Spin-½

The Stern-Gerlach experiment sends neutral atoms through a magnetic field whose strength varies across the beam, so that each atom is pushed in proportion to its magnetic-moment component μ_z, and reads the allowed values of μ_z off the spots on a screen. Use the supplied topic model Y_out = sign(j)*0.0013029*b*g/0.3532, where Y_out is SPACING ÷ SPOT WIDTH (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. g is Moment factor g, dimensionless. Keep j (Angular momentum j) = 0.5; b (Gradient ∂B_z/∂z) = 600 T m⁻¹ fixed. For this exercise use only 0.59125 ≤ g ≤ 2.00875 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 = 3.2224089, with a bounded reading uncertainty ±0.18354. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown g 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, 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.

Spin is angular momentum that belongs to a particle itself rather than to its motion: operators S that obey the same commutators as L but act on an extra discrete label instead of on the spatial wavefunction, and may take the half-integer values that L cannot. Use the supplied topic model Y_out = 1+2*sqrt(max(p*(1-p),0))*cos(2*pi*m), where Y_out is 2π|ψ|² AT φ = 2π (dimensionless). 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. p is Weight p of the trial state, dimensionless. Keep m (Trial L_z quantum number m) = 0.5 fixed. For this exercise use only 0.10625 ≤ p ≤ 0.47375 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 = 0.059267691, with a bounded reading uncertainty ±0.0086833. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown p by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00003675, 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 spin-½ state is a normalised vector α|+z⟩ + β|−z⟩ in a two-dimensional complex space, stored as the column (α, β)ᵀ; columns that differ only by an overall phase are the same state, so two real numbers fix it. Use the supplied topic model Y_out = exp(-sep^2/8), where Y_out is OVERLAP e^(−Δ²/8σ²) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. sep is Separation Δ of the components, measured in σ. Keep theta (Amplitude split θ) = 90 °; phi (Relative phase φ) = 0 ° fixed. For this exercise use only 0.6375 σ ≤ sep ≤ 5.3625 σ 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 = 0.21254958, with a bounded reading uncertainty ±0.027761. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown sep by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0004725 σ, 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 Pauli matrices σₓ, σ_y and σ_z are the three 2×2 Hermitian, traceless matrices that represent the x, y and z components of spin-½ in units of ħ/2, written in the basis |+z⟩, |−z⟩. Use the supplied topic model Y_out = (1000*cosd(beta-alpha))/1000, where Y_out is I PART a⋅b (before display rounding) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. alpha is Axis a (solid), α from +z, measured in °. Keep beta (Axis b (dashed), β from +z) = 60 ° fixed. For this exercise use only 246.15 ° ≤ alpha ≤ 321.75 ° 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 = -0.61199645, with a bounded reading uncertainty ±0.057337. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown alpha by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00756 °, 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.