Superposition Principle

The superposition principle is the rule that, in a linear system, the combined response to several inputs equals the sum of the responses that each input would produce separately. More generally, scaling and adding inputs scales and adds their responses in the same way. It connects linear algebra with applications involving fields, waves, electrical circuits, and quantum states. Its validity depends on the linearity of the relevant equations and on compatible initial and boundary conditions. (ocw.mit.edu)

Mathematical formulation

A linear map or operator $L$ satisfies

where $u$ and $v$ belong to its domain and $a,b$ are scalars. This identity combines additivity, $L(u+v)=L(u)+L(v)$, with homogeneity, $L(au)=aL(u)$. Thus, if $L(u)=f$ and $L(v)=g$, the linear combination $au+bv$ solves the equation with input $af+bg$. The coefficients may be real or complex numbers, according to the underlying mathematical setting. (ocw.mit.edu)

For a homogeneous equation $L(u)=0$, every linear combination of solutions is another solution. Consequently, the solution set forms a vector space, specifically a linear subspace of the operator’s domain. For an inhomogeneous equation $L(u)=f$, arbitrary sums of solutions do not generally solve the same equation: adding two solutions produces the input $2f$. Instead, the general solution is a particular solution plus any homogeneous solution. (ocw.mit.edu)

For example, the differential equation

has solutions $\sin x$ and $\cos x$, so $A\sin x+B\cos x$ is also a solution for arbitrary constants $A$ and $B$. This is the mathematical structure underlying a harmonic oscillator. Initial and boundary data must likewise be combined consistently; adding solutions usually adds their prescribed data rather than preserving them unchanged. (ocw.mit.edu)

Waves and interference

For a linear wave equation, overlapping disturbances add point by point. If two waves produce displacements $y_1(x,t)$ and $y_2(x,t)$, the resultant displacement is

In the ideal linear description, travelling pulses can pass through one another without permanently changing their individual forms. Small-amplitude sound and electromagnetic waves provide important examples. (feynmanlectures.caltech.edu)

Interference describes the reinforcement or cancellation that follows from this addition. Equal sinusoidal waves arriving in phase reinforce one another; waves of equal amplitude arriving half a cycle apart cancel locally. Waves with slightly different frequencies produce beats, in which the resultant amplitude varies periodically. Interference therefore follows from superposition rather than constituting a separate rule for adding waves. (feynmanlectures.caltech.edu)

The quantities being superposed are amplitudes, not generally intensities or energies. Since wave intensity commonly depends on the square of amplitude, the combined intensity includes interference terms. A linear addition rule for amplitudes does not imply a linear addition rule for every measurable quantity derived from them. (feynmanlectures.caltech.edu)

Electromagnetic fields and circuits

In classical electromagnetism, fields generated by specified sources add vectorially. For several distributions of electric charge,

Each $\mathbf E_i$ is the electric field produced by one distribution considered separately, with source locations and motions held fixed. Corresponding addition applies to the magnetic field. This field-level principle is reflected in the linear structure of Maxwell’s equations for prescribed sources in vacuum. (feynmanlectures.caltech.edu)

In linear circuit analysis, a branch voltage or current can be calculated by adding contributions from individual independent sources. When one source is considered, the others are set to zero: an ideal voltage source becomes a short circuit, while an ideal current source becomes an open circuit. Linear dependent sources remain active because they are controlled by circuit variables. Resistances and other component parameters remain unchanged. (ocw.mit.edu)

Electrical power is not normally superposed directly. For a resistor, for example, power depends quadratically on current. The total current must first be obtained by superposition, after which the power is calculated from that total. (feynmanlectures.caltech.edu)

Quantum-mechanical superposition

In quantum mechanics, quantum superposition concerns probability amplitudes rather than ordinary mechanical displacements. State vectors belong to a complex Hilbert space. Given two admissible states, a nonzero linear combination can represent another state after normalization. For orthonormal states $|1\rangle$ and $|2\rangle$,

The coefficients encode both magnitudes and relative phases. (ocw.mit.edu)

The Schrödinger equation is linear in the wave function for a specified, state-independent Hamiltonian, so combinations of solutions evolving under the same Hamiltonian remain solutions. Measurement probabilities, however, are obtained from squared amplitudes under the Born rule. When indistinguishable alternatives contribute to one outcome, their amplitudes add before squaring, producing interference. Such a coherent combination differs from merely assigning classical probabilities to alternative preparations. (feynmanlectures.caltech.edu)

Limits of applicability

Superposition is exact only for the variables and equations that are linear. A nonlinear response—for example, one containing a term proportional to the square of its input—generates cross terms when inputs are combined. These can produce additional harmonics or sum-and-difference frequencies absent from the separate inputs. (feynmanlectures.caltech.edu)

Many physical systems admit a linear approximation for sufficiently small disturbances around an equilibrium. In that regime, superposition can accurately describe their response even when the fuller equations are nonlinear. Its applicability therefore depends on the model and operating range, not simply on whether the system is described as a wave, field, or circuit. (feynmanlectures.caltech.edu)