Nonreal solutions to the TISE

[G01]

1. I am try to show that a non-real solution to the shrodinger eq., $$\psi(x)$$ can be expressed as a linear combination of real solutions.
2. Here's what I know:
Shrodinger Eq. $$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} +V\psi(x) = E\psi(x)$$

$$\psi$$ $$\psi*$$ are non-real solutions

$$(\psi + \psi*)$$ and$$i(\psi -\psi*)$$ are real solutions.


3. Well, to start, since I am given two real solutions, I am assuming $$\psi$$ 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.

 

[dextercioby]

Take $\psi (x)= \psi_{1}(x)+i\psi_{2}(x) \ , \psi_{1,2}(x)\in \mathbb{R}$. Plug the psi in the SE and and separate the real and imaginary part. That's all.

 

[G01]

Thanks Dexter.