Operators and Observables

An operator  is a rule that turns a function f(x) into a new function Âf(x); it is linear when Â(af + bg) = aÂf + bÂg for every pair of functions f, g it can act on and every pair of complex numbers a, b. Use the supplied topic model Y_out = sin(2*pi*m/n)/(2*pi*m/n), where Y_out is MATRIX ÷ EXACT sin(kh)/(kh) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; unmarked trigonometric functions use radians. n is Grid points N, dimensionless. Keep m (Wavelengths across the window m) = 2 fixed. For this exercise use only 8 ≤ n ≤ 64, at the specified test settings separated by 4. 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 = 0.98646084, with a bounded reading uncertainty ±0.028557. 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 position operator multiplies a wavefunction by x, x̂ψ = xψ, and the momentum operator differentiates it and multiplies by −iħ, p̂ψ = −iħ dψ/dx; each takes a function and returns a new function, not a number. Use the supplied topic model Y_out = 1/(2*k*sig), where Y_out is IM(p̂ψ/ψ)/ħk AT x = σ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. sig is Packet width σ, measured in nm. Keep k (Carrier wave number k) = 5 nm⁻¹; th (Prefactor phase θ in Ô = ħ e^(−iθ) d/dx) = 90 ° fixed. For this exercise use only 0.7825 nm ≤ sig ≤ 3.6175 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 = 0.039811294, with a bounded reading uncertainty ±0.0018252. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown sig by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0002835 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 Hamiltonian operator Ĥ = −(ħ²/2m) d²/dx² + V(x) is the energy written as an instruction on ψ: differentiate twice and scale by −ħ²/2m for the kinetic part, multiply by V(x) for the potential part, and add the two results. Use the supplied topic model Y_out = 1/(2*b^2), where Y_out is Ĥψ/ψ AT x = 0 (ħω₀). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. b is Trial width b, measured in ℓ₀. Keep w (Spring frequency ω) = 1 ω₀ fixed. For this exercise use only 0.70625 ℓ₀ ≤ b ≤ 1.49375 ℓ₀ 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 = 0.35509374 ħω₀, with a bounded reading uncertainty ±0.019651 ħω₀. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown b 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 ℓ₀, 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.

An eigenfunction of an operator  is a nonzero function that  returns multiplied by a single constant a, the eigenvalue, while also meeting the boundary and normalisability conditions the problem imposes. Use the supplied topic model Y_out = 1.504*q^2, where Y_out is ħ²k²/2m (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. q is Trial wave number k, measured in π/a. Keep c (Admixture c of sin 2kx) = 0.2 fixed. For this exercise use only 0.81875 π/a ≤ q ≤ 3.18125 π/a 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 = 7.6809807 eV, with a bounded reading uncertainty ±1.0009 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown q by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00023625 π/a, 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.