Quantum Harmonic Oscillator Part-5: Hermite Polynomial and Normalised Harmonic Oscillator Wavefunctions

This article is the Part 5 of the article series I wrote about Quantum Harmonic Oscillator. If you have not read the Part-1: Introduction to Quantum Harmonic Oscillator, Part 2:Schrödinger’s Equation with Dimensionless Terms, Part-3: The Asymptotic solution and Part-4: The Series Solution of Schrödinger’s Equation, then you cannot understand what I am going to explain in this article, so reading those articles is a must.

In this article, I will introduce you to Hermite Polynomials. Although, I will not discuss its full details and normalization as its an advanced mathematical topic and out of the scope of this article, rather you can just search it in the net there is ample resources available about it.

Before moving forward remember that, when we introduced H in the previous article, we declared it as an unknown variable. In this article, we will try to find out more about that.

So, if you find the family of solution of Hₙ(ξ) to equation 5 of my previous article, then those solutions are known as Hermite Polynomials. There are two major types of Hermite polynomial, one is Probabilist’s Hermite Polynomial and another one is Physicist’s Hermite Polynomial. Here we will use the second one.

For example, I am listing few of them below,

H0(ξ) =1

H1(ξ) = 2ξ

H₂(ξ) = 4ξ² – 2

H₃(ξ) = 8ξ² – 12 ξ

One thing which you can notice that the highest power of ξ appearing in the series for Hn(ξ) is ξn. If you want to get result in terms of real (x) then just substitute ξ = αx.

So, now, I hope you got the idea about how to solve the equation 5 of previous article.

Now, moving on to the normalized harmonic-oscillator wavefunctions. It is given by

ψn(ξ) = An H(ξ) e^(-ξ²/2)         (1)

where An is the normalization factor,

An = √[α/{(2^n) n! √π}]

And where,

α = (mk/ ħ²)^1/4

Equation 1 here is obtained from equation 1 of the series solution article.

Harmonic oscillator wavefunctions for n=0,1,2,3

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                                                                 -Ratnadeep Das Choudhury

                                                  Founder and writer of The Dynamic Frequency