Heisenberg Uncertainty Principle

A beam of electrons with momenta $p=6.6\cdot10^{-24}$ N$\cdot$sec is directed at a slit of width $a=10$ nm. The electrons are then observed at a screen a distance $D=10$ cm from the slit.

(a)

What is the wavelength of the electrons? Is it smaller or larger than the width of the slit?

(b)

Sketch the flux of electrons as a function of the position on the screen $y$. (The point opposite the center of the slit is at $y=0$.) On the same sketch, show the flux you would expect if electrons did not behave as waves.

(c)

How far from $y=0$ can one place an electron detector and still detect a non-zero flux?

HINT: Recall problem set # 9, problem 5 (``Alone in the Dark''.) Just like in that problem, you can assume that flux is zero outside the central intensity maximum!

(d)

Electrons hitting the screen away from $y=0$ have a non-zero momentum in the $y$ direction. Find $p_y$ of an electron hitting the screen at a point $y$. With the same assumption as in part (c), what is the maximal value of the magnitude of this momentum, $\vert p_y\vert$? What is its minimal value? The ``uncertainty'', or spread, in $p_y$ is defined as $\Delta p_y={\rm max}(\vert p_y\vert)-{\rm min}(\vert p_y\vert)$. Find $\Delta p_y$.

NOTE: Since the $y$ component of the electron momentum does not change on the way from the slit to the screen, $\Delta p_y$ can also be thought of as the uncertainty in the electron momentum at the moment when it passes through the screen.

(e)

Observing an electron on the screen, we do not know exactly its $y$ coordinate when it passed through the slit: it could be anywhere from $-a/2$ to $a/2$. Thus, the ``uncertainty'' in the position of the electron, $\Delta y$, is equal to the width of the slit $a$. Show that the uncertainties in the coordinate of the electron and its position satisfy the relation

$$\Delta y \Delta p_y = \hbar.$$

NOTE: Heisenberg uncertainty principle states that for any physical system, $\Delta$(coordinate)$\Delta$(momentum) is at least $\hbar/2$.

(f)

If the slit is made very narrow, $a\rightarrow 0$, what is the expected uncertainty in momentum, $\Delta p_y$? Based on this result, sketch the expected flux of electrons as a function of the position on the screen for such a narrow slit. Does the result agree with the expectation from our study of wave interference pattern from one narrow slit?

Figure 2: Electron diffraction experiment.