Introduction to the Hydrogen Atom

The one-electron model of hydrogen is a single spinless, non-relativistic electron moving in the Coulomb potential energy V(r) = −e²/(4πε₀r) of a fixed point proton, so the whole atom is H = −(ħ²/2mₑ)∇² − e²/(4πε₀r). Use the supplied topic model Y_out = 13.606/(mu*b^2), where Y_out is KINETIC ⟨T⟩ (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. b is Cloud size b, measured in a₀. Keep mu (Reduced mass μ) = 1 mₑ fixed. For this exercise use only 1.05062 a₀ ≤ b ≤ 3.64937 a₀ 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 = 1.9583259 eV, with a bounded reading uncertainty ±0.1405 eV. 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 0.000259875 a₀, 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.

Separating the hydrogen equation means substituting ψ = R(r)Y(θ, φ) so that one equation in three variables becomes a radial equation holding the Coulomb potential and E, and an angular equation holding neither, joined by a single constant λ = l(l + 1). Use the supplied topic model Y_out = 52.9*27.2*sqrt(max(0,(1+en*(l*(l+1))/13.6)))/(-en), where Y_out is ALLOWED REGION r₊ − r₋ (pm). 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 Trial energy E, measured in eV. Keep l (Orbital quantum number l) = 1 fixed. For this exercise use only -6.735 eV ≤ en ≤ -1.065 eV 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 = 316.14471 pm, with a bounded reading uncertainty ±44.289 pm. 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 0.000567 eV, 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 solutions of any central-potential problem are the spherical harmonics Yₗᵐ(θ, φ), labelled by an integer l ≥ 0 that fixes L² = l(l + 1)ħ² and an integer m with |m| ≤ l that fixes L_z = mħ. 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 Magnetic quantum number m, dimensionless. Keep l (Orbital quantum number l) = 2 fixed. For this exercise use only -2 ≤ m ≤ 0, 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 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 radial equation is the one-dimensional Schrödinger equation obeyed by u(r) = rR(r) on the half-line from r = 0 to ∞, with u(0) = 0 and an effective potential that adds the centrifugal barrier ħ²l(l + 1)/2μr² to the Coulomb well. Use the supplied topic model Y_out = -13.6/(nr+l+1)^2, where Y_out is ENERGY Eₙ (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. l is Orbital quantum number l, dimensionless. Keep nr (Radial nodes nᵣ) = 1 fixed. For this exercise use only 0 ≤ l ≤ 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 = -1.5111111 eV, with a bounded reading uncertainty ±0.204 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of l 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.