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Nonreal solutions to the TISE

  1. Jan 26, 2007 #1

    G01

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    1. I am try to show that a non-real solution to the shrodinger eq., [tex]\psi(x)[/tex] can be expressed as a linear combination of real solutions.



    2. Here's what I know:
    Shrodinger Eq. [tex] -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} +V\psi(x) = E\psi(x)[/tex]

    [tex]\psi[/tex] [tex] \psi*[/tex] are non-real solutions

    [tex] (\psi + \psi*)[/tex] and[tex]i(\psi -\psi*)[/tex] are real solutions.





    3. Well, to start, since I am given two real solutions, I am assuming [tex]\psi[/tex] can be expressed as a linear combination of the two real solutions above. I tried this two different ways:

    First I tried plugging the real solutions into the shrodinger eq. and seeing where it got me. I was able to split the equation into two separate equations involving only the original psi and psi* but I don't think this helps me.

    I also tried setting psi = A +Bi and seeing if I could combine the real solutions in a way so they equal psi. Also, this hasn't been of much help so far. If I am somewhat on track but am confused, please point me in the right direction. Or am I completely off track? Thank you for your insight.
     
    Last edited: Jan 26, 2007
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  3. Jan 27, 2007 #2

    dextercioby

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    Take [itex] \psi (x)= \psi_{1}(x)+i\psi_{2}(x) \ , \psi_{1,2}(x)\in \mathbb{R} [/itex]. Plug the psi in the SE and and separate the real and imaginary part. That's all.
     
  4. Jan 27, 2007 #3

    G01

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    Thanks Dexter.
     
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