The Quantum Harmonic Oscillator

The quantum harmonic oscillator is the Hamiltonian p̂²/2m + ½mω²x̂² — the leading surviving term when any potential is expanded about a stable minimum — which in its own natural units becomes ½ħω(X² + P²) with [X, P] = i. Use the supplied topic model Y_out = 1/(2*s*s), where Y_out is WELL DEPTH D (ħω). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. s is Anharmonicity a⋅x₀, dimensionless. Keep eps (Level energy E) = 2.5 ħω fixed. For this exercise use only 0.060375 ≤ s ≤ 0.359625 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 = 8.4732964 ħω, with a bounded reading uncertainty ±0.69108 ħω. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown s by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000029925, 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 annihilation operator a and the creation operator a† are the non-Hermitian combinations a = (X + iP)/√2 and a† = (X − iP)/√2 of the dimensionless quadratures, chosen so that [a, a†] = 1 and Ĥ = ℏω(a†a + ½). Use the supplied topic model Y_out = n, where Y_out is ‖a|n⟩‖² = ⟨a†a⟩ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. n is Rung n on the ladder, dimensionless. Keep k (Rungs kept above n) = 2 fixed. For this exercise use only 0 ≤ n ≤ 6, at the specified test settings separated by 1. 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 = 4, with a bounded reading uncertainty ±0.48. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of n consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The number operator N = a†a is the Hermitian, positive operator whose eigenvalue n counts the quanta in a mode; H = ħω(N + ½) then makes energy a matter of counting rungs rather than solving a differential equation. Use the supplied topic model Y_out = v, where Y_out is TRIAL ν (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. v is Trial eigenvalue ν of N, dimensionless. Keep s (Lowering steps k applied) = 4 fixed. For this exercise use only 0.371875 ≤ v ≤ 3.12812 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 = 2.0531875, with a bounded reading uncertainty ±0.16124. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown v by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000275625, 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 oscillator's energy eigenvalues are Eₙ = (n + ½)ℏω: an infinite ladder of equally spaced rungs whose lowest, the zero-point energy ℏω/2, lies above the bottom of the well and cannot be lowered at fixed ω. Use the supplied topic model Y_out = 2913.1/sqrt(max(u,0.2)), where Y_out is WAVENUMBER ω̃ (cm⁻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. u is Reduced mass μ, measured in u. Keep n (Lower rung n of the marked gap) = 0 fixed. For this exercise use only 1.03812 u ≤ u ≤ 2.06188 u 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 = 2259.227 cm⁻¹, with a bounded reading uncertainty ±34.575 cm⁻¹. 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 0.000102375 u, 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.