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Linear combination of two solutions of the simplified schroedinger equation

  1. Jun 4, 2012 #1
    This is, when it all comes around, just math. I am asked to prove that if the schroedinger equation looks like:

    [itex]\frac{d^{2}}{dx^{2}}[/itex]ψ=-[itex]\frac{4πm}{h}[/itex]*[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
     
  2. jcsd
  3. Jun 4, 2012 #2

    dextercioby

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    You already have the proof. Your "confirming" stands for proving.
     
  4. Jun 4, 2012 #3
    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-)
     
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