Superposition principle

Lore & Background

The superposition principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms. When vectors or vector fields are involved, a superposition is interpreted as a vector sum. If the superposition holds, it automatically also holds for all linear operations applied on these functions, such as gradients, differentials or integrals.

Reader's Guide

The superposition principle is fundamental to many areas of physics and engineering because it allows complex problems to be broken down into simpler parts. In Fourier analysis, a general stimulus is written as the superposition of infinitely many sinusoids, each analyzed separately, with the total response being the sum of individual sinusoidal responses. Similarly, in Green's function analysis, the stimulus is written as a superposition of impulse functions. The principle applies to waves, where the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced individually. This leads to phenomena such as constructive and destructive interference. In quantum mechanics, the Schrödinger equation is linear, allowing wave functions to be written as superpositions of stationary states. However, the article notes that in most realistic physical situations, the equation governing the wave is only approximately linear, and the superposition principle only approximately holds, with accuracy improving as wave amplitude decreases.

Did You Know?

• The superposition principle can be defined by two simpler properties: additivity and homogeneity.
• In Fourier analysis, the response to a stimulus written as a superposition of sinusoids is itself a sinusoid with the same frequency but generally different amplitude and phase.
• According to the article, Richard Feynman stated that no one has ever been able to define the difference between interference and diffraction satisfactorily.
• In quantum mechanics, the projective nature of quantum-mechanical-state space causes confusion because a quantum mechanical state is a ray in projective Hilbert space, not a vector.