The first-order state and the energy-denominator condition

The first-order state is the part of the corrected eigenstate linear in H′: every other unperturbed state mixed in with amplitude ⟨m⁽⁰⁾|H′|n⁽⁰⁾⟩/(Eₙ⁽⁰⁾ − Eₘ⁽⁰⁾), trustworthy only while each amplitude is much smaller than 1. Use the supplied topic model Y_out = exp(-(0.023846*f/hw^1.5)^2), where Y_out is P(STAY IN |0⟩), EXACT (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. hw is Level spacing ħω, measured in meV. Keep f (Field F) = 100 kV m⁻¹ fixed. For this exercise use only 2.85 meV ≤ hw ≤ 9.15 meV 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. Calculate Y_out at hw = 4.236 meV and 5.118 meV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δhw across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.

The first-order state is the part of the corrected eigenstate linear in H′: every other unperturbed state mixed in with amplitude ⟨m⁽⁰⁾|H′|n⁽⁰⁾⟩/(Eₙ⁽⁰⁾ − Eₘ⁽⁰⁾), trustworthy only while each amplitude is much smaller than 1. Use the supplied topic model Y_out = exp(-(0.023846*f/hw^1.5)^2), where Y_out is P(STAY IN |0⟩), EXACT (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. hw is Level spacing ħω, measured in meV. Keep f (Field F) = 100 kV m⁻¹ fixed. For this exercise use only 2.85 meV ≤ hw ≤ 9.15 meV 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 = 0.98121303, with a bounded reading uncertainty ±0.0017087. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown hw by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00063 meV, 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.