In an experiment, electrons are made to pass through a narrow slit of width d comparable to their de

Q: Why don't the electrons just travel in a straight line and form a shadow of the slit?

If electrons were purely classical particles, they would. However, because their de-Broglie wavelength is comparable to the slit width, their wave nature dominates, causing them to diffract and spread out.

Q: What would happen if the slit width $d$ was much larger than the electron's wavelength?

The diffraction effect would become negligible. The electrons would mostly travel straight through, and the pattern on the screen would look very much like the rectangular distribution in graph (b).

Q: Why is graph (c) incorrect? It looks like a wave pattern.

While graph (c) is curved, it restricts the entire electron distribution strictly within the physical width of the slit ($y = \pm d/2$). Diffraction inherently causes the pattern to spread *wider* than the slit.

Q: Does the distance $D$ to the screen affect the pattern?

Yes! The angular spread $\theta$ is constant, so the linear width of the pattern on the screen ($W \approx 2D\theta$) increases as the screen is moved further away.

Q: Are we detecting single electrons or a continuous wave?

We detect individual electrons as discrete hits on the screen. However, the *probability* of an electron hitting a specific spot is governed by the continuous wave intensity, which builds up the diffraction pattern over time.

Question

In an experiment, electrons are made to pass through a narrow slit of width $d$ comparable to their de-Broglie wavelength. They are detected on a screen at a distance $D$ from the slit (see figure). Which of the following graphs can be expected to represent the number of electrons $N$ detected as a function of the detector position $y$ ($y = 0$ corresponds to the middle of the slit) ?

Sigma Percentile · Accepted Answer

The wave nature of matter is one of the most profound discoveries in modern physics. When we think of electrons, we usually picture tiny, solid particles. However, under the right conditions, they behave exactly like waves!

This problem beautifully illustrates this concept through the phenomenon of single-slit diffraction.

### The Setup: Electrons as Waves

Imagine a beam of electrons traveling towards a barrier with a narrow slit of width $d$. According to de-Broglie's hypothesis, every moving particle has an associated wavelength given by:

$$ \lambda = \frac{h}{p} $$

where $h$ is Planck's constant and $p$ is the momentum of the particle.

In this experiment, the slit width $d$ is chosen to be comparable to the de-Broglie wavelength $\lambda$ of the electrons. This is the crucial condition for wave phenomena to become noticeable.

If the electrons acted purely as classical particles, they would simply travel in straight lines through the slit, creating a sharp, rectangular shadow on the screen (like in graph B).

### The Phenomenon of Diffraction

Because $d \approx \lambda$, the electron waves do not just travel straight through. Instead, they bend and spread out as they pass through the slit. This phenomenon is known as **diffraction**.

As these diffracted waves propagate towards the screen at distance $D$, they interfere with each other. This interference creates a pattern of varying intensity on the screen.

The number of electrons $N$ detected at any position $y$ is directly proportional to the intensity of the wave at that point.

### Analyzing the Graphs

The diffraction pattern consists of a bright, wide central maximum flanked by fainter secondary maxima. The angular half-width of this central maximum is approximately:

$$ 	heta \approx \frac{\lambda}{d} $$

Because the slit is very narrow, the waves spread out significantly. By the time they reach the screen, the linear width of the central maximum is much larger than the original slit width $d$.

Let's evaluate the given options:

- **Graph (a)** shows a sharp triangular peak confined exactly within the slit width $d$.
- **Graph (b)** shows a classical particle-like rectangular distribution, also confined to $d$.
- **Graph (c)** shows a curved distribution, but it abruptly drops to zero exactly at the edges of the slit ($y = \pm d/2$).
- **Graph (d)** shows a bell-shaped curve that spreads out well beyond the boundaries of the slit.

Since diffraction inherently causes the wave to spread wider than the aperture it passed through, **Graph (d)** is the only physically accurate representation of the electron distribution on the screen.

Select Answer:

Visualized Solution

Experimental Setup

Wave Nature of Electrons

Diffraction Phenomenon

Intensity Pattern on Screen

Width of Central Maximum

Conclusion

The Sigma Insight: Matter Waves and de Broglie Relation

The Setup

The Phenomenon of Diffraction

Analyzing the Graphs

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