Editing Wave function (section)

==Historical background==
{{Quantum mechanics|cTopic=Fundamental concepts}}

In 1900, [[Max Planck]] postulated the proportionality between the frequency <math>f</math> of a photon and its energy {{nowrap|<math>E</math>,}} {{nowrap|<math>E = hf</math>,}}<ref>{{cite web |title=Planck - A very short biography of Planck |url=https://spark.iop.org/planck |website=spark.iop.org |publisher=[[Institute of Physics]] |access-date=12 February 2023}}</ref><ref>{{cite book |title=C/CS Pys C191:Representations and Wave Functions 》 1. Planck-Einstein Relation E=hv |date=30 September 2008 |publisher=EESC Instructional and Electronics Support, [[University of California, Berkeley]] |page=1 |url=https://inst.eecs.berkeley.edu/~cs191/fa08/lectures/lecture8_fa08.pdf |access-date=12 February 2023}}</ref>
and in 1916 the corresponding relation between a photon's [[momentum]] <math>p</math> and [[wavelength]] {{nowrap|<math>\lambda</math>,}} {{nowrap|<math>\lambda = \frac{h}{p}</math>,}}<ref>{{harvnb|Einstein|1916|pp=47–62}}, and a nearly identical version {{harvnb|Einstein|1917|pp=121–128}} translated in {{harvnb|ter Haar|1967|pp=167–183}}.</ref>
where <math>h</math> is the [[Planck constant]]. In 1923, De Broglie was the first to suggest that the relation {{nowrap|<math>\lambda = \frac{h}{p}</math>,}} now called the [[matter wave|De Broglie relation]], holds for ''massive'' particles, the chief clue being [[Lorentz invariance]],{{sfn|de Broglie|1923|pp=507–510,548,630}} and this can be viewed as the starting point for the modern development of quantum mechanics. The equations represent [[wave–particle duality]] for both massless and massive particles.

In the 1920s and 1930s, quantum mechanics was developed using [[calculus]] and [[linear algebra]]. Those who used the techniques of calculus included [[Louis de Broglie]], [[Erwin Schrödinger]], and others, developing "[[wave|wave mechanics]]". Those who applied the methods of linear algebra included [[Werner Heisenberg]], [[Max Born]], and others, developing "[[matrix mechanics]]". Schrödinger subsequently showed that the two approaches were equivalent.{{sfn|Hanle|1977|pp=606–609}}

In 1926, Schrödinger published the famous wave equation now named after him, the [[Schrödinger equation]]. This equation was based on [[Classical physics|classical]] [[conservation of energy]] using [[operator (physics)|quantum operators]] and the de Broglie relations and the solutions of the equation are the wave functions for the quantum system.{{sfn|Schrödinger|1926|pp=1049–1070}} However, no one was clear on how to interpret it.{{sfn|Tipler|Mosca|Freeman|2008}} At first, Schrödinger and others thought that wave functions represent particles that are spread out with most of the particle being where the wave function is large.{{sfn|Weinberg|2013}} This was shown to be incompatible with the elastic scattering of a wave packet (representing a particle) off a target; it spreads out in all directions.<ref name=Born_1926_A>{{harvnb|Born|1926a}}, translated in {{harvnb|Wheeler|Zurek|1983}} at pages 52–55.</ref>
While a scattered particle may scatter in any direction, it does not break up and take off in all directions. In 1926, Born provided the perspective of [[probability amplitude]].<ref name=Born_1926_A /><ref name="Born_1926_B">{{harvnb|Born|1926b}}, translated in {{harvnb|Ludwig|1968|pp=206–225}}. Also [http://www.ymambrini.com/My_World/History_files/Born_1.pdf here] {{Webarchive|url=https://web.archive.org/web/20201201173255/http://www.ymambrini.com/My_World/History_files/Born_1.pdf |date=2020-12-01 }}.</ref>{{sfn|Young|Freedman|2008|p=1333}} This relates calculations of quantum mechanics directly to probabilistic experimental observations. It is accepted as part of the [[Copenhagen interpretation]] of quantum mechanics. There are many other [[interpretations of quantum mechanics]]. In 1927, [[Douglas Hartree|Hartree]] and [[Vladimir Fock|Fock]] made the first step in an attempt to solve the [[Many-body problem|''N''-body]] wave function, and developed the ''self-consistency cycle'': an [[Iteration|iterative]] [[algorithm]] to approximate the solution. Now it is also known as the [[Hartree–Fock method]].{{sfn|Atkins|1974}} The [[Slater determinant]] and [[Permanent (mathematics)|permanent]] (of a [[Matrix (mathematics)|matrix]]) was part of the method, provided by [[John C. Slater]].

Schrödinger did encounter an equation for the wave function that satisfied [[theory of relativity|relativistic]] energy conservation ''before'' he published the non-relativistic one, but discarded it as it predicted negative [[probability|probabilities]] and negative [[energy|energies]]. In 1927, [[Oskar Klein|Klein]], [[Walter Gordon (physicist)|Gordon]] and Fock also found it, but incorporated the [[Electromagnetic force|electromagnetic]] [[Fundamental interaction|interaction]] and proved that it was [[Lorentz covariance|Lorentz invariant]]. De Broglie also arrived at the same equation in 1928. This relativistic wave equation is now most commonly known as the [[Klein–Gordon equation]].{{sfn|Martin|Shaw|2008}}

In 1927, [[Wolfgang Pauli|Pauli]] phenomenologically found a non-relativistic equation to describe spin-1/2 particles in electromagnetic fields, now called the [[Pauli equation]].{{sfn|Pauli|1927|pp=601–623.}} Pauli found the wave function was not described by a single complex function of space and time, but needed two complex numbers, which respectively correspond to the spin +1/2 and −1/2 states of the fermion. Soon after in 1928, [[Paul Dirac|Dirac]] found an equation from the first successful unification of [[special relativity]] and quantum mechanics applied to the [[electron]], now called the [[Dirac equation]]. In this, the wave function is a [[Dirac spinor|''spinor'']] represented by four complex-valued components:{{sfn|Atkins|1974}} two for the electron and two for the electron's [[antiparticle]], the [[positron]]. In the non-relativistic limit, the Dirac wave function resembles the Pauli wave function for the electron. Later, other [[relativistic wave equations]] were found.

===Wave functions and wave equations in modern theories===
All these wave equations are of enduring importance. The Schrödinger equation and the Pauli equation are under many circumstances excellent approximations of the relativistic variants. They are considerably easier to solve in practical problems than the relativistic counterparts.

The [[Klein–Gordon equation]] and the [[Dirac equation]], while being relativistic, do not represent full reconciliation of quantum mechanics and special relativity. The branch of quantum mechanics where these equations are studied the same way as the Schrödinger equation, often called [[relativistic quantum mechanics]], while very successful, has its limitations (see e.g. [[Lamb shift]]) and conceptual problems (see e.g. [[Dirac sea]]).

Relativity makes it inevitable that the number of particles in a system is not constant. For full reconciliation, [[quantum field theory]] is needed.<ref>{{harvtxt|Weinberg|2002}} takes the standpoint that quantum field theory appears the way it does because it is the ''only'' way to reconcile quantum mechanics with special relativity.</ref>
In this theory, the wave equations and the wave functions have their place, but in a somewhat different guise. The main objects of interest are not the wave functions, but rather operators, so called ''field operators'' (or just fields where "operator" is understood) on the Hilbert space of states (to be described next section). It turns out that the original relativistic wave equations and their solutions are still needed to build the Hilbert space. Moreover, the ''free fields operators'', i.e. when interactions are assumed not to exist, turn out to (formally) satisfy the same equation as do the fields (wave functions) in many cases.

Thus the Klein–Gordon equation (spin {{math|0}}) and the Dirac equation (spin {{math|{{frac|1|2}}}}) in this guise remain in the theory. Higher spin analogues include the [[Proca equation]] (spin {{math|1}}), [[Rarita–Schwinger equation]] (spin {{math|{{frac|3|2}}}}), and, more generally, the [[Bargmann–Wigner equations]]. For ''massless'' free fields two examples are the free field [[Maxwell equation]] (spin {{math|1}}) and the free field [[Einstein equation]] (spin {{math|2}}) for the field operators.<ref>{{harvtxt|Weinberg|2002}} See especially chapter 5, where some of these results are derived.</ref>
All of them are essentially a direct consequence of the requirement of [[Lorentz invariance]]. Their solutions must transform under [[Lorentz transformation]] in a prescribed way, i.e. under a particular [[Representation theory of the Lorentz group#Common representations|representation of the Lorentz group]] and that together with few other reasonable demands, e.g. the [[cluster decomposition|cluster decomposition property]],<ref>{{harvnb|Weinberg|2002}} Chapter 4.</ref>
with implications for [[causality]] is enough to fix the equations.

This applies to free field equations; interactions are not included. If a Lagrangian density (including interactions) is available, then the Lagrangian formalism will yield an equation of motion at the classical level. This equation may be very complex and not amenable to solution. Any solution would refer to a ''fixed'' number of particles and would not account for the term "interaction" as referred to in these theories, which involves the creation and annihilation of particles and not external potentials as in ordinary "first quantized" quantum theory.

In [[string theory]], the situation remains analogous. For instance, a wave function in momentum space has the role of Fourier expansion coefficient in a general state of a particle (string) with momentum that is not sharply defined.{{sfn|Zwiebach|2009}}