Choose a path through the hydrogen atom

Route 1: understand the physical picture

Begin here if you know what an atom is but have not studied the hydrogen Schrödinger equation. The aim is to understand what the model says, what an atomic wavefunction describes and why different kinds of explanation must be kept distinct.

1. Inner generation and outer observation

   Identify the proposed generative roles and the completed quantities accessible to an observer. Keep a temporal illustration separate from an electron trajectory.

2. The proton–electron system

   Understand charge neutrality, relative separation and what is additionally required for binding.

3. The ground state

   Read the first explanation and the probability sections before opening the detailed proof.

4. Explore the probability distribution

   Compare density at a point, probability in a shell and cumulative probability. Check why their maxima differ.

5. Energy levels and spectral series

   Learn why a spectral interval depends on two states and why not every interval is an allowed one-photon transition.

6. Explore energy intervals

   Change both levels, inspect the ionization threshold and compare the Lyman, Balmer and Paschen series.

At the end, you should be able to explain why the atom has a characteristic size without describing the 1s state as a miniature planetary orbit. You should also be able to identify which part of the explanation follows from a declared Hamiltonian. [Hydrogen II]

Route 2: work through the mathematics

This route assumes elementary calculus, complex amplitudes and some familiarity with linear operators. The main proofs use an explicit continuum model. You do not need to accept an SMM/DTT source interpretation to check those calculations.

1. State the model and its origin problem

   Separate the supplied continuum operator from the proposed source construction.

2. Prove the global lower bound

   Check the square-completion identity, equality case and normalization integral.

3. Separate radial and angular structure

   Follow the role of boundary conditions, polynomial termination and angular degeneracy.

4. Derive linked scales

   Use the reduced mass consistently and distinguish relative RMS speed from electron speed in the centre-of-mass frame.

5. Supply an interaction for transitions

   Use matrix elements and angular structure to understand which perturbations connect which states.

6. Check the worked exercises

   Reproduce the integrals and ratios, then compare your calculations with the explorers.

A useful checkpoint is to derive both the energy and radius from the same coefficients. Replacing each observed quantity by an independently fitted parameter would remove the constraint that makes the exercise informative.

Route 3: identify a research contribution

Begin with the status table and choose one uncompleted correspondence. Then read the source manuscript and the website derivation together. The website gives a continuous explanation; the original manuscript identifies its version, definitions and theorem targets.

For source dynamics, follow proton–electron structure into the effective-interaction problem. For the constants programme, follow the scale dictionary into independent inputs and reproducible comparison. For correction mechanisms, start with fine structure, then radiative corrections and identifiability across intervals.

The strongest small contribution has a specific output: a proof under stated hypotheses, a counterexample that isolates a missing condition, a convergent numerical construction, or an independently specified correction operator. A result need not solve the complete atom to remove a real ambiguity. [Hydrogen synthesis]

Use the explorers as calculations you can inspect

The tools are companions to the derivations. Their controls change declared model inputs; they do not estimate new constants from measurements. Begin with the reference setting, change one parameter, predict the direction of the result and only then look at the output. This turns a slider into an experiment with a mathematical model.

In the probability explorer, ask whether the probability inside one characteristic radius exceeds one half. In the scales explorer, predict what happens to radius and binding energy when the coupling increases by ten percent. In the residual explorer, ask whether a D-state upper level removes a shift already present in the lower S state. The worked exercises explain each answer.

Keep units visible when exporting a table. A numerical difference between a gross-model value and a reference wavelength is not automatically evidence for a new effect: isotope choice, air or vacuum convention, resolved components and omitted corrections have to agree first. [NIST hydrogen data]

Continue into the wider website

The hydrogen lesson offers another accessible entry. The research chapter contains a connected operator treatment. The spectral-corrections guide explains why linked transitions must share one level model. The papers and books place the atom within the wider programme.

For Arabic reading, continue to the Arabic hydrogen lesson or the Arabic ground-state derivation. The English and Arabic routes concern the same mathematical benchmark.