Discussion
A student sets up Young’s double slit experiment with electrons of momentum \(p\) incident normally on the slits of width \(w\) separated by distance \(d\). In order to observe interference fringes on a screen at a distance \(D\) from the slits, which of the following conditions should be satisfied?
Answer: B
The condition is \(\frac{\hbar}{p} \gt \frac{dw}{D}\).
The electrons have de Broglie wavelength \(\lambda = h/p\), of order \(\hbar/p\). The interference fringes on the screen are spaced by about \(\lambda D/d\). Each slit has a finite width \(w\), and electrons passing through different parts of one slit produce fringe patterns shifted relative to each other; the fringes stay visible only if their spacing is larger than this spread, that is roughly
\[ \begin{aligned} \frac{\lambda D}{d} &\gt w \\ \Rightarrow \frac{\hbar}{p} &\gt \frac{dw}{D} \end{aligned} \]
up to a numerical factor. The other options place \(D\) and \(d\) wrongly: \(Dw/d\) would demand a longer wavelength as the screen moves away, which is the reverse of the real dependence, and the forms with \(d^{2}\) do not involve the slit width at all.
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