Atomic Physics

An atomic line spectrum is the set of sharp wavelengths an atom emits or absorbs; each line's photon energy hf equals the difference between two of the atom's allowed energy levels, so the spectrum is a map of level differences. Use the supplied topic model Y_out = 13.598*(1/n1^2 - 1/(n1+k)^2), where Y_out is PHOTON ENERGY hf (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. n1 is Lower level n₁ (series), dimensionless. Keep k (Upper level above it, n₂ − n₁) = 1 fixed. For this exercise use only 1 ≤ n1 ≤ 3, 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 = 1.8886111 eV, with a bounded reading uncertainty ±0.763 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 n1 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 Bohr model treats a one-electron atom as an electron on a classical circular Coulomb orbit whose angular momentum is restricted to whole multiples of ħ, emitting or absorbing light only when it jumps between those orbits. Use the supplied topic model Y_out = -13.6*z/r, where Y_out is ORBIT ENERGY E (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. r is Orbit radius r, measured in a₀. Keep z (Nuclear charge Z) = 1 fixed. For this exercise use only 1.92344 a₀ ≤ r ≤ 14.3266 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 = -1.4331866 eV, with a bounded reading uncertainty ±0.072585 eV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown r by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00124031 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 Coulomb-potential Schrödinger equation describes one electron bound by −ke²/r to a fixed point nucleus; separated in spherical coordinates, it has normalisable solutions only for integer m, l and n, with the energy set by n alone. Use the supplied topic model Y_out = -13.606/n^2, where Y_out is LEVEL Eₙ = −13.6 eV/n² (eV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. 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 = -1.5117778 eV, with a bounded reading uncertainty ±0.20409 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 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.

Quantum orbital angular momentum has a sharp magnitude √(l(l+1)) ħ and a sharp component mħ along one chosen axis, with l = 0, 1, 2, … and m = −l, …, +l; the angular wavefunctions carrying these values are the spherical harmonics Yₗₘ(θ, φ). Use the supplied topic model Y_out = (l*(l+1)-clamp(m,-l,l)^2)/2, where Y_out is ΔLₓ ΔL_y (ħ²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; clamp(x,a,b) confines x to [a,b]. m is Magnetic number m, held to |m| ≤ l, 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 = 2.5 ħ², with a bounded reading uncertainty ±0.16 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.