The Hydrogen Atom

A central potential depends only on the distance r between two particles, which lets the two-body problem reduce exactly to one particle of reduced mass μ moving in V(r), with a wavefunction that factors as ψ = R(r)Y_lm(θ, φ). Use the supplied topic model Y_out = 10^k, where Y_out is MASS RATIO M/m (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. k is Mass ratio, log₁₀(M/m), dimensionless. Keep t (Orbital phase) = 45 ° fixed. For this exercise use only 0.3825 ≤ k ≤ 3.2175 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 = 129.37489, with a bounded reading uncertainty ±19.044. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown k by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0002835, 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 equation is what is left of a central-potential Schrödinger equation once the radial part is separated off; its normalisable solutions are the spherical harmonics Y_l^m(θ, φ), eigenfunctions of L² with eigenvalue l(l+1)ħ² and of L_z with eigenvalue mħ. Use the supplied topic model Y_out = l*(l+1)-m*m, where Y_out is L⊥² = l(l+1) − m² (ħ²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. 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 = 5 ħ², with a bounded reading uncertainty ±0.32 ħ². 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 r ≥ 0, in an effective potential that adds the centrifugal barrier ħ²l(l+1)/2μr² to the actual central potential. Use the supplied topic model Y_out = sign(max(n^2 - l*(l+1), 0))*(n^2 - n*sqrt(max(n^2 - l*(l+1), 0))), where Y_out is INNER TURNING r− (a₀). 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. n is Principal quantum number n, dimensionless. Keep l (Orbital quantum number l) = 1 fixed. For this exercise use only 2 ≤ n ≤ 4, 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 = 1.0627461 a₀, with a bounded reading uncertainty ±0.011056 a₀. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of n 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.

Quantisation from normalisability is the demand that the power series in the hydrogen radial solution terminate, because an unterminated series rebuilds the exponentially growing solution and destroys the normalisation integral. Use the supplied topic model Y_out = -13.606/max(n*n,0.25), where Y_out is TRIAL ENERGY (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. n is Trial n (rho0 = 2n), dimensionless. Keep l (Orbital number l) = 0 fixed. For this exercise use only 1.31875 ≤ n ≤ 3.68125 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 = -1.7862887 eV, with a bounded reading uncertainty ±0.11734 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown n by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00023625, 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.