Solving the quantum harmonic oscillator via analytic method (Made Easy)

In this video I will solve the quantum harmonic oscillator using the analytic method. I tried really hard to explain every single step in depth, unlike most of what I had seen online. I really hope that this is helpful to you. My name is Nick Heumann, I am a recently graduated physicist. I love to teach physics, so I decided to give YouTube a try. English is not my first language, but I hope that you can understand me well enough regardless. ▬ Contents of this video ▬▬▬▬▬▬▬▬▬▬ 00:00 Introducing the Method 00:50 Simplifying the equation 9:00 Looking for asymptotical solutions 11:10 Solving the asymptotical equation 13:40 Checking that the wavefunction satisfies the equation 18:10 Putting it all together 18:28 Solving the Schrödinger Equation for h 22:40 Using the power series method 30:20 Discussing the recursive relation 36:10 Finding some wavefunctions 43:20 introducing the hermite polynomials 45:00 Normalizing the wavefunctions