Quantum Harmonic oscillator

The eigenfunction of the quantum harmonic oscillator can be expressed in terms of the Hermite polynomials. The n-th eigenfunction Ψn(x) is given by:


Ψn(x)=12nn!(mωπ)1/4exp(mωx22)Hn(mωx)


where:


- n is the quantum number, which takes on values of 0, 1, 2, \ldots

- m is the mass of the particle,

- ω is the angular frequency of the oscillator,

- (h-bar) is the reduced Planck's constant (=h/2π, where h is Planck's constant),

- Hn(x) is the n-th Hermite polynomial,

- The factor 12nn!(mωπ)1/4 normalizes the wave function,

- The factor exp(mωx22) is a Gaussian envelope,

- The function mωx is a scaling of the x coordinate.


The Hermite polynomials are a set of orthogonal polynomials that appear frequently in quantum mechanics. They are defined through the generating function:


g(t,x)=exp(t2+2tx)=n=0Hn(x)n!tn


where the sum is over all nonnegative integers n. The n-th Hermite polynomial Hn(x) is then given by the coefficient of tn in this series expansion. These polynomials can be computed explicitly using the recurrence relation:


H0(x)=1,

H1(x)=2x,

Hn(x)=2xHn1(x)2(n1)Hn2(x)forn2.


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