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A physicist who developed wave mechanics. In 1926, Erwin Schrödinger made the energies of chemical atoms an eigenvalue problem, rather than prescribing a separate quantization rule for each classical orbit. He shared the Nobel Prize in Physics in 1933 with Paul Dirac.^[ref]

The Schrödinger equation turns a proposed Hamiltonian operator into predictions for a wavefunction and its evolution. Boundary conditions can select discrete energy eigenvalues, while quantum superposition makes their relative quantum phases observable through interference. The infinite square well gives a transparent worked example; a hydrogen atom supplies a physical application with a different potential energy.

The Davisson–Germer experiment gives a complementary experimental route into matter waves: an accelerated Electron beam scattered from a nickel crystal produces direction-dependent collected electric current. Its apparatus and angular scans make the wave prediction testable. Interpreting a wavefunction as a probability amplitude also requires the Born rule; that statistical interpretation deserves its own history alongside Erwin Schrödinger's dynamics.

A discovery becomes more intelligible when its motivating question, proposed explanation and actual measurement can be followed together. The photoelectric effect connects Albert Einstein's light-quantum reasoning with the Millikan photoelectric experiment. The Schrödinger equation and the Davisson–Germer experiment connect mathematical wave mechanics with a laboratory test of matter waves.

The Nobel Prize in Physics provides one route through these subjects. Following the original papers and the people who built the apparatus also brings collaborators into view: Lester Halbert Germer appears alongside Clinton J. Davisson in the Davisson–Germer experiment. The history belongs in the articles on those people and concepts as well as in award records.

In 1926, Erwin Schrödinger constructed arrays from differential operators and complete orthogonal function systems, showing how their products reproduce the array calculus. His paper connected wave mechanics with matrix mechanics; the translation's sections 2–3 develop the product and commutation rules.^[ref]

The modern proof is a change of orthonormal basis. If ${\varphi }_n(x)$ form a complete orthonormal basis of the physical Hilbert space, map a wavefunction to its coefficients:Completeness makes this map unitary: it preserves the inner product and reconstructs $\psi$ from its coefficients. Inserting the complete orthonormal basis proves the matrix multiplication rule. The Schrödinger equation becomesFor a time-independent Hamiltonian operator, the Heisenberg picture transfers the same evolution to the observables, preserving measurement predictions. Unitary equivalence requires matching operator domains as well as formal entries. A finite selection of eigenfunctions gives a compression, not an exact replacement of the full Hilbert space; a continuous spectrum needs its spectral integrals. This equivalence concerns representations of specified operators, and does not remove the ordering question when quantizing an arbitrary classical expression, as Weyl ordering illustrates.