Superposition (physics)

In physics, superposition , also known as the superposition principle, is understood to be a superposition of identical physical quantities that do not interfere with one another. This superposition principle is used for linear problems in many areas of physics and only differs in the type of superimposed quantities. Often the phrase "several sizes superpose each other" is used.

Important areas of application of the superposition principle are electromagnetic waves in optics and radio technology , forces in classical mechanics and states in quantum mechanics .

One area in which the principle of superposition does not apply due to the mathematical structure of the theory is the general theory of relativity to describe gravitation .

Math background

Mathematically, a superposition can be described as a linear combination

${\ displaystyle x (t) = \ sum _ {i = 1} ^ {n} {\ alpha _ {i} x_ {i} (t)}}$

represent. The sum formula states that any functions or quantities of the same kind can be added to a new quantity . The factor indicates the weighting of the respective component. ${\ displaystyle x_ {i} (t)}$${\ displaystyle x (t)}$${\ displaystyle \ alpha _ {i}}$

The validity of the principle in many physical systems is a consequence of the fact that they obey linear differential equations . If a homogeneous linear differential equation has the two solutions and , then their sum is also a solution due to the sum rule . In general terms: ${\ displaystyle f_ {1}}$${\ displaystyle f_ {2}}$${\ displaystyle f_ {1} + f_ {2}}$

If bis are solutions of a homogeneous linear differential equation, then every sum of these solutions is also a solution of the differential equation.${\ displaystyle f_ {1}}$${\ displaystyle f_ {n}}$

Wave theory

Two waves penetrate each other without affecting each other.

In wave theory, superposition means the undisturbed superposition ( interference ) of several waves of the same type. The relevant size of the superposition is the amplitude (the "height") of the individual waves. For example, several electromagnetic waves can be superimposed on one another, whereby their amplitudes at the same time reinforce each other at some points and weaken each other at others.

However, the resulting amplitude curve - apart from possible energy losses - does not affect the individual amplitude curves on which it is based. It is only the overall result of the "superimposed" individual courses. The waves cross each other without affecting each other. They only influence their medium of propagation.

Mathematically, the relationship results for the resulting wave function ${\ displaystyle \ Psi ({\ vec {x}}, t)}$

${\ displaystyle \ Psi ({\ vec {x}}, t) = \ sum _ {i = 1} ^ {n} \ Psi _ {i} ({\ vec {x}}, t)}$,

where are the wave functions of the original individual waves. ${\ displaystyle \ Psi _ {i} ({\ vec {x}}, t)}$

Classic mechanics

Powers

Superposition of two forces

Mechanical forces can also be superimposed. In this context, one speaks of the principle of undisturbed superposition of forces , the principle of the resulting force, or Newton's fourth law .

In mathematical terms, the relationship results

${\ displaystyle {\ vec {F}} = \ sum _ {i = 1} ^ {n} {\ vec {F}} _ {i}}$.

This expression says that different forces, all individually acting on the same body, have the same effect, as if only their sum were acting on the body.

The pushing of a box can be cited as an example: With regard to the end result, it does not matter whether a box is pushed first forwards and then to the left or whether it is pushed diagonally to the left and front.

Several load cases can be superimposed, but in the case of non-linear problems, for example in the (linearized) second order theory , this is not possible simply by adding the individual forces of the respective load cases, but requires a new determination of the internal forces under the action of all loads, since the forces are different rearrange, and there is also a softening (or stiffening) of the system in the deformed position. In the second order theory, loads also have a systemic character, since the stiffnesses depend in particular on the normal force.

Quantum mechanics

Superposition in quantum mechanics is comparable to that in classical wave theory, since quantum mechanical states are also described by wave functions . It should be noted, however, that the quantum mechanical wave functions, in contrast to the classical ones, do not yet have a "real" or unambiguous meaning. In the equivalent representation with state vectors , superposition simply means the addition (or linear combination) of vectors .

Mathematically, this is done in the Bra-Ket notation

${\ displaystyle | \ psi \ rangle = \ sum \ limits _ {i = 1} ^ {n} c_ {i} | \ varphi _ {i} \ rangle}$

expressed. This equation states that the overall state can be described by superimposing the possible individual states . It is therefore also called the superposition state . If these are all orthogonal to one another (and normalized), then the squares of the magnitude of the complex probability amplitudes indicate the probability of finding the associated state in a measurement that specializes in this state . ${\ displaystyle | \ psi \ rangle}$${\ displaystyle | \ varphi _ {i} \ rangle}$${\ displaystyle | \ varphi _ {i} \ rangle}$ ${\ displaystyle | c_ {i} | ^ {2}}$ ${\ displaystyle c_ {i}}$${\ displaystyle | \ varphi _ {i} \ rangle}$

Schrödinger's cat is often cited as an example . But the wave function of a particle can also be understood as a superposition state. It is the superposition of states in which the particle is localized in one place.

thermodynamics

Superposition principle in a transient heating process

The superposition principle is used in thermodynamics to calculate transient heating processes. All processes that contribute to the dissipation and supply of heat are superimposed. For example, you can determine the temperature of a power semiconductor at a certain point in time when a power pulse has acted on this component. ${\ displaystyle t}$

In the example on the left, an output acts from the point in time to , which causes the component to heat up. The temperature increases according to an exponential function (red curve): ${\ displaystyle t = 0}$${\ displaystyle t = t_ {1}}$

${\ displaystyle \ Delta T = k \, \ left (1-e ^ {- {\ frac {t} {t_ {1}}}} \ right)}$.

In order to determine the temperature of the component after the end of heating, the power pulse is allowed to continue to act and an equally large negative power pulse is applied at the end of the heating. This results in a “negative” warming curve (green curve). The sum of the two heating curves then gives the cooling function (blue curve).

Electrical engineering

In Electrical Engineering is meant by superimposing set the overlay method according to Helmholtz . It is a simplified method for calculating linear electrical circuits with multiple voltage and / or current sources. The superposition theorem states that the calculation can be carried out separately for each source, with all other (ideal) sources being set to the value zero. Voltage sources are replaced by short circuits (0 V) and current sources by interruptions (0 A), but the internal resistances of the sources remain in the circuit. At the end, the linear superimposition takes place by adding the calculated partial results with the correct sign.

Originally, the superposition theorem was formulated only for direct current and direct voltage . However, its validity is also transferred to alternating current and alternating voltage within the framework of the complex AC calculation . By using the operator calculation , for example the Laplace transform , it is even valid for any signal shape. In general, however, the superposition theorem only applies to circuits made up of linear components .

literature

Electrodynamics:

Quantum Mechanics: