Heisenberg and Schrodinger Complete Quantum Mechanics
Two competing mathematical descriptions of the atom turn out to be exactly the same theory
Quick facts
- Matrix mechanics
- Heisenberg, 1925
- Uncertainty principle
- Heisenberg, 1927
- Wave mechanics
- Schrodinger, 1926, six papers
- Status
- Shown mathematically equivalent shortly after publication
What happened
In 1925, Werner Heisenberg wrote his first paper on quantum mechanics, developing what became known as matrix mechanics, an abstract mathematical formulation built entirely from measurable quantities rather than pictures of electron orbits. Two years later he stated his uncertainty principle, that the position and momentum of a particle cannot both be measured with arbitrary precision at once, expressed as the relation Delta x times Delta p is greater than or equal to h-bar. In 1926, working independently, Erwin Schrodinger published a series of papers introducing an entirely different approach, wave mechanics, in which each particle is described by a wave function governed by what is now called the Schrodinger equation. MacTutor's history of the period records that mathematicians quickly showed the two theories were equivalent: to each function of the position and momentum coordinates in wave mechanics there may be related a matrix in such a way that these matrices, in every case, satisfy the formal calculation rules of Born and Heisenberg, meaning Schrodinger's differential equation and Heisenberg's matrix algebra described exactly the same physics from two different mathematical directions.
Why it matters
Between matrix and wave mechanics, quantum theory went from Bohr's patchwork model of fixed orbits to a complete, self-consistent mathematical framework capable of describing any atomic or subatomic system, still the operating theory of quantum mechanics used today. The equivalence of the two formulations was itself an early sign of a deeper mathematical unity underlying quantum physics, and Heisenberg's uncertainty principle introduced a form of built-in unpredictability that some physicists, including Einstein, never fully accepted, a dispute over interpretation that continues among physicists and philosophers of science today.
How we know
Heisenberg's and Schrodinger's original 1925 and 1926 papers were published in Zeitschrift fur Physik and Annalen der Physik respectively, and their mathematical equivalence was demonstrated in follow-up papers by Schrodinger and by Paul Dirac shortly afterward; MacTutor's biographies and its dedicated history of the quantum era document this sequence directly from the published record.
Sources
- MacTutor History of Mathematics, University of St Andrews. Quantum mechanics history · Primary source (author-declared)mathshistory.st-andrews.ac.uk · Cited as a "primary" source (no stronger domain match). · Link is live and its text matches the event's key terms (Jul 2026)
- MacTutor History of Mathematics, University of St Andrews. Erwin Schrodinger (1887 - 1961) · Primary source (author-declared)mathshistory.st-andrews.ac.uk · Cited as a "primary" source (no stronger domain match). · Link is live and its text matches the event's key terms (Jul 2026)
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