Schrödinger equation

The Schrödinger equation specifies how a quantum system’s wave function evolves in time or varies across space, given its energy and interactions.

Time-dependent Schrödinger equation: The equation that determines a quantum state’s change over time from its Hamiltonian. This form gives the equation’s central rule for temporal evolution.
Wave function: A mathematical description of a quantum state whose amplitudes encode probabilities for measurement outcomes. It is the unknown function whose evolution or spatial form the equation determines.
Particle in a box: A quantum model confining a particle to a region with impenetrable boundaries. Its exact solutions show how confinement produces discrete energy levels.
Newton's laws of motion: Classical laws relating forces to changes in the motion of macroscopic bodies. They predict definite trajectories, unlike the equation’s evolution of quantum states.
Time-independent Schrödinger equation: The eigenvalue equation used to find stationary quantum states and their allowed energies. Separating time from space turns the equation into this energy-eigenvalue problem.
Quantum state: The mathematical specification of a quantum system’s physical condition. The equation describes how this broader object changes under a system’s dynamics.
Quantum harmonic oscillator: A quantum system with a potential energy proportional to the square of displacement from equilibrium. Its solvable spectrum illustrates the equation’s use for bound motion.
Klein–Gordon equation: A relativistic wave equation for spin-zero fields. It addresses relativistic dynamics where the Schrödinger equation is not fundamental.
Hamiltonian operator: The quantum operator corresponding to a system’s total energy. It determines how the wave function evolves and which energies are allowed.
Linear operator: A map between vector spaces that preserves addition and scalar multiplication. The Hamiltonian acts on wave functions as a linear operator.

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