WKB Approximation: The Connection Formulas (Case E=V) Derivation (Upwards Sloping)

In this video I will explain the semiclassical approximation (WKBJ) for the case E=V. We will find the connection formulas for the downsloping potential by using the solutions of the Airy Equation.

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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 Explaining the situation
00:50 What is the problem here?
02:20 Explaining the steps that we will take
04:00 What about the down-sloping potential?
05:00 Approximating the Potential in the patching region
06:00 Solving the Schrödinger Equation in this region
08:58 Explaining how to solve the Airy Differential Equation (not necessary)
11:02 Explaining the Solutions to the Airy Differential Equation
15:45 Applying the border conditions (Equate the wavefunctions in the asymptotic limit)
20:35 Calculating the momentum
23:40 Calculating the integral of the momentum
25:55 Equating wavefunction 3 to patching wavefunction
31:15 Equating wavefunction 1 to patching wavefunction
39:00 Finding B and C in terms of D
41:30 Plugging everything into the wavefunction
43:30 Simplifying the result
46:39 Please check out my patreon!

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