In 1926, Austrian physicist Erwin Schrödinger developed wave mechanics, which became one of the fundamental pillars of quantum mechanics. His revolutionary equation describes how quantum systems evolve over time, providing a mathematical framework that transformed our understanding of atomic and subatomic physics.
The time-dependent Schrödinger equation is written as i h-bar partial psi partial t equals H psi. Here, psi represents the wave function that describes the quantum state, h-bar is the reduced Planck constant, H is the Hamiltonian operator representing energy, and i is the imaginary unit that makes the equation fundamentally different from classical physics.
The wave function psi describes the quantum state of a particle. The blue curve shows the wave function oscillating in time, while the red curve shows its probability density - the square of the wave function magnitude. This probability density tells us where we are most likely to find the particle when we measure its position.
The particle in a box is a fundamental quantum mechanical system. A particle confined between two walls can only exist in specific energy states, labeled by quantum number n. Each state has a characteristic wave function - n equals 1 gives one half wavelength, n equals 2 gives one full wavelength, and so on. The energy increases as n squared, demonstrating quantum energy levels.