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Hi! I'm currently studying Griffith's fantastic book on QM, and I'm confused for a bit about the wave function for a free particle.

Here's what I think so far; for a free particle, there are no stationary states, so therefore we can't solve the SE with

ψ(x)=Ae

That is, we can't write a discrete sum. But We can have solutions as:

ψ(x,t)=∫dkφ(k)e

I don't know if my understanding is correct, so please tell me so. Now, I assume that this understanding is correct and get to the question: If the solutions can only be the latter, then why was the solution from the book for the scattering states in the delta function potential a sum of stationary states and not the continuous sum? Also, why is it the same for the bound states if we are solving for the free particle when x<0 and x>0? Is it because it has a potential at x=0?

Thanks in advanced!

Here's what I think so far; for a free particle, there are no stationary states, so therefore we can't solve the SE with

ψ(x)=Ae

^{ikx}+Be^{-ikx}That is, we can't write a discrete sum. But We can have solutions as:

ψ(x,t)=∫dkφ(k)e

^{i(kx-ωt)}I don't know if my understanding is correct, so please tell me so. Now, I assume that this understanding is correct and get to the question: If the solutions can only be the latter, then why was the solution from the book for the scattering states in the delta function potential a sum of stationary states and not the continuous sum? Also, why is it the same for the bound states if we are solving for the free particle when x<0 and x>0? Is it because it has a potential at x=0?

Thanks in advanced!