Editing Superposition principle (section)

==Wave superposition==
{{further|Wave|Wave equation}}

[[File: Standing wave 2.gif|thumb|right|Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a [[standing wave]].]] <!-- same as below! -->
[[File:Standing_waves1.gif|thumb|two waves permeate without influencing each other]]

Waves are usually described by variations in some parameters through space and time—for example, height in a water wave, [[pressure]] in a sound wave, or the [[electromagnetic field]] in a light wave. The value of this parameter is called the [[amplitude]] of the wave and the wave itself is a [[function (mathematics)|function]] specifying the amplitude at each point.

In any system with waves, the waveform at a given time is a function of the [[wave equation|sources]] (i.e., external forces, if any, that create or affect the wave) and [[initial condition]]s of the system. In many cases (for example, in the classic [[wave equation]]), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at the top.)

===Wave diffraction vs. wave interference{{anchor|Diffraction vs. interference}}{{anchor|Interference vs. diffraction}}===
With regard to wave superposition, [[Richard Feynman]] wrote:<ref>Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [https://books.google.com/books?id=S-JFAgAAQBAJ&dq=feynman+interference+and+diffraction&pg=SA30-PA1]</ref>
{{blockquote|No-one has ever been able to define the difference between [[interference (wave propagation)|interference]] and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do, roughly speaking, is to say that when there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used.|author=|title=|source=}}

Other authors elaborate:<ref>N. K. VERMA, ''Physics for Engineers'', PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [https://books.google.com/books?id=kY-7AQAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA361]</ref>
{{blockquote|The difference is one of convenience and convention. If the waves to be superposed originate from a few coherent sources, say, two, the effect is called interference. On the other hand, if the waves to be superposed originate by subdividing a wavefront into infinitesimal coherent wavelets (sources), the effect is called diffraction. That is the difference between the two phenomena is [a matter] of degree only, and basically, they are two limiting cases of superposition effects.}}

Yet another source concurs:<ref>Tim Freegarde, ''Introduction to the Physics of Waves'', Cambridge University Press, Nov 8, 2012. [https://books.google.com/books?id=eMMgAwAAQBAJ&dq=feynman+interference+and+diffraction&pg=PA106]</ref>
{{blockquote|In as much as the interference fringes observed by Young were the diffraction pattern of the double slit, this chapter [Fraunhofer diffraction] is, therefore, a continuation of Chapter 8 [Interference]. On the other hand, few opticians would regard the Michelson interferometer as an example of diffraction. Some of the important categories of diffraction relate to the interference that accompanies division of the wavefront, so Feynman's observation to some extent reflects the difficulty that we may have in distinguishing division of amplitude and division of wavefront.}}

===Wave interference===
{{Main|Interference (wave propagation)}}

The phenomenon of [[Interference (wave propagation)|interference]] between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in [[noise-canceling headphones]], the summed variation has a smaller [[amplitude]] than the component variations; this is called ''destructive interference''. In other cases, such as in a [[line array]], the summed variation will have a bigger amplitude than any of the components individually; this is called ''constructive interference''.
[[File:Waventerference.gif|thumb|green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.]]  <!-- same as above! -->
{|
|- style="border-bottom: solid thin black"
| | combined<br> waveform
| colspan="2" rowspan="3" | [[File:Interference of two waves.svg]]
|- style="border-bottom: solid thin black"
| wave 1
|-
| wave 2
|-
|
| Two waves in phase
| Two waves 180° out <br>of phase
|}

===Departures from linearity===
In most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles [[nonlinear optics]] and [[nonlinear acoustics]].

===Quantum superposition===
{{main|Quantum superposition}}

In [[quantum mechanics]], a principal task is to compute how a certain type of wave [[wave propagation|propagates]] and behaves. The wave is described by a [[wave function]], and the equation governing its behavior is called the [[Schrödinger equation]]. A primary approach to computing the behavior of a wave function is to write it as a superposition (called "[[quantum superposition]]") of (possibly infinitely many) other wave functions of a certain type—[[stationary state]]s whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.<ref name="QuaMech">Quantum Mechanics, [[Hendrik Anthony Kramers|Kramers, H.A.]] publisher Dover, 1957, p. 62 {{ISBN|978-0-486-66772-0}}</ref>

{{Anchor|projective2016-01-30}}The projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a ''ray'' in [[projective Hilbert space]], not a ''vector''.
According to [[Paul Dirac|Dirac]]: "''if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state'' [italics in original]."<ref>[[Paul Adrien Maurice Dirac|Dirac, P. A. M.]] (1958). ''The Principles of Quantum Mechanics'', 4th edition, Oxford, UK: Oxford University Press, p.&nbsp;17.</ref>
However, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himself
uses ket vector representations of states to decompose or split,
for example, a ket vector <math>|\psi_i\rangle</math>
into superposition of component ket vectors <math>|\phi_j\rangle</math> as:
<math display="block">|\psi_i\rangle = \sum_{j}{C_j}|\phi_j\rangle,</math>
where the <math>C_j\in \textbf{C}</math>.
The equivalence class of the <math>|\psi_i\rangle</math> allows a well-defined meaning to be given to the relative phases of the <math>C_j</math>.,<ref>{{cite journal|last1=Solem|first1=J. C.|last2=Biedenharn|first2=L. C.|year=1993|title=Understanding geometrical phases in quantum mechanics: An elementary example|journal=Foundations of Physics|volume=23|issue=2|pages=185–195|bibcode = 1993FoPh...23..185S |doi = 10.1007/BF01883623 |s2cid=121930907}}</ref> but an absolute (same amount
for all the <math>C_j</math>) phase change on the <math>C_j</math>
does not affect the equivalence class of the <math>|\psi_i\rangle</math>.

There are exact correspondences between the superposition presented in the main on this page and the quantum superposition.
For example, the [[Bloch sphere]] to represent [[pure state]] of a [[two-level system|two-level quantum mechanical system]]
([[qubit]]) is also known as the [[Bloch sphere|Poincaré sphere]] representing different types of classical
pure [[Polarization (waves)|polarization]] states.

Nevertheless, on the topic of quantum superposition, [[Hans Kramers|Kramers]] writes: "The principle of [quantum] superposition ... has no analogy in classical physics"{{Citation needed|date=March 2023|reason=hopefully not only conclusion but also reasoning}}.
According to [[Paul Dirac|Dirac]]: "''the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory'' [italics in original]."<ref>[[Paul Adrien Maurice Dirac|Dirac, P. A. M.]] (1958). ''The Principles of Quantum Mechanics'', 4th edition, Oxford, UK: Oxford University Press, p.&nbsp;14.</ref>
Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase,
they actually mean phase translation symmetry derived from [[time translation symmetry]], which is also
applicable to classical states, as shown above with classical polarization states.