Animated Solution for Chemistry - Atomic Structure: The electrons are more likely to be found

Where is the Electron?

Imagine you are trying to locate a firefly in a dark room. You don't know exactly where it is, but you know the areas it prefers to fly around. In the quantum world, electrons behave somewhat like this firefly. We can never pinpoint their exact location, but we can determine the probability of finding them in a specific region.

This problem presents us with a graph of the wave function, $\Psi (x)$, plotted against the distance $x$ from the nucleus. The graph shows a curve that rises to a positive peak in region a, crosses the horizontal axis at point b, and dips into a negative trough in region c. The question asks us a fundamental physical question: where are the electrons more likely to be found?

Probability

To answer this, we must rely on the brilliant insight of physicist Max Born. He proposed that the wave function $\Psi$ itself doesn't have a direct physical meaning. It is merely a mathematical amplitude, which can be positive, negative, or even complex.

However, the square of the wave function, $|\Psi {|}^2$, holds the key to reality. It represents the probability density—the likelihood of finding an electron at a given point in space.

Let's apply Born's interpretation to our graph.

In region a, the wave function $\Psi (x)$ is positive. When we square a positive number, the result is positive. Therefore, $|\Psi (x){|}^2>0$. This indicates a high probability of finding the electron in this region.

Now, look at region c. Here, the wave function $\Psi (x)$ dips below the axis, meaning it is negative. But remember, probability can never be negative! When we square a negative number, it becomes positive. Thus, $|\Psi (x){|}^2>0$ in region c as well. The negative sign of the wave function simply represents the phase of the wave, not a negative probability. So, there is a significant chance of finding the electron in region c too.

Finally, let's examine point b. At this exact location, the wave function crosses the x-axis. This means $\Psi (x)=0$.

If we square zero, we still get zero. Therefore, the probability density $|\Psi (x){|}^2=0$ at point b. This specific point is known as a radial node (or a radial nodal surface in 3D space). It is a region where the probability of finding an electron is absolutely zero. The electron can exist on either side of the node, but never exactly on it.

Putting all the pieces together, the probability density is non-zero in both regions a and c, while it is strictly zero at point b. Therefore, the electrons are more likely to be found in regions a and c.

This elegant problem reminds us that in quantum mechanics, both the peaks and the troughs of a wave function are equally important when it comes to finding the elusive electron.