= 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.<On the Relation between the Quantum Mechanics of Heisenberg, Born, and Jordan, and that of Schrödinger by Erwin Schrödinger>[Collected Papers on Wave Mechanics, On the Relation between the Quantum Mechanics of Heisenberg, Born, and Jordan, and that of Schrödinger, pp. 48–51, equations (6)–(10)]{ref}
The modern proof is a change of <orthonormal basis>. If $\phi_n(x)$ form a complete <orthonormal basis> of the physical <Hilbert space>, map a <wavefunction> to its coefficients:
$$
c_n=\int\phi_n(x)^*\psi(x)\,dx,\qquad A_{mn}=\int\phi_m(x)^*(A\phi_n)(x)\,dx.
$$
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> becomes
$$
i\hbar\dot c_m=\sum_nH_{mn}c_n.
$$
For 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.
Back to article page
