Editing Fourier transform (section)

=== Quantum mechanics ===
The Fourier transform is useful in [[quantum mechanics]] in at least two different ways. To begin with, the basic conceptual structure of quantum mechanics postulates the existence of pairs of [[complementary variables]], connected by the [[Heisenberg uncertainty principle]]. For example, in one dimension, the spatial variable {{mvar|q}} of, say, a particle, can only be measured by the quantum mechanical "[[position operator]]" at the cost of losing information about the momentum {{mvar|p}} of the particle. Therefore, the physical state of the particle can either be described by a function, called "the wave function", of {{mvar|q}} or by a function of {{mvar|p}} but not by a function of both variables. The variable {{mvar|p}} is called the conjugate variable to {{mvar|q}}. In classical mechanics, the physical state of a particle (existing in one dimension, for simplicity of exposition) would be given by assigning definite values to both {{mvar|p}} and {{mvar|q}} simultaneously. Thus, the set of all possible physical states is the two-dimensional real vector space with a {{mvar|p}}-axis and a {{mvar|q}}-axis called the [[phase space]].

In contrast, quantum mechanics chooses a polarisation of this space in the sense that it picks a subspace of one-half the dimension, for example, the {{mvar|q}}-axis alone, but instead of considering only points, takes the set of all complex-valued "wave functions" on this axis. Nevertheless, choosing the {{mvar|p}}-axis is an equally valid polarisation, yielding a different representation of the set of possible physical states of the particle. Both representations of the wavefunction are related by a Fourier transform, such that 
<math display="block">\phi(p) = \int dq\, \psi (q) e^{-i pq/h} ,</math>
or, equivalently, 
<math display="block">\psi(q) = \int dp \, \phi (p) e^{i pq/h}.</math>

Physically realisable states are {{math|''L''<sup>2</sup>}}, and so by the [[Plancherel theorem]], their Fourier transforms are also {{math|''L''<sup>2</sup>}}.  (Note that since {{mvar|q}} is in units of distance and {{mvar|p}} is in units of momentum, the presence of the Planck constant in the exponent makes the exponent [[Nondimensionalization|dimensionless]], as it should be.)

Therefore, the Fourier transform can be used to pass from one way of representing the state of the particle, by a wave function of position, to another way of representing the state of the particle: by a wave function of momentum. Infinitely many different polarisations are possible, and all are equally valid. Being able to transform states from one representation to another by the Fourier transform is not only convenient but also the underlying reason of the Heisenberg [[#Uncertainty principle|uncertainty principle]].

The other use of the Fourier transform in both quantum mechanics and [[quantum field theory]] is to solve the applicable wave equation. In non-relativistic quantum mechanics, the [[Schrödinger equation]] for a time-varying wave function in one-dimension, not subject to external forces, is
<math display="block">-\frac{\partial^2}{\partial x^2} \psi(x,t) = i \frac h{2\pi} \frac{\partial}{\partial t} \psi(x,t).</math>

This is the same as the heat equation except for the presence of the imaginary unit {{mvar|i}}. Fourier methods can be used to solve this equation.

In the presence of a potential, given by the potential energy function {{math|''V''(''x'')}}, the equation becomes
<math display="block">-\frac{\partial^2}{\partial x^2} \psi(x,t) + V(x)\psi(x,t) = i \frac h{2\pi} \frac{\partial}{\partial t} \psi(x,t).</math>

The "elementary solutions", as we referred to them above, are the so-called "stationary states" of the particle, and Fourier's algorithm, as described above, can still be used to solve the boundary value problem of the future evolution of {{mvar|ψ}} given its values for {{math|''t'' {{=}} 0}}. Neither of these approaches is of much practical use in quantum mechanics. Boundary value problems and the time-evolution of the wave function is not of much practical interest: it is the stationary states that are most important.

In relativistic quantum mechanics, the Schrödinger equation becomes a wave equation as was usual in classical physics, except that complex-valued waves are considered. A simple example, in the absence of interactions with other particles or fields, is the free one-dimensional Klein–Gordon–Schrödinger–Fock equation, this time in dimensionless units,
<math display="block">\left (\frac{\partial^2}{\partial x^2} +1 \right) \psi(x,t) = \frac{\partial^2}{\partial t^2} \psi(x,t).</math>

This is, from the mathematical point of view, the same as the wave equation of classical physics solved above (but with a complex-valued wave, which makes no difference in the methods). This is of great use in quantum field theory: each separate Fourier component of a wave can be treated as a separate harmonic oscillator and then quantized, a procedure known as "second quantization". Fourier methods have been adapted to also deal with non-trivial interactions.

Finally, the [[Quantum harmonic oscillator#Ladder operator method|number operator]] of the [[quantum harmonic oscillator]] can be interpreted, for example via the [[Mehler kernel#Physics version|Mehler kernel]], as the [[Symmetry in quantum mechanics|generator]] of the [[#Eigenfunctions|Fourier transform]] <math>\mathcal{F}</math>.<ref name="auto"/>