The Klein-Gordon Equation

The Klein-Gordon equation, (1/c²)∂²φ/∂t² − ∇²φ + (mc/ħ)²φ = 0, is the wave equation obtained by making E²/c² − p² = m²c² an operator identity with E = iħ∂/∂t and p = −iħ∇; written as (□ + (mc/ħ)²)φ = 0 it has the same form in every inertial frame. Use the supplied topic model Y_out = 197.327/m, where Y_out is COMPTON LENGTH ħ/mc (fm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. m is Rest energy mc², measured in MeV. Keep p (Momentum pc) = 100 MeV fixed. For this exercise use only 30.1875 MeV ≤ m ≤ 179.813 MeV 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 = 1.6246421 fm, with a bounded reading uncertainty ±0.079086 fm. 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 0.0149625 MeV, 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 Klein-Gordon current j^μ = (iħ/2m)(φ*∂^μφ − φ∂^μφ*) is the four-vector the equation conserves; its time component, the density ρ, depends on ∂φ/∂t as well as on φ and can be positive, negative or zero. Use the supplied topic model Y_out = 1/sqrt(1+q^2), where Y_out is THRESHOLD r = E₁/|E₂| (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. q is Momentum of wave 2, q = p₂/mc, dimensionless. Keep r (Amplitude ratio r = a₂/a₁) = 0.7; s (Sign of wave 2's energy) = 1 fixed. For this exercise use only 0.871875 ≤ q ≤ 3.62812 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 = 0.36469239, with a bounded reading uncertainty ±0.014846. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown q by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000275625, 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 non-relativistic limit of the Klein-Gordon equation is what remains once the rest-energy oscillation e^(−imc²t/ħ) is factored out: Schrödinger's equation plus a correction smaller by T/2mc², whose leading term is −p⁴/(8m³c²). Use the supplied topic model Y_out = 100*((q^2/2 - clamp(n,0,1)*q^4/8 + clamp(n-1,0,1)*q^6/16 - clamp(n-2,0,1)*5*q^8/128)-(sqrt(1+q^2)-1))/(sqrt(1+q^2)-1), where Y_out is TRUNCATION ERROR (%). 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]. q is Momentum q = p/mc, dimensionless. Keep n (Correction terms kept n) = 1 fixed. For this exercise use only 0.259375 ≤ q ≤ 1.44063 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 = -8.8045595 %, with a bounded reading uncertainty ±1.3124 %. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown q by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000118125, 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.

Minimal coupling puts a charge q into the Klein-Gordon equation by replacing every four-momentum p^μ with the kinetic four-momentum p^μ − qA^μ (iħ∂^μ − qA^μ as an operator), which leaves the equation unchanged when the potentials are gauge-transformed and φ gains the phase e^(iqΛ/ħ). Use the supplied topic model Y_out = sqrt(max((en-v0)^2-1,0))/(en-v0), where Y_out is VELOCITY v = c²P/(E − V₀) (c). 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. en is Particle energy E, measured in mc². Keep v0 (Plateau potential energy V₀ = qΦ₀) = 0.2 mc²; qa (Vector potential qA₀ (pure gauge)) = 0.4 mc fixed. For this exercise use only 1.2165 mc² ≤ en ≤ 1.85437 mc² 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 = 0.70274881 c, with a bounded reading uncertainty ±0.014083 c. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown en by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000637875 mc², 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.