Quantum numbers


Quantum numbers can be used to understand the electronic structure of atoms.

Quantum numbers come from Quantum Mechanics which was the successor to the Bohr Model.

The Bohr model traces to the early 20th century physicist Niels Bohr, who realized that he could use ideas from classical mechanics to formulate a simple picture of an atom.

His main idea was that an electron orbits around the nucleus at a specific radius in the same way that planets orbit around the Sun.

Unfortunately, the Bohr model does not remain accurate when dealing with atoms with more electrons than hydrogen.

For many electron atoms, more sophisticated theories based on quantum mechanics have been developed.

A main point of these theories is that, rather than imagining an atom as a collection of electrons orbiting the nucleus, it is better to think of the electrons as a diffuse cloud around the nucleus -- corresponding to a probability of electrons existing in various locations.

Therefore, rather than stating that an electron lies at a given, fixed distance from the nucleus like the Bohr model, these more sophisticated models provide the probability of finding electrons in various locations.

These clouds of likely electron locations have a name -- they are called orbitals.

Looking at a picture of an electron density model, we can see that all these dots indicate the probability of finding an electron.

We don't know exactly where the electron is, but it is somewhere in this cloud around the nucleus.

If we think about this orbital as being a three-dimensional sphere, we would say there is a 90% probability that the electron is somewhere in that sphere around the nucleus.

Electrons in orbitals are described by four quantum numbers.

The first quantum number is called the principal quantum number, and it is symbolized by N.

N is a whole number, like 1, 2, 3, or higher. In a diagram of an atom, electrons with higher values of N -- and thus higher principal quantum numbers -- are found further away from the nucleus.

We can think of electrons that are further away from the nucleus as having higher energy, because they have a higher potential energy relative to the nucleus.

And so the principal quantum number N tells us the energy level of an electron.

Also, an electron’s principal quantum number is sometimes referred to as a shell.

The second quantum number is called the angular momentum quantum number.

It is symbolized by “L”, and it refers to the shape of the orbital.

“L” can be any whole number between 0 and N -1 -- so the energy level, or first quantum number, determines the possible values of the angular momentum quantum number.

So for the case of the first energy level, N = 1, we know that L can be at most equal to L-1.

So that means that when N = 1, L always equals 0.

An orbital with L = 0 corresponds to a spherically shaped orbital called an S orbital.

So the first energy level has only one orbital, an s orbital shaped like a sphere.

In the case of the second energy level, N is equal to 2.

When N is equal to 2, our values for L can be both 0 or N minus one which equals one.

When L = zero, we are dealing with an S orbital which is shaped like a sphere.

When L = one, we are dealing with a P orbital.

P orbitals are shaped like dumbbells.

Since we can refer to energy levels as shells, we can also refer to our S and P orbitals as subshells.

So in the second shell, there are two subshells.

The third quantum number is called the magnetic quantum number.

It is symbolized by “m sub l,” and it tells us the orientation of an orbital. “m sub l” can be any whole number between negative L and positive L.

So if L equals 0, the only possible value for “m sub L” is zero.

If L equals 1, then “m sub L” can equal -1, 0, or 1.

If L equals zero, we know that we are dealing with an S orbital which is shaped like a sphere.

And since we have only one value for “m sub L,” then we have only one orientation.

There is no way to change the orientation of a sphere -- it’s still a sphere if you turn it upside down -- and so it makes sense that m sub L would only have one possible value.

If L equals one, we know that we are dealing with a P orbital which is shaped like a dumbbell.

If we have three values for “m sub L,” then we have three possible orientations.

In a 3D coordinate system, we would refer to these three differently-oriented p orbitals as px, py, and pz.

This is consistent with the rule that m sub L has three allowed values: -1, 0, 1.

The fourth and final quantum number is the spin quantum number.

It is symbolized by “m sub s”, and it tells is the spin of an electron.

For electrons, m sub s can be either positive ½, or negative ½.

To get some intuition for what this number means, we can visualize electrons as constantly spinning.

A spinning charge creates a magnetic field, which points upwards or downwards depending on whether the electron is spinning clockwise or counterclockwise.

These two orientations correspond to the two possible values for m sub s of positive ½ or negative ½

Let’s use the quantum numbers to determine the total number of electrons in the first four energy levels.