Editing Hermite polynomials (section)

===Definition===
One can define the '''Hermite functions''' (often called Hermite-Gaussian functions) from the physicist's polynomials:
<math display="block">\psi_n(x) = \left (2^n n! \sqrt{\pi} \right )^{-\frac12} e^{-\frac{x^2}{2}} H_n(x) = (-1)^n \left (2^n n! \sqrt{\pi} \right)^{-\frac12} e^{\frac{x^2}{2}}\frac{d^n}{dx^n} e^{-x^2}.</math>
Thus, 
<math display="block">\sqrt{2(n+1)}~~\psi_{n+1}(x)=    \left ( x- {d\over dx}\right ) \psi_n(x).</math>

Since these functions contain the square root of the [[weight function]] and have been scaled appropriately, they are [[Orthonormality|orthonormal]]:
<math display="block">\int_{-\infty}^\infty \psi_n(x) \psi_m(x) \,dx = \delta_{nm},</math>
and they form an orthonormal basis of {{math|''L''<sup>2</sup>('''R''')}}. This fact is equivalent to the corresponding statement for Hermite polynomials (see above).

The Hermite functions are closely related to the [[Whittaker function]] {{Harv|Whittaker|Watson|1996}} {{math|''D''<sub>''n''</sub>(''z'')}}:
<math display="block">D_n(z) = \left(n! \sqrt{\pi}\right)^{\frac12} \psi_n\left(\frac{z}{\sqrt 2}\right) = (-1)^n e^\frac{z^2}{4} \frac{d^n}{dz^n} e^\frac{-z^2}{2}</math>
and thereby to other [[parabolic cylinder function]]s.

The Hermite functions satisfy the differential equation
<math display="block">\psi_n''(x) + \left(2n + 1 - x^2\right) \psi_n(x) = 0.</math>
This equation is equivalent to the [[Schrödinger equation]] for a harmonic oscillator in quantum mechanics, so these functions are the [[eigenfunctions]].

[[Image:Herm5.svg|thumb|center|450px|Hermite functions: 0 (blue, solid), 1 (orange, dashed), 2 (green, dot-dashed), 3 (red, dotted), 4 (purple, solid), and 5 (brown, dashed)]]
<math display="block">\begin{align}
 \psi_0(x) &= \pi^{-\frac14} \, e^{-\frac12 x^2}, \\
 \psi_1(x) &= \sqrt{2} \, \pi^{-\frac14} \, x \, e^{-\frac12 x^2}, \\
 \psi_2(x) &= \left(\sqrt{2} \, \pi^{\frac14}\right)^{-1} \, \left(2x^2-1\right) \, e^{-\frac12 x^2}, \\
 \psi_3(x) &= \left(\sqrt{3} \, \pi^{\frac14}\right)^{-1} \, \left(2x^3-3x\right) \, e^{-\frac12 x^2}, \\
 \psi_4(x) &= \left(2 \sqrt{6} \, \pi^{\frac14}\right)^{-1} \, \left(4x^4-12x^2+3\right) \, e^{-\frac12 x^2}, \\
 \psi_5(x) &= \left(2 \sqrt{15} \, \pi^{\frac14}\right)^{-1} \, \left(4x^5-20x^3+15x\right) \, e^{-\frac12 x^2}.
\end{align}</math>
[[Image:Herm50.svg|thumb|center|680px|Hermite functions: 0 (blue, solid), 2 (orange, dashed), 4 (green, dot-dashed), and 50 (red, solid)]]