Schrodinger Equation

The Schrödinger equation is the fundamental dynamical law of quantum mechanics. It describes how the wave function of a quantum system evolves over time. First formulated by Erwin Schrödinger in 1926, it is to quantum mechanics what Newton's second law is to classical mechanics: the equation that governs the motion of the system.

The time-dependent Schrödinger equation for a single particle in a potential V(x) is:

iℏ ∂ψ/∂t = Ĥψ

where Ĥ is the Hamiltonian operator, ψ is the wave function, i is the imaginary unit, ℏ is the reduced Planck constant, and t is time. The Hamiltonian encodes the total energy of the system — kinetic plus potential — as an operator acting on the wave function.

The equation is linear and deterministic. Given the wave function at any initial time, the Schrödinger equation predicts the wave function at all future times. There is no randomness in the dynamics. The apparent randomness of quantum measurement arises not from the Schrödinger equation but from the measurement process, which is not described by the Schrödinger equation.

This linearity is the source of the superposition principle: if ψ₁ and ψ₂ are solutions, then any linear combination αψ₁ + βψ₂ is also a solution. The Schrödinger equation preserves superposition for all time. It does not collapse wave functions. It does not select outcomes. It evolves superpositions into superpositions, correlations into correlations, entanglement into entanglement.

The measurement problem is precisely the gap between what the Schrödinger equation does and what we observe. The equation predicts that a measuring device interacting with a quantum system will itself enter a superposition. We observe a single definite outcome. The interpretations of quantum mechanics are competing theories of what happens in this gap. The Copenhagen interpretation adds a collapse postulate that is not part of the Schrödinger equation. The many-worlds interpretation denies that anything non-unitary happens and asserts that all branches of the superposition are real. The pilot wave theory adds hidden variables that determine which branch is realized. The decoherence program explains why the superposition becomes unobservable without explaining why a single outcome occurs.

The Schrödinger equation is not the final word. It is non-relativistic; it does not incorporate the principles of special relativity. The relativistic generalization is the Dirac equation, which describes electrons and other spin-1/2 particles. The quantum field-theoretic generalization replaces the wave function with an operator-valued field and the Schrödinger equation with the equations of quantum field theory. In quantum gravity, it is not yet known whether the Schrödinger equation survives in any form, or whether the concept of a wave function evolving in time is itself an approximation valid only in regimes where spacetime is approximately classical.

The Schrödinger equation is the most successful equation in the history of physics. It predicts the behavior of atoms, molecules, solids, and quantum fields with extraordinary precision. Yet it does not describe measurement, and it does not describe gravity. The equation that governs the quantum world is incomplete — not because it is wrong, but because it is not the whole story.

See also

• Wave Function — the object whose evolution the Schrödinger equation governs
• Quantum Mechanics — the theory whose fundamental law is the Schrödinger equation
• Quantum Superposition — the principle that follows from the equation's linearity
• Measurement Problem — the puzzle that arises because the equation does not describe measurement
• Decoherence — how the equation produces effective classicality
• Copenhagen Interpretation — the interpretation that adds collapse to the equation
• Many-Worlds Interpretation — the interpretation that takes the equation literally
• Pilot Wave Theory — the interpretation that adds hidden variables to the equation
• Dirac Equation — the relativistic generalization
• Quantum Field Theory — the quantum field-theoretic generalization