The orbit picture had more detail than the evidence
A spectrograph records line frequencies and intensities. Old quantum theory connected some frequencies to differences between stationary energies, but retained classical orbits with selected quantization rules. Success for hydrogen did not make that collection of rules a general mechanics.
Heisenberg’s 1925 paper begins by challenging the use of unobservable orbital quantities and the restricted applicability of the old rules. Its practical question is how to calculate with quantities tied to transitions. This is a proposal for a new theoretical language, not a newly photographed electron trajectory.
Read the evidence: W. Heisenberg (1925), On the Quantum Reinterpretation of Kinematical and Mechanical Relationships — English translation by Luca Doria (2014)
A product has to pass through intermediate states
Classical Fourier amplitudes carry harmonic labels for one motion. Transition amplitudes instead carry pairs of stationary-state labels. To form the analogue of a product, the intermediate state is summed over. Reversing the factors can change the result.
Born later recalled recognizing Heisenberg’s multiplication rule as matrix multiplication; Born and Jordan, followed by the three-author work with Heisenberg, developed the formalism. The modern commutator below expresses its departure from commuting classical variables. It is not a rule that arbitrary finite matrices can satisfy exactly.
Read the evidence: M. Born (1954), The Statistical Interpretation of Quantum Mechanics — Nobel lecture, December 11W. Heisenberg (1925), On the Quantum Reinterpretation of Kinematical and Mechanical Relationships — English translation by Luca Doria (2014)
A wave equation makes quantization a boundary problem
Schrödinger approached from the relation between mechanics and optics. In his first 1926 paper, the hydrogen problem becomes an eigenvalue problem: acceptable functions must meet conditions at the origin and far away. Discrete bound energies emerge from those requirements rather than an orbit being selected afterward.
The notation below is the modern stationary equation. The Hamiltonian specifies the model; its domain and boundary conditions are part of the problem. The same equation can also possess a continuum of scattering energies. A wave equation does not imply that every physical energy is discrete.
Read the evidence: E. Schrödinger (1926), Quantisierung als Eigenwertproblem, Erste Mitteilung, Ann. Phys. 79, 361–376 — German scan
Equivalent calculations need not share a picture
Vienna’s publication record lists Schrödinger’s 1926 comparison with Heisenberg–Born–Jordan mechanics separately from his sequence of wave-mechanics papers. In an appropriate complete energy basis, an operator can be represented by matrix elements between wavefunctions. Changing the representation changes the working language.
Agreement requires matching the Hamiltonian, admissible states, observables and approximation. A small truncated matrix and an untruncated differential equation are not automatically identical calculations. Nor does mathematical agreement force agreement about what the wavefunction means physically.
Read the evidence: University of Vienna, Erwin Schrödinger: Publications — 1926 wave-mechanics and equivalence papersM. Born (1954), The Statistical Interpretation of Quantum Mechanics — Nobel lecture, December 11
A line position is only one test
Schrödinger’s English account separates the hydrogen equation, characteristic frequencies, light intensities and the difficulties of relativity and magnetic fields. Matching gross hydrogen energies is a major achievement, but it leaves further questions about transition strengths and additional structure.
A fair comparison holds measured quantities fixed and asks which predictions follow without adjusting a rule for each case. Later precision spectroscopy probes terms absent from a simple nonrelativistic Hamiltonian. Such discrepancies locate missing physics; they do not turn an orbital probability plot into a failed photograph.
Read the evidence: E. Schrödinger (1926), An Undulatory Theory of the Mechanics of Atoms and Molecules, Phys. Rev. 28, 1049 — author abstract
Keep the successful abstraction visible
The hydrogen Lab connects selected stationary states to spectral transitions. The wave-packet Lab explores preparation and spread in a different model. Neither is a reconstruction of the historical calculation notebooks, and a rendered surface is not a direct observation of an atomic wave.
The gain was a consistent way to compute relations among preparations and observable results. Read this article with the Born and Pauli bridges: representation, probability and many-electron symmetry solve connected problems, but one should not be silently substituted for another.
Read the evidence: E. Schrödinger (1926), An Undulatory Theory of the Mechanics of Atoms and Molecules, Phys. Rev. 28, 1049 — author abstractUniversity of Vienna, Erwin Schrödinger: Publications — 1926 wave-mechanics and equivalence papers