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Irid

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## Homework Statement

This is from

*Griffiths*Introduction to Quantum Mechanics, Problem 2.21.

Suppose a free particle, which is initially localized in the range -a<x<a, is released at time t=0:

[tex] \Psi(x,0) = \begin{cases}

\frac{1}{\sqrt{2a}}, & \text{if } -a<x<a,\\

0, & \text{otherwise},\end{cases} [/tex]

Determine the Fourier transform [tex]\phi(k)[/tex] of [tex]\Psi(x,0)[/tex] and comment on the behavior of [tex]\phi(k)[/tex] for very small and very large values of a. How does this relate to the uncertainty principle?

## Homework Equations

[tex] \phi(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \Psi(x,0) e^{-ikx}\, dx = \frac{1}{2\sqrt{\pi a}} \int_{-a}^{+a} e^{-ikx}\, dx = -\frac{1}{2ik\sqrt{\pi a}} \left(e^{-ika}-e^{ika}\right) = \frac{\sin (ka)}{k\sqrt{\pi a}}[/tex]

## The Attempt at a Solution

The one Fourier transform with a greater [tex]a[/tex] has more rapid oscillations and a higher peak at k=0, when compared to the one with a smaller [tex]a[/tex]. A big value of [tex]a[/tex] corresponds to a widely spread particle, while a small [tex]a[/tex] stands for a particle well localized at x=0.

My problem is that I don't see the connection between my Fourier transform and uncertainty. As far as I understand, rapid oscillations serve to pick up some specific values of energy, and the resulting wave function is close to a discrete linear combination, while slow oscillations pick up a wide spectrum of energies. But what does this have to do with uncertainty? Both seem quite uncertain to me...

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