Quantum Potentials

A piecewise-constant potential is flat inside each of a few regions and jumps between them, so the Schrödinger equation is solved separately in every region and the pieces are joined by demanding that ψ and ψ′ agree at each boundary. Use the supplied topic model Y_out = 5.1227*sqrt(en)*cos(2.56135*sqrt(en))+5.1227*sqrt(max(v0-en,0.02))*sin(2.56135*sqrt(en)), where Y_out is ψ′ MISMATCH AT STEP (nm⁻¹). 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; unmarked trigonometric functions use radians. en is Energy E, measured in eV. Keep v0 (Step height V₀) = 6 eV fixed. For this exercise use only 0.67625 eV ≤ en ≤ 2.3825 eV 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 = -8.8173564 nm⁻¹, with a bounded reading uncertainty ±0.554 nm⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown en by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000170625 eV, 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 infinite square well is the model of a particle that moves freely inside a segment of length L and is forbidden outside it, so its stationary states are the standing sine waves that vanish at both walls. Use the supplied topic model Y_out = 0.376*(2*n+1)/max(w*w,0.0025), where Y_out is GAP TO n+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. w is Well width L, measured in nm. Keep n (Quantum number n) = 2 fixed. For this exercise use only 0.46375 nm ≤ w ≤ 0.93625 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 = 3.3246891 eV, with a bounded reading uncertainty ±0.17699 eV. 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 0.00004725 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.

A superposition in a box is a normalised sum Ψ = Σ cₙψₙ over the well's energy eigenfunctions, in which |cₙ|² is the probability that an energy measurement returns Eₙ and each term evolves by its own phase factor e^(−iEₙt/ħ). Use the supplied topic model Y_out = cosd(a)^2 + n^2*sind(a)^2, where Y_out is MEAN ENERGY ⟨H⟩ (E₁). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. a is Mixing angle θ, measured in °. Keep n (Upper level n) = 2; t (Elapsed time in beat periods) = 0 fixed. For this exercise use only 9.5625 ° ≤ a ≤ 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 = 2.9031906 E₁, with a bounded reading uncertainty ±0.1521 E₁. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown a 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.

For a well eigenstate, the expectation value of an observable is the probability-weighted mean of the values a measurement could return, and its uncertainty is the standard deviation of that same distribution, σ_A = √(⟨A²⟩ − ⟨A⟩²). Use the supplied topic model Y_out = 0.3760*n*n/(w*w), where Y_out is ELECTRON ENERGY E_n (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. w is Well width L, measured in nm. Keep n (Quantum number n) = 1 fixed. For this exercise use only 0.30625 nm ≤ w ≤ 1.09375 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 = 0.6076484 eV, with a bounded reading uncertainty ±0.044001 eV. 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 0.00007875 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.