In quantum mechanics, a wave function, represented by the Greek letter psi, is a mathematical description that contains all information about a quantum system. Unlike classical waves that we can observe directly, wave functions describe the quantum state of particles and determine their behavior at the microscopic level.
Wave functions are complex-valued functions of position and time. They have both real and imaginary parts, and must satisfy the normalization condition where the integral of the probability density equals one. Wave functions can also be combined through superposition, allowing multiple quantum states to exist simultaneously.
Max Born's interpretation gives physical meaning to wave functions. The square of the wave function's magnitude represents probability density - the likelihood of finding a particle at a specific location. This probabilistic nature is fundamental to quantum mechanics and explains why we cannot predict exact positions, only probabilities.
The time evolution of wave functions is governed by the Schrödinger equation. This fundamental equation contains the imaginary unit i, the reduced Planck constant, and the Hamiltonian operator. It determines how wave functions change shape over time while preserving their normalization, ensuring probability conservation.
Wave functions appear in many quantum systems. In a particle in a box, standing wave patterns form with specific boundary conditions. The quantum harmonic oscillator uses Hermite polynomials. Hydrogen atom orbitals combine radial and angular components, creating the familiar s, p, d, and f orbital shapes that determine atomic structure and chemical bonding.