Two level system Schrodinger Equation problem

[TheDestroyer]

Hello guys,

in the attached file, I can't understand how the guy arrived to the equation in the red rectangle.

My problem is: how could there not be dC2/dt term. Why only dC1/dt term?
ψ contains both C1 and C2, and when the derivative is applied, both have to be influenced, and both are functions of time. Why does C2 disappear?

Could someone please explain how this last step is exactly taken?

Thank you for any efforts.

 

[Bill_K]

Because it had been something like (dC₁/dt) ψ₁ + (dC₂/dt) ψ₂. But then he says "multiply by ψ₁* and integrate over all space." Well, ∫ψ₁*ψ₁ dx = 1 so you don't see it explicitly in the result, just dC₁/dt is left in the first term. But ∫ψ₁*ψ₂ dx = 0, by orthogonality, so the dC₂/dt term drops out completely.

 

[TheDestroyer]

Got it. Thanks man :)