The Schrödinger Equation

In the winter of 1925–26, the Austrian physicist Erwin Schrödinger — vacationing in the Alps with a mistress whose identity has never been confirmed — wrote down the wave equation of quantum mechanics. He was looking for an equation whose solutions would be the de Broglie waves of matter, just as Maxwell's equations have electromagnetic waves as solutions. iℏ ∂ψ/∂t = Ĥψ became the central equation of quantum mechanics, and ψ — the wavefunction — became its central object, the complete description of a quantum system. The equation tells ψ how to evolve in time, smoothly and deterministically; it is the equation of motion of the quantum world. The physical meaning of ψ is another matter. Max Born proposed that |ψ|² is a probability density — the bridge from a deterministic wave to a probabilistic measurement — and quantum mechanics has been irreducibly probabilistic at its foundations ever since.

The time-dependent Schrödinger equation iℏ ∂ψ/∂t = Ĥψ has ψ(x, t) as the wavefunction, ℏ as the reduced Planck constant, and Ĥ as the Hamiltonian — the quantum translation of total energy. It is first-order in time, second-order in space, and linear, so solutions superpose: any two valid states add to a third, and a system can sit in a superposition of possibilities at once. The time-independent form Ĥψ = Eψ is an eigenvalue equation, and for a bound system only certain solutions survive — the eigenstates, stationary states with quantized energies E. This is where discreteness enters physics: solving it for the hydrogen atom (Schrödinger himself did this weeks after writing the equation) yields exactly the allowed energy levels, reproducing every spectral line and its relative intensity, and the same shell structure ordered into the periodic table of the elements. That was the triumph that made quantum mechanics suddenly persuasive.

The mathematical setting is linear algebra on infinite-dimensional Hilbert spaces: wavefunctions are vectors, observables are Hermitian operators, measurement projects onto eigenspaces. Position x̂ and momentum p̂ = −iℏ ∂/∂x do not commute — x̂p̂ − p̂x̂ = iℏ — and that non-commutation is the origin of the Heisenberg uncertainty principle: observables whose operators fail to commute cannot be sharply defined at the same time. The two forms work in concert — solve the time-independent form for the eigenstates, then the time-dependent form propagates any superposition of them forward by a simple phase. That evolution is unitary; it preserves total probability and runs reversibly, like a wave rolling on — until measurement. What happens then — how a smeared-out superposition yields a single definite outcome, the collapse of the wavefunction in Copenhagen language — is not described by the Schrödinger equation at all. This unresolved measurement problem is where the various interpretations of quantum mechanics differ, each trying to explain the one thing the equation leaves silent.