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GR9277 #27
Problem
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\prob{27}
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?

1. $\Delta k = 1/x_0$
2. $\Delta k = \frac{1}{\Delta x_0}$
3. $\Delta k = \frac{\Delta x_0}{x_0^2}$
4. $\Delta k = k_0\frac{\Delta x_0}{x_0}$
5. $\Delta k = \sqrt{k_0^2+(1/x_0)^2}$

Quantum Mechanics$\Rightarrow$}Uncertainty

This problem looks much more complicated than it actually is. Since $k$ and $x$ are fourier variables, their localization would vary inversely, as in choice (B).

Alternate Solutions
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94709
2007-10-16 23:57:54
uncertainty principle.

$\Delta x\Delta p\approx \hbar$

In Q.M., for any matter,

$p=\hbar k$

Thus, $\Delta x \Delta p = \hbar \left(\Delta x \Delta k\right)=\hbar$.

Above equations can be done in 5 seconds without writing them down.

Therefore, $\Delta k=\frac{1}{\Delta x}$. 5 seconds in one's head.
 zaijings2009-03-13 15:21:33 like this solution!
 Professor Plum2011-07-14 14:38:57 $\Delta$k should not depend on $x_0$. Eliminate everything but B.
gray
2007-10-03 07:26:07
What are Fourier variables??
 underboywonder2007-10-03 17:43:54 I thought about it in context of the uncertainty principle. The better the localization of the particle in space, delta x, the worse the localization in momentum space, delta k. This implies an inverse relationship, like answer B.
 craklyn2007-10-24 21:45:39 To quote my professor: The uncertainty principle isn't anything mysterious or magical. It's simply a property of Fourier analysis!
 Poop Loops2008-10-25 21:07:03 What Yosun means by "Fourier variables" is that the Fourier transform of x-space takes you into k-space. This is because the Fourier transform of distance-space takes you into momentum-space. And since the "k" we are referring to is from the wave equation, we can use $p = \hbar k$. If you don't know what Fourier analysis is, then you won't understand this, though, and it isn't something I can explain in a few sentences. I think my class spent a few weeks on it. Pretty important to know, though.

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