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CuriousQuark

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## Homework Statement

Normalize: [tex]\Psi_1 (x,t) = N_1 \cos(\frac{\pi x}{L}) e^{-\frac{iE_1t}{\hbar}}[/tex]

Where [itex]N_1[/itex] and [itex]E_1 [/itex] are the normalization constant and energy for the ground state of a particle in an infinite square well.

## Homework Equations

Normalization Condition:

[tex]\int_\infty^\infty P(x,t) dx = \int_\infty^\infty \Psi(x,t)^* \Psi(x,t) dx = 1[/tex]

(sorry... that’s –inf to +inf for the integrals, but I can’t quite get it right in latex )

## The Attempt at a Solution

So, I apply the normalization condition. My exponential term and its complex conjugate cancel each other out, which leaves me with:

[tex]N^2_1 \int_\infty^\infty {\cos(\frac{\pi x}{L})}^2 dx = 1[/tex]

I use a handy trig power reducing identity, and when I integrate I get:

[tex]N^2_1 (\frac{x}{2} +\frac{L}{4\pi}\sin(\frac{2\pi x}{L}))[/tex]

But... then things start to go off the rails.

I don't know if I got my calculus wrong (pretty sure I didn't) or just completely misunderstood the physics (probably), but I don't know how to evaluate this integral without it blowing up to infinity! And it's absolutely not supposed to, because it's described as a 'physically admissible' state function.

I know my normalization constant

*should*equal [itex]\sqrt{\frac{2}{L}}[/itex], and I assumed that my terms in psi*psi could commute, and that my imaginary exponential terms should cancel each other out prior to integration. But just looking at a graph of cos^2 shows my integrand clearly won't converge to a finite value.

Does anybody know what I'm doing wrong?

Much thanks in advance!

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