Angular Momentum and Spin

The angular-momentum commutation relations say that the three components of L = r × p do not commute with one another, [L_x, L_y] = iℏL_z and cyclically, while L² commutes with each of them. Use the supplied topic model Y_out = sqrt((l*(l+1)-clamp(m,-l,l)^2)/2), where Y_out is ΔL_x = ΔL_y (ℏ). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; clamp(x,a,b) confines x to [a,b]. m is Projection m (clamped to |m| ≤ l), dimensionless. Keep l (Angular momentum number l) = 3 fixed. For this exercise use only 0 ≤ m ≤ 3, 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 = 2 ℏ, with a bounded reading uncertainty ±0.09798 ℏ. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of m 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 ladder operators J± = Jx ± iJy raise or lower the Jz eigenvalue by one unit of ħ without changing J², and demanding that every stepped state keep a non-negative norm forces the eigenvalues J² = j(j+1)ħ² and Jz = mħ with m running from −j to +j. Use the supplied topic model Y_out = sqrt(max(j*(j+1)-min(k-j,j)^2,0)/2), where Y_out is TRANSVERSE RMS ΔJx (ħ). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; min(a,b) selects the smaller value; max(a,b) selects the larger value. k is Raising steps from the bottom rung, dimensionless. Keep j (Quantum number j) = 2 fixed. For this exercise use only 0 ≤ k ≤ 2, 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 = 1.5811388 ħ, with a bounded reading uncertainty ±0.058564 ħ. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of k 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 spherical harmonics Y_l^m(θ, φ) are the simultaneous eigenfunctions of L̂² and L̂_z on the unit sphere, and they exist only for integer l ≥ 0, with m running in integer steps from −l to +l. Use the supplied topic model Y_out = -max(sind(t),0.02)^(j+0.5)+2*(j-1)*cosd(t)^2*max(sind(t),0.02)^(j-1.5), where Y_out is RUNG 2 AT θ₀ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value; sind/cosd/tand use degrees. t is Marker polar angle θ₀, measured in °. Keep j (Trial angular momentum j) = 0.5 ħ fixed. For this exercise use only 21.4875 ° ≤ t ≤ 85.4325 ° 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 = -1.1490242, with a bounded reading uncertainty ±0.021876. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0063945 °, 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 electron's orbital motion gives it a magnetic moment μ = −(e/2mₑ)L, so a field B shifts each sublevel by m_l μ_B B and splits every level into 2l+1 equally spaced rungs — the normal Zeeman effect is the three-line spectrum those rungs produce in singlet atoms. Use the supplied topic model Y_out = 13.996*b, where Y_out is π–σ SPACING μ_B B/h (GHz). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. b is Magnetic field B, measured in T. Keep gu (Upper-level g factor) = 1 fixed. For this exercise use only 0.2125 T ≤ b ≤ 1.7875 T 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 = 16.420807 GHz, with a bounded reading uncertainty ±1.2896 GHz. 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 0.0001575 T, 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.