The early 20th century marked a profound shift in physics when classical mechanics failed to describe matter at the atomic level. Phenomena like the way hot objects radiate energy or the stability of electron orbits could not be reconciled with the old physics, which was based on continuous, predictable motion. This crisis prompted a search for a new mathematical language capable of modeling the strange, discontinuous reality of the subatomic world.
Werner Heisenberg and Erwin Schrödinger independently pioneered two fundamentally different mathematical frameworks to solve this problem. Heisenberg focused on measurable quantities and algebraic discontinuity, while Schrödinger pursued a theory based on continuous waves. Their resulting theories, though seemingly incompatible, provided the first complete, logically consistent descriptions of quantum mechanics.
Heisenberg’s Focus on Observables
Werner Heisenberg’s approach, developed in 1925, became known as Matrix Mechanics. He argued that a scientific theory should only concern itself with quantities that could be directly measured, such as the frequencies and intensities of light emitted by atoms. This focus on “observables” led him to abandon the classical idea of electrons following defined orbits, as they could not be experimentally seen.
Heisenberg’s mathematics represented physical properties using arrays of numbers called matrices, rather than continuous variables. The physical state of a particle was described by how these matrices evolved over time. This algebraic formulation naturally incorporated “quantum jumps,” where electrons instantaneously transition between energy states.
This framework led directly to the Heisenberg Uncertainty Principle, which established a fundamental limit on measurement precision. The principle states that the more accurately one measures a particle’s position, the less accurately one can simultaneously know its momentum, and vice versa. This uncertainty is a deep-seated feature of nature, arising because subatomic particles exhibit wavelike behavior.
Schrödinger’s Description of Waves
In contrast to Heisenberg’s algebraic approach, Erwin Schrödinger developed Wave Mechanics in 1926. He treated particles like electrons as continuous waves, inspired by Louis de Broglie’s hypothesis that all matter has an associated matter wave. Schrödinger sought a wave equation to govern their behavior.
The result was the Schrödinger Equation, a foundational equation in quantum mechanics that describes how a system’s wave function ($\Psi$) changes over time. This equation is the quantum equivalent of Isaac Newton’s second law, but it predicts the evolution of the wave function, rather than the exact future path of a particle.
The wave function is a mathematical description of the quantum state. Max Born provided the accepted interpretation: the square of its absolute value gives the probability density of finding the particle at that location. This means the wave function is a wave of probability, not a physical wave.
Before measurement, the particle lacks a definitive location, and the function contains all possible positions and their likelihoods. The equation’s solutions correctly predict the quantized energy levels observed in atoms.
The Mathematical Equivalence of the Theories
In 1926, the scientific community faced two radically different theories—Heisenberg’s discontinuous Matrix Mechanics and Schrödinger’s continuous Wave Mechanics—that both accurately predicted the same experimental results. Heisenberg’s system used non-commuting matrices, while Schrödinger’s utilized continuous partial differential equations.
Despite these conflicting starting points, Schrödinger demonstrated in 1926 that his wave formulation and Heisenberg’s matrix formulation were mathematically equivalent. One could be transformed into the other, guaranteeing identical predictions for all observable phenomena.
This discovery confirmed the validity of quantum theory by showing that the physical reality described was independent of the specific mathematical representation used. Paul Dirac later demonstrated that both approaches were simply different “pictures” of a more general theoretical structure.
In the Schrödinger picture, the particle’s state evolves over time while the operators remain constant. Conversely, in the Heisenberg picture, the state remains fixed, and the operators evolve in time. Both frameworks calculate the same expectation values for measurements.
Defining Reality: The Interpretational Debate
The mathematical equivalence did not resolve the philosophical disagreement about what the mathematics actually described. This debate centered on the nature of the wave function and the effect of measurement. Heisenberg, along with Niels Bohr, championed the Copenhagen Interpretation, which became the dominant view.
The Copenhagen view holds that a quantum system exists in a “superposition” of all possible states simultaneously until observation occurs. The act of measurement forces the wave function to “collapse” instantaneously, selecting one possibility based on the encoded probabilities. This interpretation introduced the idea that nature is intrinsically indeterministic, governed by probability rather than strict cause and effect.
Schrödinger found this probabilistic interpretation deeply troubling, especially the idea that physical reality could exist in a blurred, indeterminate state. He devised the famous thought experiment known as Schrödinger’s Cat in 1935 to illustrate the absurdity of applying quantum superposition to the macroscopic world.
The thought experiment places a cat in a box where its life or death is linked to a random subatomic event. According to the Copenhagen view, until the box is opened, the cat exists in a superposition of both “alive” and “dead” states. Schrödinger intended this scenario as a reductio ad absurdum to show the interpretation led to a ridiculous conclusion.
Despite his objection, the thought experiment has become a popular way to explain the peculiar concept of quantum superposition and the measurement problem at the heart of modern physics.