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# How Quantum Physics Predicts the Size of an Atom

How Quantum Physics Predicts the Size of an Atom

Atoms are tiny particles that make up everything. But exactly how big (or rather how small) are they?

In this video, we’ll start by defining the « size » of an atom as the distance between the center of the nucleus, and the furthest (outermost) electron. This can also be known as the radius of the atom. If we study a simple hydrogen atom, then this radius simply becomes the distance between the proton forming the nucleus, and the electron in the lowest energy level.

According to quantum mechanics, it is not necessary that these two particles will be found exactly the same distance apart. Instead, the wave function of the system tells us the probability of finding the electron a certain distance away from the proton. And from the wave function we can see that the electron is most likely to be found a certain distance away from the proton.

This most likely distance is known as the « Bohr Radius » after Niels Bohr. It can be calculated in terms of some important physical constants (such as the permittivity of free space, Planck’s constant, mass of the electron, and charge of the electron). It can also be rewritten as a simpler expression in terms of the reduced Planck constant, electron rest mass, speed of light in a vacuum, and the fine structure constant.

Ultimately though, the numerical value of the Bohr Radius in meters is about 5.29 x 10^(-11) m. This is roughly 0.0529 nm, or about 100 million times smaller than the width of an average pencil. In other words, atoms are extremely tiny, and the hydrogen atom is (of course) the tiniest of them all.

It’s also worth noting that the Bohr Radius is calculated using an assumption about the proton, which is that it is completely stationary at the center of the atom. This is equivalent to assuming the proton has infinite mass. This assumption is necessary to make the math easier, and is reasonable since the mass of the proton is much larger than the mass of the electron. So the result we get using this assumption is quite close to the actual most likely size of the atom.

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