Quantum Numbers — Overview & Types

Quantum Numbers and Atomic Orbitals

Electrons reside in atomic orbitals. Each orbital can hold up to two electrons. Each orbital type (s, p, d, f) has a different number of possible orbitals.

Every electron in atomic orbitals can be described by a set of quantum numbers. Quantum numbers are values which give information on electron energy ($n$), angular momentum ($l$), magnetic vector ($m_l$), and spin number ($m_s$).

Notice in the chart above, if n=3, the largest l value possible is 2, which corresponds to a d orbital. When l=2, $m_l$ can be -2, -1, 0, 1, 2. There are five possible $m_l$ values and five d orbitals. The spin quantum number has two possible values because each orbital can hold two electrons, usually opposite spin directions.

(Video) Why Real Atoms Don't Look Like This - Quantum Numbers to Understand Atomic Structure

by Parth G

In this video, Parth translates the first two quantum numbers into an atom's structure. But, what are quantum numbers? He defines them as quantities that we can know alongside energy. But doesn't that violate the uncertainty principle? Thankfully, no! Next, he examines the principal quantum number. It's essentially the Bohr atomic model. But, Parth soon explains the angular momentum quantum number. It generates the shapes of the atomic orbitals. Why? Because angular momentum defines the electron's path. But remember, quantum mechanics is all about probability. Parth does an excellent job of translating these two quantum numbers into probabilities.

We know that the Schrodinger equation results in a wavefunction. That's the heart of quantum mechanics. But, what is a wavefunction? Do you remember the de Broglie equation? It showed that all matter had a wavelength. The wave-particle duality applies to everything! So, we can consider electrons as waves. But, waves behave differently than particles. They don't have exact locations or momentum. Instead, we have to use probability. The wavefunction is a probability function from statistics. But, we apply it to atoms. That's why wavefunctions describe the atomic orbitals.

When Schrodinger solved his equation for the hydrogen atom, he got three wavefunctions. Each one describes a different feature of hydrogen's atomic orbitals. But, each function depended on a unique component. We call the three components quantum numbers. Each quantum number describes a unique aspect of atomic orbitals.

How Do We Get Quantum Numbers?

The math used to solve the Schrodinger equation is very complicated. Below is an explanation of the result of that math. It's okay if it does not make sense. When you read the descriptions of each quantum number, you will know what each one describes. If you are curious about how we got the quantum numbers, read the next paragraph.

Whenever we solve the Schrodinger equation, we get a wavefunction. The wave function is a mathematical function that describes the atomic orbitals. Remember that an atom is three-dimensional. So, when we solve the Schrodinger equation, we solve it in three dimensions. It turns out that we can separate the functions that correspond to each dimension. What happens when we isolate them? Each function describes one dimension. Each quantum number is the quantized component of one of the dimensions.

In the quantum mechanical model, we consider the electron cloud a sphere. Why? Our assumption lets us solve the Schrodinger equation in spherical coordinates. The dimensions of spherical coordinates are based on the radius and two angles.

The Principal Quantum Number ($n$)

The principal quantum number defines how far an electron is from the nucleus. It's the atomic radius! The distance from the nucleus also tells us about the orbital's energy. The electrons closest to the nucleus have the lowest energy. Electrons further from the nucleus have higher energy. The principal quantum number is the basis for the Aufbau principle. Electrons fill lower energy orbitals before higher energy ones.

Theoretically, the principal quantum number can have values from one to infinity. But, the highest principal quantum number for know elements is seven. On the periodic table, the period beginning with the metals Francium and Radium fills the 7s orbital. Here, seven is the principal quantum number ($n$=7). The principal quantum number corresponds to the electron shell.

Azimuthal Quantum Number ($l$)

The azimuthal quantum number is also called the angular momentum quantum number. It defines the shape of the orbital. It may seem strange that angular momentum describes an orbital's shape. But, angular momentum has a direction. So, it tells us about the electron's movement. The orbital's shape is the most likely space to find the electron.

The possible values of the $l$ are always integers of 0 to $n-1$. For example, if $n$ is 4, $l$ is 0, 1, 2, and 3. Zero corresponds to the s orbital. One is the p orbital. Two is the d orbital. Three is the f orbital. The angular quantum number corresponds to the electron subshells.

Magnetic Quantum Number ($m_{l}$)

The magnetic quantum number tells us the orientation of the orbitals. It has integer values between -$l$ and $l$. The total number of orientations is $2l-1$. For example, for a d orbital $l$ is 3, and has five different values for $m_{l}$. This means that the d orbitals have five different orientations. For a p orbital, $l$ is 2 and $m_{l}$ has three values. The orientation of the three p orbitals is along the x, y, and z-axis. The magnetic quantum number is the basis of Hund's rule. Each orbital has the same energy. The only difference is the orientation. So, the electrons half-fill each orbital orientation before pairing.

Spin Quantum Number ($m_{s}$)

Originally, the Schrodinger equation was only solved for the hydrogen atom. The hydrogen atom has one electron. Schrodinger could ignore the repulsion between electrons. Electron repulsion increases the energy of an orbital. Later other physical chemists accounted for electron repulsion by adding a final quantum number.

The final quantum number is the electron spin quantum number. Electrons don't actually spin. But, they do have angular momentum. It's analogous to a spinning object. So we call it spin. In reality, they have an electromagnetic field. The field is a circular flow of energy that creates angular momentum.

Why is spin important? The electrons must have opposite spins to fill an atomic orbital. The opposite spins result in the magnetic fields canceling. The Pauli exclusion principle tells us that an orbital can only hold two electrons, and they must have opposite spins. Electron spin values are +$\frac{1}{2}$ or -$\frac{1}{2}$.