; unmarked trigonometric functions use radians. u is Drive frequency ω/ω₀, dimensionless. Keep n (Drive duration ω₀t/2π) = 2 cycles fixed. For this exercise use only 1.105 ≤ u ≤ 1.4725 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 = 0.15635231, with a bounded reading uncertainty ±0.02275. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown u by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00003675, 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.