Expectation values, the virial split, and uncertainty

An expectation value in a Fock state is the diagonal matrix element ⟨n|Â|n⟩; writing  through a and a† reduces it to whichever term raises and lowers equally often, so the calculation becomes counting rather than integration. Use the supplied topic model Y_out = 78.07*sqrt((n+0.5)/max(w,10)), where Y_out is SPREAD Δx (pm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. w is Level spacing ħω, measured in meV. Keep n (Quantum number n) = 3 fixed. For this exercise use only 96.125 meV ≤ w ≤ 363.875 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 w = 155.03 meV and 192.515 meV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δw across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.

An expectation value in a Fock state is the diagonal matrix element ⟨n|Â|n⟩; writing  through a and a† reduces it to whichever term raises and lowers equally often, so the calculation becomes counting rather than integration. Use the supplied topic model Y_out = 78.07*sqrt((n+0.5)/max(w,10)), where Y_out is SPREAD Δx (pm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. w is Level spacing ħω, measured in meV. Keep n (Quantum number n) = 3 fixed. For this exercise use only 96.125 meV ≤ w ≤ 363.875 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 = 9.0675356 pm, with a bounded reading uncertainty ±0.21162 pm. 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 0.026775 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.