The probability density associated with the wave function is obtained by taking the modulus squared, |ψ(x, t)|². This quantity is interpreted as the probability per unit length of locating the particle within a specific interval. Because probabilities must add to one, the wave function is usually normalized so that the integral of |ψ|² over all space equals one. Normalization does not alter the physical predictions, but it is a convenient way to ensure consistent interpretation of the amplitude.
The dynamics of the wave function are governed by Schrödinger’s equation, a linear partial differential equation. In its time-dependent form, iħ∂ψ/∂t = Ĥψ, the Hamiltonian operator Ĥ determines how the state evolves. For systems with stationary states, solutions can be expressed as ψ(x, t) = φ(x)e⁻ⁱEt/ħ, where φ(x) satisfies the time-independent Schrödinger equation and E is the energy eigenvalue. The eigenfunctions φ(x) form a complete basis set that can be used to expand any arbitrary wave function through superposition, reflecting the principle that quantum states can exist as linear combinations of basis states.
The wave function also encapsulates key quantum phenomena such as interference and tunneling. For example, when two wave functions overlap, their amplitudes can add constructively or destructively, leading to observable interference patterns, as seen in the double-slit experiment. Quantum tunneling, wherein particles penetrate potential barriers higher than their classical kinetic energy, is directly described by the non-zero probability density inside the barrier region.
In quantum field theory, the notion of a wave function generalizes to field operators acting on vacua, yet the underlying idea of a probability amplitude remains. Despite its abstract nature, the lainefunktsioon remains the most fundamental tool for predicting measurable properties of microscopic systems and continues to serve as the backbone of modern physics.