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* Wave packet (20 min)

We assume that a particle with energy $ E=p^2/2m$ can be described by a function that is a superposition of plane waves,
$\displaystyle \Psi(x,t)$ $\displaystyle =$ $\displaystyle \int_{-\infty}^{\infty}dk a(k) e^{i(kx-\omega(k) t)},\quad
\hbar\omega(k)=E=\hbar^2k^2/(2m).$ (4)

Use

$\displaystyle a(k)=C\sqrt{\sigma^2/(2\pi)}e^{-k^2\sigma^2/2}$

to calculate the wave packet $ \Psi(x,t)$. Here, $ C$ is a constant. Show that

$\displaystyle \Psi(x,t)= \frac{C}{\sqrt{1+i(\hbar t/m\sigma^2)}}
\exp{\left(-\frac{x^2}{2\sigma^2[1+i(\hbar t/m\sigma^2)]}\right)}.
$

To simplify your calculation, you can set $ \hbar=2m=1$ during your calculation and re-install it in the result. Why does this `trick' work? Discuss $ \Psi(x,t)$ as a function of time.



Tobias Brandes 2004-02-04