I recently came across an intriguing question on Twitter that sparked my interest: why don’t electrons fall into the nucleus of an atom while orbiting it? This is not a trivial matter, as pondering the same question about the moon led to the first great unification of physics. However, it appears that electrons may not behave as one would expect with the typical image of an atom in mind.

This question delves into the heart of atomic structure, and understanding it requires a fundamental grasp of the behavior of electrons. In the following paragraphs, I will explore this question and provide some insight into why electrons do not fall into the nucleus of an atom while orbiting it.

## Unanswered Questions

One of the most important questions in the history of science was undoubtedly why the Moon doesn’t fall to Earth due to gravity when rocks do. The answer gave rise to the theory of universal gravitation, where the force with which two masses attract each other at a distance is given by the equation F = G(m1m2)/r^2.

The answer is “simple”: the Moon does “fall,” but it has enough horizontal velocity to not collide with Earth in its fall. This can be seen with Newton’s cannon thought experiment: if we launch a cannonball from a certain height with increasing velocity, it will eventually advance enough to compensate for the curvature of the Earth in its fall and never stop falling. This is what we call an orbit.

### The constitution of matter and the concept of the atom

After centuries, scientists began to wonder about the constitution of matter, arriving (or rediscovering) the concept of the atom, and with it, the first models for it. Thomson discovered negative particles in matter. He proposed that atoms consist of negatively charged particles (electrons) immersed in a positively charged fluid so that the whole is neutral. Rutherford attempted to explain the scattering of alpha particles and proposed the concept of the nucleus of atoms. For Rutherford, the atom has the appearance seen in the logo of the blog.

However, these models were incorrect. The problem is that when electric charges undergo accelerations (and something has to accelerate to orbit something else, as bodies naturally move in a straight line), they radiate electromagnetic waves, losing energy and spiraling toward the nucleus. This was known by Rutherford, but he was more interested in explaining the scattering experiments that Thomson’s model did not explain.

### Bohr and the electrons in stable orbits

Bohr came up with the idea that electrons could be in stable orbits as long as their angular momentum was a multiple of Planck’s constant. This explained the size of atoms: the size of the electron orbits is proportional to a constant called Bohr’s radius. However, it is curious that Bohr, who was willing to give up the conservation of energy principle to explain some experimental results, in his youth was so strict and forced electrons to remain constrained in narrow orbits.

In summary, the unanswered questions are: Why don’t electrons fall into the nucleus due to the attractive force between the positively charged nucleus and negatively charged electrons? Why do electrons not lose energy and spiral into the nucleus? Why are the orbits of electrons quantized? Why is the angular momentum of electrons quantized?

## Intuitive Answer: The Uncertainty Principle

One way to understand why electrons do not fall into the nucleus is due to the Uncertainty Principle. This principle states that we cannot simultaneously determine the position and momentum (mass times velocity) of a particle with arbitrary precision. This principle was formulated as part of the development of quantum mechanics, which followed the Bohr atom.

The Uncertainty Principle implies that atoms must have a minimum size (and therefore a minimum average distance of the electron from the nucleus) to satisfy this principle. The principle is based on the idea that particles are dispersed throughout space with a certain probability of being in each point in space, with a certain distribution of velocities. Electrons can be imagined as a cloud around the nucleus.

### Position of the electron

When we measure the position of the electron, we find it in different points with a certain probability. We can assume that the electron is dispersed around the nucleus with an uncertainty, which gives us an idea of its average distance from the nucleus. Similarly, if we measure the linear momentum, we will find different values with a certain probability of measuring one or the other. We can assume that these values are distributed with an uncertainty, which gives us an idea of the average momentum of the electron.

The Uncertainty Principle tells us that the product of the uncertainties in position and momentum must be at least . We can use this to find an approximate value for the uncertainty in momentum.

To find the value of , we can require that the total energy of the electron is at a minimum. In nature, systems tend towards the minimum energy. The energy of the electron is given by the equation:

To minimize the energy, we can take the derivative with respect to position and set it equal to zero. This gives us the value of the radius, which is the same as the Bohr radius.

**It is interesting that atoms have a minimum size due to the Uncertainty Principle. **This explains why we cannot easily compress atoms and why we do not fall through the ground when we walk. Our mass tries to compress the atoms in our shoes and the ground, but this would localize the electrons even more, increasing their momentum and energy, which they avoid.

Quantum mechanics has an answer to why electrons do not fall into the nucleus, which is surprising and counterintuitive. However, this answer will be discussed in a later section.

## The pragmatic answer: they are already in the nucleus

Electrons can be found anywhere in space, including the nucleus. The functions of the electron wave obtained by solving the Schrödinger equation for hydrogen (the simplest atom) depend on three quantum numbers: n, l, and m. For the hydrogen atom’s minimum energy state, called 1s, the function of the electron wave is such that the maximum probability of finding the electron is in the nucleus. This is also true for other states, such as 2s and 3s.

Therefore, it is pointless to ask why electrons do not fall into the nucleus since there is a certain probability of finding them there. However, electrons can be captured by nuclei with an excess of protons relative to neutrons, causing a proton to convert into a neutron and emitting a neutrino in the process. This is known as electron capture.

### The probability of finding an electron at a certain distance from the nucleus

The probability of finding an electron at a certain distance from the nucleus can be studied by examining the radial probability density, which has maxima at the distances predicted by the Bohr model for states with n > 1. However, in quantum mechanics, the way to predict results is to perform experiments multiple times and study the average values, standard deviations, etc.

In quantum mechanics, operators are associated with the variables of interest in a system, and the position operator is used to study the possible values of a position measurement. The average value of the position of an electron in the ground state of hydrogen is slightly higher than that predicted by the Bohr model, but the order of magnitude is similar.

## Conclusion

In summary, the reason why electrons do not fall into the nucleus is due to the Heisenberg uncertainty principle, which requires atoms to have a minimum size. If the electron gets too close to the nucleus, it gets localized in a smaller region of space, which makes its linear momentum indeterminate and thus gains enough velocity to escape. However, it is important to note that electrons are not defined orbiting balls but rather probability clouds with certain amplitude of probability to be in the nucleus, but also a kilometer away from it. Therefore, the stability of atoms is due to the probability of finding the electrons in a certain region rather than their actual position.