# 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 H****ₙ****(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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