Normalization of wave function (Griffiths QM, 2.5)

AI Thread Summary
The discussion focuses on normalizing the wave function for a particle in an infinite square well, given as an even mixture of the first two stationary states. The user correctly identifies that the integral simplifies due to the orthonormality of the wave functions, leading to the equation |A|^2 ∫ (|ψ1|^2 + |ψ2|^2) dx = 1. It is clarified that both wave functions are already normalized, which simplifies the calculation. The final integral results in a value of 2 over the specified interval, confirming the normalization process. Understanding the orthonormality of the stationary states is crucial for this normalization.
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Homework Statement


A particle in the infinite square well has its initial wave function an even mixture of the first two stationary states:

\Psi(x,0) = A\left[ \psi_1(x) + \psi_2(x) \right]

Normalize \Psi(x,0). Exploit the orthonormality of \psi_1 and \psi_2

Homework Equations


\psi_n(x) = \sqrt{\frac{2}{a}} \sin \left( \frac{n\pi}{a}x\right),
where a is the width of the infinite square well.

The Attempt at a Solution


I've managed to eliminate the orthogonal parts of my integral, so I'm now left with
|A|^2 \int |\psi_1|^2 + |\psi_2|^2 dx = 1

I have the feeling that I now have to exploit the fact that they are both normalized, but why is that so? What's the logic here?

EDIT: I had written a wrong expression for \psi_n. Sorry! :(
 
Last edited:
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Solution: The expression for \psi_n is already normalized. I should have realized this. Therefore, the integral yields 2 over the interval
 
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