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Poop-Loops
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Homework Statement
I need to prove that. I know conceptually how that is. If it's complex, I can split it up into a bunch of real solutions that have complex constants or what not, so Psi(x) is still always real. That much makes sense.
I also see why it has to be used as real, because then |Psi(x,t)|^2 is real. Otherwise it would be complex and the Terrorists win.
Homework Equations
The book (Griffith's Intro to QM) tells me to use
[tex]\frac{-\hbar^2}{2m}\frac{d^2\Psi}{dx^2} + V(x)\Psi = E\Psi[/tex]
"If Psi(x) works for the equation, then so does its complex conjugate and any linear combination of them." Great.
The Attempt at a Solution
I don't think simple plugging in Psi(x)* for Psi(x) will give me anything useful.
Although, I think I see why it plugging in the complex conjugate would work if Psi(x) works. It means Psi(x) is real and so is the CC, right? If it wasn't real, then you'd get a complex energy.
Or something...
I just don't understand how I can prove this mathematically.
Only thing I can think of right now is saying that if Psi(x) is complex, I can just make a complex constant in front of a real Psi(x) (more if I have to), then show that even though I have a bunch of complex constants, it will all cancel out once I multipy Psi(x)* by Psi(x).
Although that would mean that either all the constants are complex or all of them are real, or else I would still end up with complex stuff at the end.
Then again, the only time you would even write Psi(x) in the form of an infinite sum with constants is if you wanted the constants to be complex, right? There's no point in writing it out if it's already real...
This is the point of QM, right? To make your head explode?