Time-Independent Perturbation Theory

Rayleigh–Schrödinger perturbation theory writes H = H₀ + λH′, expands an eigenvalue and its eigenstate in powers of λ, and solves the eigenvalue equation one power of λ at a time, starting from the known eigenstates of H₀. Use the supplied topic model Y_out = -(lam*v)^2 + clamp((kmax-2)/2,0,1)*(lam*v)^4 - 2*clamp((kmax-4)/2,0,1)*(lam*v)^6 + 5*clamp((kmax-6)/2,0,1)*(lam*v)^8, where Y_out is SERIES TO ORDER K, E/Δ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; clamp(x,a,b) confines x to [a,b]. lam is Bookkeeping parameter λ, dimensionless. Keep v (Coupling V/Δ) = 0.4; kmax (Series truncated at order K) = 2 fixed. For this exercise use only 0.10625 ≤ lam ≤ 0.89375 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 = -0.055060622, with a bounded reading uncertainty ±0.0079882. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown lam by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00007875, 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 shift of a non-degenerate level is the expectation value of the perturbation in the unperturbed state, Eₙ⁽¹⁾ = ⟨n⁽⁰⁾|H′|n⁽⁰⁾⟩; for a potential it is H′ averaged over the unperturbed probability density. Use the supplied topic model Y_out = 0.37603*n^2+0.001*20/max(w,0.05)*(w-cos(2*pi*n*x0)*sin(pi*n*w)/(pi*n)), where Y_out is LEVEL Eₙ⁽⁰⁾ + E⁽¹⁾ (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value; unmarked trigonometric functions use radians. x0 is Bump centre x₀, measured in nm. Keep n (Level n) = 1; w (Bump width w) = 0.1 nm fixed. For this exercise use only 0.513125 nm ≤ x0 ≤ 0.696875 nm 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 = 0.40992207 eV, with a bounded reading uncertainty ±0.00085706 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown x0 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000018375 nm, 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 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.

The second-order energy shift Eₙ⁽²⁾ is the part of a level's energy quadratic in the perturbation: the sum, over every other unperturbed state m, of |⟨m⁽⁰⁾|H′|n⁽⁰⁾⟩|² divided by the energy gap Eₙ⁽⁰⁾ − Eₘ⁽⁰⁾. Use the supplied topic model Y_out = -(n+1)*d^2/2, where Y_out is PUSH FROM LEVEL n + 1 (ħω). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is Field as displacement d/x₀ = qF/(mω²x₀), dimensionless. Keep n (Level n examined) = 0 fixed. For this exercise use only 0.14875 ≤ d ≤ 1.25125 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 = -0.33724631 ħω, with a bounded reading uncertainty ±0.048927 ħω. 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 0.00011025, 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.