Twelve worked hydrogen exercises

Normalization gives 1 = 4π|C|²∫₀∞r²exp(−2r/a)dr. The integral is a³/4, so |C|² = 1/(πa³). Choose a real positive C = 1/√(πa³); an overall constant phase does not change position probabilities. The dimensions of ψ are length^−3/2. This is why a three-dimensional density has dimensions length⁻³.

The derivative is p′(x) = 8x(1 − x)exp(−2x). It is positive on 0 < x < 1 and negative for x > 1, so the radial maximum is x = 1. The decreasing volume density has its maximum at x = 0. The shell factor x² explains the difference; neither maximum specifies an electron orbit.

Integrating p(x) gives F(x) = 1 − exp(−2x)(1 + 2x + 2x²). Hence F(1) = 1 − 5exp(−2) ≈ 0.323324 and F(2) = 1 − 13exp(−4) ≈ 0.761897. Check these in the probability explorer. The most probable radial separation and the radius containing a specified probability are different measures.

Using ∫₀∞xᵏexp(−2x)dx = k!/2^k+1, one obtains ⟨r⟩ = 3a/2 and ⟨r²⟩ = 3a². Therefore Var(r) = 3a² − 9a²/4 = 3a²/4 and the standard deviation is √3a/2. The distribution is broad; a single characteristic length does not mean that all detections occur near one radius.

The identity is ⟨ψ,Hψ⟩ + (D/a²)‖ψ‖² = D∫|∇ψ + r̂ψ/a|²d³r. The right-hand side is nonnegative. The normalized exponential obeys ∇ψ = −r̂ψ/a and attains zero. Thus E₁ = −D/a² = −A²/(4D). Existence of this state for every A > 0 also demonstrates why binding by itself does not select a unique coupling.

The first interval is K(1/4 − 1/9) = 5K/36; the second is K(1 − 1/4) = 3K/4. Their ratio is 5/27 ≈ 0.185185. The common K cancels. This checks the shape of the gross spectrum, while an absolute interval additionally checks its scale. It does not address whether a particular one-photon transition is allowed.

The radius is multiplied by 1/1.1 ≈ 0.909091, the binding magnitude by 1.1² = 1.21, and the relative RMS speed by 1.1. The same changed coupling determines all three. In the scales explorer these are illustrative model settings, not alternative measured values for ordinary hydrogen.

One has μ/mₑ = M/(M + 1), aμ/a₀ = (M + 1)/M and |E₁|/|E₁,∞| = M/(M + 1). For M ≈ 1836.15, the radius grows by about 0.05446 percent and the binding magnitude decreases by about 0.05443 percent. These are leading nonrelativistic kinematic effects; extra recoil terms belong to the precision treatment.

The values are ℓ = 0, 1, 2, giving 1, 3 and 5 values of m respectively. The scalar count is 1 + 3 + 5 = 9 = n². Adding a spin-1/2 factor doubles this count to 18 before spin-dependent interactions lift some degeneracies. State counting is not itself a proof of the physical origin of the representation.

The level shifts are 8 kHz for 1S, 1 kHz for 2S, 0.015625 kHz for 8S and zero for 8D. Endpoint subtraction gives −7 kHz, −0.984375 kHz and −1 kHz respectively. The last interval retains the lower 2S contribution. The signed 1S→8S shift is −7.984375 kHz, equal to the sum of the first two intervals. These are synthetic model values.

No. Proportional columns in the comparison model have the same direction, so only a combination of their amplitudes is constrained by those observables. One needs an independently fixed coefficient or observables for which the response shapes differ. This is an identifiability question, distinct from the numerical precision of a fit.

The reproduction checks a calibrated model or its internal algebra. It is not an independent prediction of the interval used to define the tick. A stronger check would fix the tick and other model inputs without that interval, then calculate the interval and additional linked observables with a declared error budget. A useful source result may also explain why the calibration applies to another preparation without retuning.