Editing Wave function (section)

=== Relations between position and momentum representations ===
The {{math|''x''}} and {{math|''p''}} representations are
<math display="block">\begin{align}
|\Psi\rangle = I|\Psi\rangle &= \int |x\rangle \langle x|\Psi\rangle dx = \int \Psi(x) |x\rangle dx,\\
|\Psi\rangle = I|\Psi\rangle &= \int |p\rangle \langle p|\Psi\rangle dp = \int \Phi(p) |p\rangle dp.
\end{align}</math>

Now take the projection of the state {{math|Ψ}} onto eigenfunctions of momentum using the last expression in the two equations,
<math display="block">\int \Psi(x) \langle p|x\rangle dx = \int \Phi(p') \langle p|p'\rangle dp' = \int \Phi(p') \delta(p-p') dp' = \Phi(p).</math>

Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of the [[Free_particle#Mathematical_description|free Schrödinger equation]]
<math display="block">\langle x | p \rangle = p(x) = \frac{1}{\sqrt{2\pi\hbar}}e^{\frac{i}{\hbar}px} \Rightarrow \langle p | x \rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{-\frac{i}{\hbar}px},</math>
one obtains
<math display="block">\Phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int \Psi(x)e^{-\frac{i}{\hbar}px}dx\,.</math>

Likewise, using eigenfunctions of position,
<math display="block">\Psi(x) = \frac{1}{\sqrt{2\pi\hbar}}\int \Phi(p)e^{\frac{i}{\hbar}px}dp\,.</math>

The position-space and momentum-space wave functions are thus found to be [[Fourier transform]]s of each other.{{sfn|Griffiths|2004}} They are two representations of the same state; containing the same information, and either one is sufficient to calculate any property of the particle.

In practice, the position-space wave function is used much more often than the momentum-space wave function. The potential entering the relevant equation (Schrödinger, Dirac, etc.) determines in which basis the description is easiest. For the [[harmonic oscillator]], {{mvar|x}} and {{mvar|p}} enter symmetrically, so there it does not matter which description one uses. The same equation (modulo constants) results. From this, with a little bit of afterthought, it follows that solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in {{math|''L''<sup>2</sup>}}.<ref group=nb>The Fourier transform viewed as a unitary operator on the space {{math|''L''<sup>2</sup>}} has eigenvalues {{math|±1, ±''i''}}. The eigenvectors are "Hermite functions", i.e. [[Hermite polynomials]] multiplied by a [[Gaussian function]]. See {{harvtxt|Byron|Fuller|1992}} for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9.</ref>