Quantum harmonic oscillator

Quantum harmonic oscillator is one of the most important concept for graduate Physics and chemistry students as well as huge group of engineers working around the world in various fields such as material science and semiconductor industry. But unfortunately even this simple problem of quadratic potential,without any perturbations is difficult to deal analytically. The solution for Schrödinger equation for Harmonic oscillator in general is very difficult,for the reason it involves Forbenius method of series solution(brute force solution). But there's a cute trick which most of the people deploy inorder to study Harmonic oscillator. Eventhough it does not do a complete job,it suffices for most of the part like exploiting ground state wavefunction and energy,how energy gets incremented with quantum number n,etc. But even this method is difficult for most students as well as faculty because of the wierd definitions that come up all the way till the end of the treatment(hand wavings). Even I felt the same while reading it from Griffiths. But I bridged that gap wherever necessary and re written the whole approach with my own touch. I'm uploading that PDF below. Those who find this helpful, please give me a feedback in comment section. If you have any questions,you are free to comment them too. Here goes the link