logo

Hermite Functions 📂Functions

Hermite Functions

Definitions

Hermite functions are defined as follows: yn=(Dx)nex22=ex22Dnex2 \begin{align} y_{n} &= \left( D-x \right)^{n} e^{-\frac{x^{2}}{2}} \\ &=e^{\frac{x^{2}}{2}} D^{n} e^{-x^{2}} \end{align} where D=ddxD=\frac{d}{dx} is the differential operator.

Description

Hermite functions are solutions to the differential equation ynx2yn=(2n+1)yn,n=0,1,2, y_{n}^{\prime \prime}-x^{2}y_{n}=-(2n+1)y_{n},\quad n=0,1,2,\cdots and represent the solution to the one-dimensional harmonic oscillator Schrödinger equation in physics, i.e., the wave function of a one-dimensional harmonic oscillator. (1)(1) can be obtained directly by solving the differential equation. That (2)(2) is identical to (1)(1) can easily be shown through the following theorem.

Theorem

For any f(x)f(x), (Dx)nf(x)=ex22Dn[ex22f(x)],n=0,1,2, \begin{align} (D-x)^{n}f(x)=e^{\frac{x^{2}}{2}}D^{n}\left[ e^{-\frac{x^{2}}{2}}f(x) \right] ,\quad n=0,1,2,\cdots \end{align} holds.

By substituting f(x)=ex22f(x)=e^{-\frac{x^{2}}{2}}, it can be shown that (1)(1) and (2)(2) are identical.

Proof

  • Part 1 Proof for when n=0n=0 (Dx)0f(x)=f(x)=ex22D0[ex22f(x)] (D-x)^{0}f(x)=f(x)=e^{\frac{x^{2}}{2}}D^{0}\left[ e^{-\frac{x^{2}}{2}}f(x) \right]

  • Part 2 Proof that if it holds for nn, it also holds for n+1n+1

    Assuming it holds for nn, then for n+1n+1 the right side of (3)(3) is ex22Dn+1[ex22f(x)]=ex22DnD[ex22f(x)]=ex22Dn[xex22f(x)+ex22Df(x)]=ex22Dn[ex22(Dx)f(x)] \begin{align*} e^{\frac{x^{2}}{2}}D^{n+1} \left[ e^{-\frac{x^{2}}{2}}f(x) \right] &=e^{\frac{x^{2}}{2}}D^{n} D\left[ e^{-\frac{x^{2}}{2}}f(x) \right] \\ &=e^{\frac{x^{2}}{2}}D^{n} \left[ -xe^{-\frac{x^{2}}{2}}f(x) +e^{-\frac{x^{2}}{2}}Df(x)\right] \\ &=e^{\frac{x^{2}}{2}}D^{n} \left[ e^{-\frac{x^{2}}{2}}(D-x)f(x) \right] \end{align*} Substituting with (Dx)f(x)=g(x)(D-x)f(x)=g(x) yields ex22Dn+1[ex22f(x)]=ex22Dn[ex22(Dx)f(x)]=ex22Dn[ex22g(x)]=(Dx)ng(x)=(Dx)n(Dx)f(x)=(Dx)n+1f(x) \begin{align*} e^{\frac{x^{2}}{2}}D^{n+1} \left[ e^{-\frac{x^{2}}{2}}f(x) \right] &=e^{\frac{x^{2}}{2}}D^{n} \left[ e^{-\frac{x^{2}}{2}}(D-x)f(x) \right] \\ &=e^{\frac{x^{2}}{2}}D^{n} \left[ e^{-\frac{x^{2}}{2}}g(x) \right] \\ &= (D-x)^{n}g(x) \\ &= (D-x)^{n}(D-x)f(x) \\ &= (D-x)^{n+1}f(x) \end{align*} The third equality holds since we assumed it holds for nn.

  • Part 3.

    As it holds for both n=0n=0 and nn, it consequently holds for n+1n+1 by mathematical induction, thus valid for all n=0,1,2,n=0,1,2,\cdots.