###### By Joshua Winn, Princeton University

## Zooming in on the fundamental particles of a matter, we see the individual atoms that make up molecules. But, why atoms are as big as they are. Why are they 10 to the minus-10 meters, and not much bigger or smaller? Who or what determines their size?

### The Case of a Shrinking Cloud

Imagine letting go of an electron, and watching it fall toward a proton. It accelerates as it falls, causing it to radiate and lose energy.

A more accurate picture is an electron cloud that’s contracting around the proton. So, the radius of the cloud, *r*, is shrinking. And because it’s losing energy to radiation, the cloud will end up with whatever size corresponds to the minimum possible energy. And what would that be?

How do we calculate the minimum possible energy? Well, if the electron were a classical particle, the energy would have 2 parts. The first part is the kinetic energy, a positive number, which is usually written 1/2 *mv*-squared. For this problem, it’ll be more useful to write it in terms of momentum, *p*. We can do that by inserting *v* equals *p *over* m*, giving *p*-squared over 2*m*.

The second part is the electrical potential energy, that’s a negative number that varies as one over *r*. In quantum theory, the electron cloud extends over a range of *r*, and has a range of possible values of *p*. But still, for any particular combination, this equation for the total energy still holds.

Now, to minimize *E,* we should keep *p* as low as possible, that’ll reduce the kinetic part. And we should also make *r *as small as possible, because that’ll reduce the potential energy, it will make it more negative.

### Effect of Heisenberg’s Uncertainty Principle

Keeping in mind the Heisenberg’s uncertainty principle, we can’t make both *r* and *p *arbitrarily small. If the cloud has radius of *r*, the uncertainty in the electron’s position is on the order of *r*. Which means the uncertainty in its momentum can be no smaller than *h-*bar over 2 divided by *r*.

So, if we shrink *r* too much, the momentum goes up, and the kinetic energy term grows out of control. But if we increase *r* too much, the potential energy rises and it starts to dominate. To get the minimum energy, we need to balance the kinetic against the potential.

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### Understanding the Formula

For *p*, we’ll insert the minimum possible value, *h-*bar over *r*. Officially, it should be 1/2 *h-*bar over *r*, but this is just an order-of-magnitude calculation, we haven’t been precise enough about the meaning of *Delta*–*x* and *Delta*–*p* to justify worrying about factors of 2. That’s why there’s a squiggle in the equation, instead of an equals sign—a squiggle means “has the same order of magnitude”.

With that substitution, we get *E* equals *h-*bar over *r*-squared over 2*m* minus *eta-e*-squared over *r*. That’s a function of one variable,* r*. So, one way to find the minimum is graphically. We can plug in the numerical values of the constants, and then plot *E* against *r*. Let’s use a logarithmic x-axis to help us see what’s going on.

For small values of *r*, the one over *r*-squared term is dominant, causing the energy to shoot up as we approach zero. But for large values of *r*, the negative one over *r* term is more important. That causes the energy to rise from negative values toward zero. And in between, there’s a minimum.

The minimum occurs when *r* is about 1/2 of 10 to the minus-10 meters, or 0.05 nanometers, which is about equal to the observed size of a hydrogen atom! So, the calculation succeeded in explaining the size of the hydrogen atom.

### Calculating the Minimum Size Possible

Those of you who know calculus can also locate the minimum through a direct calculation. You take the derivative of the energy with respect to *r*, and set it equal to zero. You’ll find that at the minimum, *r* equals *h*-bar-squared over *eta-me*-squared.

That combination of constants comes up so often in atomic physics we give it a special name, the Bohr radius, and a special symbol, an *a* with subscript zero, *a*_{0}. Its value, as we’ve seen, is about a 20th of a nanometer.

### Value of the energy

And what about the value of the energy? That turns out to be minus- *eta* *e*-squared over 2* a_{0}*, which has the numerical value of 2.2 times 10 to the minus-18 joules, as shown on the chart. That energy scale, too, is central to atomic physics, so it’s useful to express it in different units: electron volts, or eV.

One eV is defined as 1.6 times 10 to the minus-19 joules. That way, the minimum energy comes out to be minus-13.6 eV—which is, indeed, the measured binding energy of an electron in a hydrogen atom. So now we understand the sizes of atoms, and their energies.

The combination of Coulomb’s Law, and the Heisenberg uncertainty principle, singles out a characteristic size for atoms—the Bohr radius—and a characteristic energy for the electrons, a few electron volts.

### Common Questions about Why Are Atoms as Big as They Are?

**Q: What is kinetic energy of an electron?**

The electrons that are a classical particle, have 2 parts of energy. The first part is the kinetic energy, a positive number, which is usually written 1/2 *mv*-squared. For this problem, it’ll be more useful to write it in terms of momentum, *p*. We can do that by inserting *v* equals *p *over* m*, giving *p*-squared over 2*m*.

**Q: How do you define 1 eV (Electron Volt)?**

One Electron Volt is defined as 1.6 times 10 to the minus-19 joules.

**Q: What are the factors determining the size of an atom?**

The combination of Coulomb’s Law, and the Heisenberg uncertainty principle, singles out a characteristic size for atoms—the Bohr radius—and a characteristic energy for the electrons, a few electron volts.