Angular Momentum

The orbital angular momentum operators are the components of L = r × p with p = −iħ∇, such as L_z = −iħ(x ∂/∂y − y ∂/∂x); each is Hermitian, and each is measured about a chosen origin. Use the supplied topic model Y_out = (100*s*cosd(theta))/100, where Y_out is ⟨x⟩ FROM O (before display rounding) (nm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. theta is Direction of k, θ, measured in °. Keep a (Origin O raised by a) = 0 nm; s (Packet position along its path s) = -1 nm fixed. For this exercise use only -23.625 ° ≤ theta ≤ -1.575 ° 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 = -0.98427432 nm, with a bounded reading uncertainty ±0.0023022 nm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown theta by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.002205 °, 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.

In spherical coordinates the orbital angular-momentum operators act on the angles alone: L_z = −iħ ∂/∂φ, and L² is −ħ² times the angular part of r²∇², with no r and no ∂/∂r anywhere in either. Use the supplied topic model Y_out = 2*pi*(2/(2*l+1)+b^2*(1.6449*max(0,1-abs(l))+1.2150*max(0,1-abs(l-1))+0.8290*max(0,1-abs(l-2)))), where Y_out is NORM ∫f² dΩ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; abs(x) means |x|; max(a,b) selects the larger value. b is Singular admixture B, dimensionless. Keep l (Angular number l) = 1 fixed. For this exercise use only -0.7875 ≤ b ≤ -0.0525 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 = 5.0668817, with a bounded reading uncertainty ±0.12856. 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 0.0000735, 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 angular-momentum commutation relations [L̂_x, L̂_y] = iħL̂_z, [L̂_y, L̂_z] = iħL̂_x and [L̂_z, L̂_x] = iħL̂_y say that two components applied in opposite orders differ by iħ times the third, so no two components are sharp together unless all three are zero. Use the supplied topic model Y_out = abs(sind(theta))*sqrt(1-m*m/2), where Y_out is SPREAD ΔL_z (ħ). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; abs(x) means |x|; sind/cosd/tand use degrees. m is Sharp value of L along n̂, measured in ħ. Keep theta (Tilt θ of the sharp axis n̂ from z) = 30 ° fixed. For this exercise use only -0.7875 ħ ≤ m ≤ -0.0525 ħ 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 = 0.48540927 ħ, with a bounded reading uncertainty ±0.0021369 ħ. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown m by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000735 ħ, 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.

L² and L_z are compatible because [L², L_z] = 0: a complete set of states exists in which both have definite values, and measuring one never disturbs a value already found for the other. Use the supplied topic model Y_out = sqrt(max(m*m*(1+cosd(beta)^2)/2+(1-m*m)*sind(beta)^2-(m*cosd(beta))^2, 0)), where Y_out is SPREAD ΔL_n (ħ). 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; sind/cosd/tand use degrees. beta is Tilt β of the measuring axis n from z, measured in °. Keep m (Prepared state m, with L_z = mħ) = 1 fixed. For this exercise use only 19.125 ° ≤ beta ≤ 85.275 ° 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 = 0.60911657 ħ, with a bounded reading uncertainty ±0.014338 ħ. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown beta by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.006615 °, 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.