The Quantum Wavefunction

The state of the particle is a unit vector |ψ⟩ in the Hilbert space L²(ℝ), defined only up to a global phase; its position representation ψ(x) = ⟨x|ψ⟩ lists its components along the delta-normalised position basis. Use the supplied topic model Y_out = (2*s1*s2/(s1^2+s2^2))*exp(-d^2/(s1^2+s2^2)), where Y_out is PROBABILITY |⟨φ|ψ⟩|² (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. d is Separation d, dimensionless. Keep s1 (Width σ₁ of φ) = 1; s2 (Width σ₂ of ψ) = 1 fixed. For this exercise use only 0.425 ≤ d ≤ 3.575 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 = 0.063734381, with a bounded reading uncertainty ±0.0093085. 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 0.000315, 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.

Born's rule reads |ψ(x)|² as a probability density: integrating it across an interval gives the chance of finding the particle there, and normalisation sets the whole-line integral to one. Use the supplied topic model Y_out = (1-w)*(c-sin(2*pi*c)/(2*pi))+w*(c-sin(4*pi*c)/(4*pi))+2*sqrt(w*(1-w))*cosd(phi)*(sin(pi*c)/pi-sin(3*pi*c)/(3*pi)), where Y_out is P(0 < x < cL) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; sind/cosd/tand use degrees; unmarked trigonometric functions use radians. c is Interval edge c, fraction of L, dimensionless. Keep w (Weight w of state |2⟩) = 0.5; phi (Relative phase φ) = 0 ° fixed. For this exercise use only 0.145625 ≤ c ≤ 0.854375 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 = 0.96990184, with a bounded reading uncertainty ±0.0033865. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown c by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000070875, 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 observable is a self-adjoint operator: a rule A together with a dense domain D(A) on which ⟨φ|Aψ⟩ = ⟨Aφ|ψ⟩ for every pair, and for which the adjoint's domain D(A†) is no larger than D(A) itself. Use the supplied topic model Y_out = 6.626/w, where Y_out is LEVEL SPACING h / L (×10⁻²⁵ kg m/s). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. w is Box length L, measured in nm. Keep t (Boundary twist θ) = 0 °; n (Level index n) = 1 fixed. For this exercise use only 1.76563 nm ≤ w ≤ 3.73438 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 = 2.2335616 ×10⁻²⁵ kg m/s, with a bounded reading uncertainty ±0.068885 ×10⁻²⁵ kg m/s. 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 0.000196875 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 expectation value of an observable A in the state |ψ⟩ is the number ⟨A⟩ = ⟨ψ|A|ψ⟩: the mean of the outcomes A would give over a large ensemble of systems all prepared in |ψ⟩, weighted by the Born probabilities. Use the supplied topic model Y_out = 0.5-0.1801265*sind(2*th)*cosd(ph), where Y_out is MEAN <x> / L (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. th is Mixing angle θ, measured in °. Keep ph (Relative phase φ) = 0 ° fixed. For this exercise use only 9.5625 ° ≤ th ≤ 42.6375 ° 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 = 0.34483526, with a bounded reading uncertainty ±0.0036524. 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 0.0033075 °, 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.