# Superposition Principle

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## Superposition principle

The principle, obeyed by many equations describing physical phenomena, that a linear combination of the solutions of the equation is also a solution.

An effect is proportional to a cause in a variety of phenomena encountered at the level of fundamental physical laws as well as in practical applications. When this is true, equations which describe such a phenomenon are known as linear, and their solutions obey the superposition principle. Thus, when *f*, *g*, *h*, · · ·, solve the linear equation, then *s*, where α, β, γ, · · ·, are coefficients) also satisfies the same equation.

For example, an electric field is proportional to the charge that generates it. Consequently, an electric force caused by a collection of charges is given by a superposition—a vector sum—of the forces caused by the individual charges. The same is true for the magnetic field and its cause—electric currents. Each of these facts is connected with the linearity of Maxwell's equations, which describe electricity and magnetism. *See* Electric field, Maxwell's equations

The superposition principle is important both because it simplifies finding solutions to complicated linear problems (they can be decomposed into sums of solutions of simpler problems) and because many of the fundamental laws of physics are linear. Quantum mechanics is an especially important example of a fundamental theory in which the superposition principle is valid and of profound significance. This property has proved most useful in studying implications of quantum theory, but it is also a source of the key conundrum associated with its interpretation.

Its effects are best illustrated in the double-slit superposition experiment, in which the wave function representing a quantum object such as a photon or electron can propagate toward a detector plate through two openings (slits). As a consequence of the superposition principle, the wave will be a sum of two wave functions, each radiating from its respective slit. These two waves interfere with each other, creating a pattern of peaks and troughs, as would the waves propagating on the surface of water in an analogous experimental setting. However, while this pattern can be easily understood for the normal (for example, water or sound) waves resulting from the collective motion of vast numbers of atoms, it is harder to understand its origin in quantum mechanics, where the wave describes an individual quantum, which can be detected, as a single particle, in just one spot along the detector (for example, photographic) plate. The interference pattern will eventually emerge as a result of many such individual quanta, each of which apparently exhibits both wave (interference-pattern) and particle (one-by-one detection) characteristics. This ambivalent nature of quantum phenomena is known as the wave-particle duality. *See* Interference of waves, Quantum mechanics

## Superposition Principle

a method of calculating linear electrical circuits in which the current of each branch is considered to be the algebraic sum of the currents produced in the branch by the action of each source of electromotive force separately.

## Superposition Principle

(1) The superposition principle states that if the components of a complex process of action on a system do not influence each other, then the resultant effect is the sum of the effects produced by each component acting separately. The principle is only applicable to linear systems, that is, systems whose behavior can be described by linear equations. Suppose, for example, the medium in which a wave *S* propagates is linear; in other words, the properties of the medium do not change under the influence of the disturbances produced by the wave. Then all effects caused by an anharmonic wave can be defined as the sum of the effects produced by each of the wave’s harmonic components: *S* = *S*_{1} + *S*_{2} + *S*_{3} + ..., The superposition principle plays an important role in mechanics (for example, vector addition according to the parallelogram law), oscillation theory, network theory, quantum mechanics, and other branches of physics and technology.

(2) In the theory of classical fields and in quantum theory, the superposition principle asserts that when any states of a physical system that are permissible under given conditions are superposed, or added to each other, the superposition (that is, the result of the addition) is also a permissible state. With respect to processes that are possible in the system, the principle asserts that the superposition of the processes is also a possible process. For example, the superposition principle is satisfied by the classical electromagnetic field in a vacuum: the sum of any number of physically realizable electromagnetic fields is also a physically realizable electromagnetic field. It follows from the superposition principle that the electromagnetic field produced by a set of electric charges and currents is equal to the sum of the fields generated by the charges and currents taken separately. A weak gravitational field also obeys the superposition principle to a high degree of accuracy.

In classical physics, the superposition principle is an approximate principle that follows from the linearity of the equations of motion of suitable systems (this linearity is usually a good approximation for describing real systems), such as Maxwell’s equations for the electromagnetic field. Thus, the superposition principle stems from deeper dynamic principles and, consequently, is not fundamental. Moreover, it is not universal. For example, a sufficiently strong gravitational field does not satisfy the superposition principle, since such a field is described by Einstein’s nonlinear equations. Another example is a macroscopic electromagnetic field in a substance; the field, strictly speaking, does not obey the superposition principle, since the dielectric constant and magnetic permeability are dependent, sometimes to a considerable degree, on the external field. This situation is characteristic, for example, of a ferromagnetic material.

In quantum mechanics, the superposition principle is a fundamental postulate. Together with the uncertainty relation, the principle determines the structure of the theory’s mathematical apparatus. It follows from the superposition principle, for example, that the states of a quantum-mechanical system should be represented by vectors in a linear space, in particular, by wave functions (*see*QUANTUM MECHANICS) and that the operators of physical quantities must be linear. The superposition principle asserts that if a quantum-mechanical system can be in states describable by the wave functions ψ_{1}, ψ_{2},..., ψ_{n}, then the superposition of these states is also physically permissible. The superposition is the state represented by the wave function

ψ = *c*_{1}ψ_{1} + *c*_{2}ψ_{2} ... + *c _{n}*ψ

_{n}

where *c*_{1}, *c*_{2}, ..., *c _{n}* are arbitrary complex numbers.

It follows from the superposition principle that any wave function can be expanded into a sum (generally an infinite sum) of the eigenfunctions of the operator of any physical quantity. The squares of the moduli of the coefficients in the expansion can be understood as the probabilities of detecting experimentally the corresponding values of the quantity. The superposition of the states ψ_{i} is determined, however, not only by the moduli of the coefficients *c _{1}*, but also by the relative phases (for different relative phases of the c,, the resultant states are different). The superposition ψ = Ʃ

_{i}

*c*

_{i}ψ

_{i}, is therefore a result of the interference of the states ψ

_{i}(

*see*DIFFRACTION OF PARTICLES). The quantum superposition principle is not as easy to visualize as the superposition principle in classical physics, since quantum theory makes use of the superposition of alternative states that are mutually exclusive from the classical standpoint. The superposition principle reflects the wave nature of microparticles and is satisfied without exception in nonrelativistic quantum mechanics.

In the relativistic quantum theory that considers processes in which the transmutations of particles can occur, the superposition principle must be supplemented by superselection rules. Thus, superpositions of states with different values of the charges known as electric charge, baryon number, and lepton number are not taken to be physically realizable. The realizability of such superpositions would mean, for example, that the physical properties of a particle beam containing electrons and positrons in some proportion are not unambiguously determined by the dynamic characteristics of the particles. In other words, interference between states with different charge values would be possible. Such interference, however, has never been observed in experiments. The operators of physical quantities must therefore conserve charges. This refinement of the superposition principle in relativistic quantum theory imposes certain restrictions on the matrix elements of operators. These restrictions are the superselection rules.

### REFERENCES

Dirac, P. A. M.*Prinlsipy kvantovoi mekhaniki*. Moscow, 1960. (Translated from English.)

Landau, L. D., and E. M. Lifshits.

*Kvantovaia mekhanika*, 3rd ed. Moscow, 1974.

Schweber, S.

*Vvedenie v reliativistskuiu kvantovuiu teoriiu polia*. Moscow, 1963. (Translated from English.)

O. I. ZAV’IALOV