The Hydrogen Atom

A central-field Hamiltonian is H = p²/2μ + V(r) with V depending only on the distance |r|; because it commutes with L² and L_z, its eigenvectors carry the labels |n l m⟩ and the three-dimensional eigenproblem splits into one half-line radial problem for each l. Use the supplied topic model Y_out = -13.606/max(a,0.05)^2, where Y_out is SPURIOUS E −ħ²/2μa² (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. a is Decay length a of e^(−r/a), measured in a₀. Keep k (Wavenumber k of sin(kr)) = 1.5 1/a₀ fixed. For this exercise use only 0.586875 a₀ ≤ a ≤ 2.71312 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 = -3.8337213 eV, with a bounded reading uncertainty ±0.2978 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown a by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000212625 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.

The spherical harmonics Y_l^m(θ, φ) = ⟨n̂|l m⟩ are the simultaneous eigenfunctions of L² and L_z on the unit sphere, with eigenvalues ħ²l(l+1) and mħ, and they form a complete orthonormal basis for every square-integrable function of direction. Use the supplied topic model Y_out = deg(acos(clamp(m,-l,l)/max(sqrt(l*(l+1)),0.01))), where Y_out is TILT θ FROM z (°). 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; clamp(x,a,b) confines x to [a,b]; deg converts radians to degrees; unmarked trigonometric functions use radians. l is Angular-momentum label l, dimensionless. Keep m (Requested rung m (the ladder sticks at |m| = l)) = 2 fixed. For this exercise use only 2 ≤ l ≤ 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 = 54.73561 °, with a bounded reading uncertainty ±2.2536 °. 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.

The radial Hamiltonian is the operator −(ℏ²/2μ)d²/dr² + V(r) + ℏ²l(l+1)/(2μr²) acting on u = rR on the half-line r > 0, made self-adjoint by the domain condition u(0) = 0. Use the supplied topic model Y_out = s*(s-1)-l*(l+1), where Y_out is INDICIAL RESIDUAL (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. s is Trial exponent s in u ~ r^s, dimensionless. Keep l (Orbital quantum number l) = 1; n (Principal quantum number n) = 2 fixed. For this exercise use only -2.25625 ≤ s ≤ 0.31625 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 = -0.84097165, with a bounded reading uncertainty ±0.20629. 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 0.00025725, 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 radial wavefunction u_nl = r R_nl is the half-line eigenfunction of the radial Hamiltonian at fixed l: an r^(l+1) rise at the origin, an e^(−Zr/na₀) tail, and between them an associated Laguerre polynomial of degree n − l − 1. Use the supplied topic model Y_out = abs((2/(n*sqrt(n)))*rb*(1+(1-n)*rb/n+(1-n)*(2-n)*rb^2/(3*n^2)+(1-n)*(2-n)*(3-n)*rb^3/(18*n^3))*exp(-rb/n)), where Y_out is WALL AMPLITUDE |u(R)| (a_0^(-1/2)). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; exp(x) means eˣ; abs(x) means |x|. rb is Grid box radius R, measured in a₀. Keep n (Principal quantum number n) = 3 fixed. For this exercise use only 14.5725 a₀ ≤ rb ≤ 54.2625 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 = 0.0031394565 a_0^(-1/2), with a bounded reading uncertainty ±0.00046292 a_0^(-1/2). Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown rb by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.003969 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.