Quantum Uncertainty and Commutation Relations

The commutator of two operators, [A, B] = AB − BA, is the operator measuring how much their order matters; it is linear in each argument, antisymmetric, and obeys the product rule [A, BC] = [A, B]C + B[A, C]. Use the supplied topic model Y_out = 100*(d-1)/max(d,1), where Y_out is RUNGS OBEYING THE ALGEBRA (%). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. d is States kept in the truncation d, measured in states. Keep n (Rung probed n) = 3 fixed. For this exercise use only 2 states ≤ d ≤ 10 states, 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 = 85.714286 %, with a bounded reading uncertainty ±3.2 %. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of d 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 canonical commutation relation says position and momentum fail to commute by a fixed amount: x̂p̂ − p̂x̂ is iħ times the identity operator — the same number in every state, in every representation, and at every time. Use the supplied topic model Y_out = (b*b/(a*a))*exp(-b*b/(2*a*a)) - (b*b/(a*a) - 1)*exp(-b*b/(2*a*a)), where Y_out is DIFFERENCE ÷ iħ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. a is Gaussian width a, measured in nm. Keep b (Probe position x) = 1 nm fixed. For this exercise use only 0.79125 nm ≤ a ≤ 2.20875 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 = 0.83331561, with a bounded reading uncertainty ±0.010391. 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 0.00014175 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 Robertson relation says that for any two Hermitian observables and any normalised state, the product of their standard deviations is at least half the modulus of the expectation of their commutator, taken in that same state. Use the supplied topic model Y_out = min(hypot(0.5*sind(th)^2*sind(2*ph), cosd(th))/max(abs(cosd(th)), 0.0005), 99), where Y_out is PRODUCT / FLOOR (×). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; hypot(a,b) means √(a²+b²); abs(x) means |x|; min(a,b) selects the smaller value; max(a,b) selects the larger value; sind/cosd/tand use degrees. th is Polar angle θ from +z, measured in °. Keep ph (Azimuth φ in the xy-plane) = 35 ° fixed. For this exercise use only 9.5625 ° ≤ th ≤ 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 = 1.1149035 ×, with a bounded reading uncertainty ±0.017223 ×. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown th 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.

The position–momentum uncertainty relation says that for one prepared state, the standard deviation of x̂ and the standard deviation of p̂ — each read from its own ensemble of identical copies — satisfy σ_x σ_p ≥ ħ/2, whatever the state and whatever the apparatus. Use the supplied topic model Y_out = q*q/(2*s*s), where Y_out is KINETIC ⟨T⟩ (ħω). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. s is Position spread σx / √(ħ/mω), dimensionless. Keep q (Uncertainty product σx σp / ħ) = 0.5 fixed. For this exercise use only 0.480625 ≤ s ≤ 1.81937 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 = 0.074276988 ħω, with a bounded reading uncertainty ±0.0054771 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 0.000133875, 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.