---
title: "Matrices and waves: two routes to the same predictions"
description: "What changed when atomic theory stopped assigning a path to every electron?"
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language: "en"
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  zh-CN: "https://sandboxphysics.com/zh/stories/quantum-evidence/matrix-wave-mechanics-observables/"
last_modified: "2026-10-15"
---

[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# Matrices and waves: two routes to the same predictions

Sandbox Physics · 2026-10-15

What changed when atomic theory stopped assigning a path to every electron?

## The orbit picture claimed more detail than the evidence showed

A spectrograph measures line frequencies and intensities. The old quantum theory tied some of those frequencies to differences between stationary energies, but it kept classical orbits and picked quantization rules to select among them. Its success for hydrogen did not turn that set of rules into a general mechanics.

Heisenberg’s 1925 paper opens by objecting to unobservable orbital quantities and to how narrowly the old rules applied. Its practical question was how to calculate with quantities tied to transitions. It proposed a new theoretical language; it did not report a newly observed 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

A classical Fourier amplitude is labeled by one harmonic of a single motion. A transition amplitude is labeled by a pair of stationary states. To form the analogue of a product, one sums over the intermediate state, and swapping the order of the factors can change the result.

Born later recalled recognizing Heisenberg’s multiplication rule as matrix multiplication. Born and Jordan, and then the three of them with Heisenberg, developed the formalism. The modern commutator below shows how it departs from classical variables, which commute. Truncated finite matrices cannot be expected to satisfy it exactly.

$$
[\widehat{x},\widehat{p}]=i\hbar\widehat{I}
$$

Read the evidence: M. Born (1954), The Statistical Interpretation of Quantum Mechanics — Nobel lecture, December 11 W. 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 started from the connection between mechanics and optics. In his first 1926 paper, hydrogen becomes an eigenvalue problem: acceptable functions must meet conditions at the origin and far away. The discrete bound energies follow from those conditions; no orbit is selected afterward.

Below is the modern stationary equation. The Hamiltonian defines the model, and its domain and boundary conditions are part of the problem. The same equation can also have a continuum of scattering energies, so a wave equation does not make every physical energy discrete.

$$
\widehat{H}\psi_n=E_n\psi_n
$$

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 list gives Schrödinger’s 1926 comparison with Heisenberg–Born–Jordan mechanics as a separate paper from his wave-mechanics series. In a suitable complete energy basis, an operator can be written as matrix elements between wavefunctions. A change of representation changes the working language.

The two agree only when the Hamiltonian, the allowed states, the observables and the approximation all match. A small truncated matrix and the full differential equation are not automatically the same calculation. And agreeing on the mathematics does not force agreement on what the wavefunction means physically.

Read the evidence: University of Vienna, Erwin Schrödinger: Publications — 1926 wave-mechanics and equivalence papers M. Born (1954), The Statistical Interpretation of Quantum Mechanics — Nobel lecture, December 11

## A line position is only one test

Schrödinger’s English account treats several things separately: the hydrogen equation, the characteristic frequencies, the light intensities and the difficulties with relativity and magnetic fields. Getting hydrogen’s gross energy levels right was a major achievement, but transition strengths and further structure remained open questions.

A fair test takes the measured quantities as fixed and asks what follows without a new rule for each case. Later precision spectroscopy probes terms that a simple nonrelativistic Hamiltonian leaves out. Such discrepancies point to 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

## What the new abstraction achieved

The hydrogen Lab links selected stationary states to spectral transitions, and the wave-packet Lab explores preparation and spreading in a different model. Neither reconstructs the historical calculations, and a rendered surface is not a direct observation of an atomic wave.

The gain was a consistent way to calculate how preparations relate to observed results. Read this article with the Born and Pauli articles: representation, probability and many-electron symmetry solve connected problems, but none of them can quietly stand in for another.

Read the evidence: E. Schrödinger (1926), An Undulatory Theory of the Mechanics of Atoms and Molecules, Phys. Rev. 28, 1049 — author abstract University of Vienna, Erwin Schrödinger: Publications — 1926 wave-mechanics and equivalence papers

## Continue with the models

These teaching Labs produce simulated data, which are separate from the historical observations discussed above.

- [Explore hydrogen’s stationary states and spectrum](https://sandboxphysics.com/labs/hydrogen-orbitals-spectra/)
- [Compare how localized a wave packet is and how it spreads](https://sandboxphysics.com/labs/wave-packet-uncertainty/)

## Sources and reading notes

Our sources are Heisenberg’s 1925 paper in Luca Doria’s English translation, a German scan of Schrödinger’s first 1926 paper with his English author abstract, Vienna’s publication list and Born’s retrospective lecture. The article compares the methods of calculation. It does not offer a new proof of operator equivalence or reproduce all the 1926 spectra.

1. [M. Born (1954), The Statistical Interpretation of Quantum Mechanics — Nobel lecture, December 11](https://www.nobelprize.org/uploads/2018/06/born-lecture.pdf)
2. [W. Heisenberg (1925), On the Quantum Reinterpretation of Kinematical and Mechanical Relationships — English translation by Luca Doria (2014)](https://quantasc.ir/files/hp4-paper%28eng%29.pdf)
3. [E. Schrödinger (1926), Quantisierung als Eigenwertproblem, Erste Mitteilung, Ann. Phys. 79, 361–376 — German scan](https://gilles.montambaux.com/files/histoire-physique/Schrodinger-1926-1.pdf)
4. [E. Schrödinger (1926), An Undulatory Theory of the Mechanics of Atoms and Molecules, Phys. Rev. 28, 1049 — author abstract](https://journals.aps.org/pr/abstract/10.1103/PhysRev.28.1049)
5. [University of Vienna, Erwin Schrödinger: Publications — 1926 wave-mechanics and equivalence papers](https://zbphysik.univie.ac.at/ausstellung/schroedinger/ebibliographie/publications.htm)

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