Proving linear combinations of Schrödinger solutions are also solutions

[freddyfish]

This is, when it all comes around, just math. I am asked to prove that if the schroedinger equation looks like:

$\frac{d^{2}}{dx^{2}}$ψ=-$\frac{4πm}{h}$*[E-U(x)]ψ(x)

and ψ₁ and ψ₂ are two separate solutions for the same potential energy U(x), then Aψ₁ + Bψ₂ is also a solution of the equation.

I am asking this because I think it is obvious that the last solution also satisfies the S.E. if the two terms of it are two individual solutions.

To show that the linear combination is also a solution all you have to do is replace ψ by Aψ₁ + Bψ₂ in the differential equation, but this feels more like confirming rather than proving. Since I'm pretty new to this quantum mechanic discipline, I would prefer a simple proof, and if there is no simple proof that is more of a proof than a confirmation of the statement, then I have probably already answered the question in the way intended.

Thanks //F

 

[dextercioby]

You already have the proof. Your "confirming" stands for proving.

 

[freddyfish]

Thanks for your answer. I came here now to actually post that I found it to be the desired solution :p

I was not too satisfied finding out that I already had come up with the answer according to the solutions manual, since I prefer to start at the opposite end and prove the statement without using the fact that I know where I should end up after carrying through the proving process.

The Schroedinger equation can't be proven, I know that. But the problem I was asked to solve was just about the mathematical representation of the S.E. so I gave it a shot B-)