Spin and Charge in Electromagnetic Fields

In the Heisenberg picture the state stays at |ψ(0)⟩ and each observable carries the time dependence, A_H(t) = U†(t)AU(t), evolving by dA_H/dt = (i/ħ)[H_H, A_H] + (∂A/∂t)_H, with every prediction the same as in the Schrödinger picture. Use the supplied topic model Y_out = -(x0+b*(x0^3+3*x0*s^2)), where Y_out is MEAN FORCE ⟨F⟩ = d⟨p⟩/dt (kℓ). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. s is Packet width σ, measured in ℓ. Keep x0 (Packet centre ⟨x⟩) = 0.8 ℓ; b (Anharmonicity β = 4λℓ²/k) = 0.6 fixed. For this exercise use only 0.153125 ℓ ≤ s ≤ 0.546875 ℓ 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 = -1.3299604 kℓ, with a bounded reading uncertainty ±0.028349 kℓ. 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 0.000039375 ℓ, 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 spin magnetic moment is the vector operator μ = γS through which a particle's spin couples to a magnetic field, H = −μ⋅B; for the electron γ = −g_e e/2mₑ with g_e = 2.0023, so μ points opposite to S. Use the supplied topic model Y_out = 9.2848*field*sind(theta), where Y_out is TORQUE |⟨μ⟩ × B| (× 10⁻²⁴ N m). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. theta is Tilt of ⟨S⟩ from B, θ, measured in °. Keep field (Field B at the spin) = 1 T; grad (Gradient ∂B_z/∂z) = 0 T m⁻¹ fixed. For this exercise use only 19.125 ° ≤ theta ≤ 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 = 7.998121 × 10⁻²⁴ N m, with a bounded reading uncertainty ±0.18827 × 10⁻²⁴ N m. 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 0.006615 °, 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.

Larmor precession is the rotation of a spin's expectation value ⟨S⟩ about a uniform magnetic field at the angular frequency ω₀ = |γ|B₀, with its component along the field, and so its tilt from the field, held fixed. Use the supplied topic model Y_out = -0.042577*bp*t, where Y_out is PROTON TURNS (+ = CCW) (turns). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. bp is Field on the proton, measured in T. Keep be (Field on the electron) = 1 mT; t (Time elapsed t) = 5 ns fixed. For this exercise use only 0.2125 T ≤ bp ≤ 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 = -0.24976733 turns, with a bounded reading uncertainty ±0.019615 turns. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown bp 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.

Magnetic resonance is the flipping of a spin in a static field B₀ by a weak field B₁ rotating about B₀ near the Larmor frequency; in the frame that turns with B₁ the Hamiltonian is static, so the flip probability follows exactly. Use the supplied topic model Y_out = (0.042577*b)^2/((0.042577*b)^2+d^2), where Y_out is FLIP CEILING (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is Detuning f − f₀, measured in kHz. Keep b (Rotating field B₁) = 50 μT; t (Pulse length t) = 0.23 ms fixed. For this exercise use only 0.525 kHz ≤ d ≤ 7.875 kHz 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 = 0.15301992, with a bounded reading uncertainty ±0.012738. 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 0.000735 kHz, 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.