Hydrogen Atom Part 3: the Angular equation (deriving spherical harmonics, solving Legendre Dif. Eq.)
Автор: Nick Heumann University
Загружено: 2021-02-28
Просмотров: 4818
In this video I will solve the angular equation for the hydrogen atom. In the process we will solve Legendre's Differential Equation and derive the Legendre Polynomials, we will also solve the Associated Legendre Differential Equation and derive (and normalize) the spherical harmonics
In the next video we will begin to solve the radial equation
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 ▬▬▬▬▬▬▬▬▬▬
00:00:00 Introducing the work ahead
00:02:10 substituting x=cos theta
00:10:00 Solving the Associated Legendre Differential Equation
00:12:10 Using substitution
00:20:00 Substituting first half into the Equation
00:26:00 Finishing the substitutions
00:30:22 Comparing to the Legendre Differential Equation (case m=0)
00:32:50 Solving the Legendre Dif. Eq. by power series
00:36:10 Determining the recursion relation
00:41:00 Understanding the recursion relation
00:48:30 Solving for other m
00:53:40 Introducing the Associated Legendre Polynomials
00:56:20 Putting solutions to theta and phi back to Y (spherical harmonics)
00:57:10 Normalizing the spherical harmonics
01:06:50 The Result!
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