The Schrödinger equation was formulated by Austrian physicist Erwin Schrödinger in late 1925 and published in 1926.
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system.[1]: 1–2 Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.
Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equation predicted bound states of the atom in agreement with experimental observations.
Schrödinger’s equation connects energy with the way a quantum state changes in time. The left-hand side describes the change of the wavefunction. On the right, the Hamiltonian, the operator representing the system’s total energy, acts on that same wavefunction. The beautiful connection is that the operator used to describe energy also determines how the state evolves.
For a closed quantum system, knowing its initial state and Hamiltonian lets us calculate its later state. Yet this precise evolution does not generally give us a definite outcome for an individual measurement. The wavefunction gives us probability amplitudes, from which we calculate the probabilities of possible outcomes. There is an exact law governing the evolution of those amplitudes.
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