Thursday, December 22, 2011

Quantum Mechanics (Uncertainity principle)

Question If a freely moving electron is localized in space to within $\Delta x_0$ of $x_0$, its wave function can be described by a wave packet $\psi(x,t)=\int_\infty^{-\infty}e^{i(kx-\omega t)}f(k)dk$, where $f(k)$ is peaked around a central value $k_0$. Which of the following is most nearly the width of the peak in $k$?
A. $\Delta k = 1/x_0$
B. $\Delta k = \frac{1}{\Delta x_0}$
C. $\Delta k = \frac{\Delta x_0}{x_0^2}$
D. $\Delta k = k_0\frac{\Delta x_0}{x_0}$
E. $\Delta k = \sqrt{k_0^2+(1/x_0)^2}$
Solution:
In quantum mechanics, the momentum $(p=\hbar{k})$ and position $(x)$ wave functions are Fourier transform pairs and the relation between $p$ and $x$ representations forms the Heisenberg uncertainty relation:
$\Delta{x}\Delta{k}\geq1 \Rightarrow \Delta k \geq \frac{1}{\Delta x}$