Matrix and wave mechanics equivalence

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.