why do we use hermite polynomials in harmonic oscilator wave function?

Answer 1

Hermite polynomials form a complete orthogonal basis. What this means is that you can take any function, it can be as complex as you want, and you can express that function as a sum of Hermite polynomials. Usually only the first few terms will be significant and you can ignore lower order terms.

This technique is can be applied to a variety of different sets of orthogonal functions to solve difficult problems. You may have heard of other such sets like Legendre polynomials, Bessel functions, Fourier series (sines and cosines), etc. Which functions you choose is usually determined by the geometry and physics of interest.

Answer 2

The harmonic oscillator Schroedinger equation is difficult to solve. What people do is assume the solution has the form of a polynomial times an error function. When this is done the polynomials turn out to be Hermite polynomials.

Answer 3

Therefore I will divide Him a portion with the great, And He shall divide the spoil with the strong, Because He poured out His soul unto death, And He was numbered with the transgressors, And He bore the sin of many, And made intercession for the transgressors.