Hermite Functions
📂FunctionsHermite Functions
Definitions
Hermite functions are defined as follows:
yn=(D−x)ne−2x2=e2x2Dne−x2
where D=dxd is the differential operator.
Description
Hermite functions are solutions to the differential equation
yn′′−x2yn=−(2n+1)yn,n=0,1,2,⋯
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) can be obtained directly by solving the differential equation. That (2) is identical to (1) can easily be shown through the following theorem.
Theorem
For any f(x),
(D−x)nf(x)=e2x2Dn[e−2x2f(x)],n=0,1,2,⋯
holds.
By substituting f(x)=e−2x2, it can be shown that (1) and (2) are identical.
Proof
Part 1 Proof for when n=0
(D−x)0f(x)=f(x)=e2x2D0[e−2x2f(x)]
Part 2 Proof that if it holds for n, it also holds for n+1
Assuming it holds for n, then for n+1 the right side of (3) is
e2x2Dn+1[e−2x2f(x)]=e2x2DnD[e−2x2f(x)]=e2x2Dn[−xe−2x2f(x)+e−2x2Df(x)]=e2x2Dn[e−2x2(D−x)f(x)]
Substituting with (D−x)f(x)=g(x) yields
e2x2Dn+1[e−2x2f(x)]=e2x2Dn[e−2x2(D−x)f(x)]=e2x2Dn[e−2x2g(x)]=(D−x)ng(x)=(D−x)n(D−x)f(x)=(D−x)n+1f(x)
The third equality holds since we assumed it holds for n.
Part 3.
As it holds for both n=0 and n, it consequently holds for n+1 by mathematical induction, thus valid for all n=0,1,2,⋯.
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