# Quantum Number Calculator

Created by Davide Borchia
Last updated: Sep 14, 2022

Calculating the quantum number is not complicated; after all: the formulas are straightforward, and the values are small. However, the reasoning behind them and the meaning carried by those small integers is worth an entire book. We will condense it by a lot! Follow us, and learn how to calculate the quantum numbers:

• Learn how to find the principal quantum number $n$;
• Find out what is the azimuthal quantum number and how to find it from the value of $n$;
• Learn the formulas to find the magnetic quantum number and how it relates to the orientation of the orbitals;
• Discover our quantum number calculator and how to use it.

## Quantum numbers

According to the latest atomic model, electrons move around the atom as a delocalized cloud where the double nature of wave and particle allows them to exist without dissipating energy (we talked about this duplicity in our De Broglie wavelength calculator); the inaccuracies that still flawed Bohr's atomic model — we found them in our Bohr model calculator — are finally gone.

Electrons are fermionic particles: other than obeying the Fermi-Dirac statistics (we outlined it in the Fermi level calculator), they follow what's known as Pauli exclusion principle: every electron occupies alone a specific orbital.

We specify the orbital using a collection of numbers, each indicating a peculiar characteristic of the orbital itself. Each orbital can contain two electrons: at that point, a split happens thanks to the fermionic nature of electrons, and two particles with opposite spin fill the shell. We will dwell on this last detail later.

Each combination of quantum numbers defines the energy of the electron in the orbital. For an atom, we have four quantum numbers. In the following sections, we will meet them and, with due detail, learn:

• How to find the principal quantum number $n$;
• How to find the azimuthal quantum number $l$;
• Ho to find the magnetic quantum number $m$; and
• How to complete the description of an electron orbital with the spin number $s$.

## How to find the principal quantum number

The first and most important quantum number is the principal quantum number $n$. It indicates the energy of the shell (and the electron in it). The larger the value of $n$, the farther the electron is from the nucleus — though there are exceptions for heavier atoms.

How do you find the principal quantum number? You can start by checking the periodic table: the value of $n$ corresponds to the period of the table where the desired electron lies in the outer shell (hydrogen and helium have electrons in the first shell, hence $n=1$, oxygen's outer electron lies at higher energy and has $n=2$. After learning how to find the azimuthal quantum number, we will expand this technique.

$n$ assumes positive, integer, and non-zero values:

$n = 0, 1, 2 ,3,...$

🙋 In our current chemistry understanding, the maximum principal quantum number allowed is $n=7$.

## How to find azimuthal quantum number

The next step is defining the shape of the orbital. The azimuthal quantum number $l$ also separates electrons in energetic sublevels. It has multiple values, whose amount depends on one of the principal quantum numbers. For each energetic shell determined by $n$, we have:

$l = 0,1,2,...,n-1$

For example:

• To $n=1$ we find $l=0$, a number associated to spherical shells;
• To $n=2$ we find $l=0,1$: on top of $l=0$ we have $l=1$, that we associate to bilobated shells;
• For $n=3$ the values of $l$ are $l=0,1,2$. $l=2$ corresponds to a "cloverleaf" shape and a bilobated-plus-doughnut shape.

Higher values of $n$ add more complex shapes that require a more detailed description.

To accelerate the identification of such shapes, chemists associated letters with values of $l$:

Value of $l$

Letter

Shape

$0$

s

Sphere

$1$

p

Bilobated

$2$

d

Cloverleaf*

$3$

f

Unique shapes

$4$

g

$5$

h

🙋 *The $d$ orbital also assumes a complex bilobated shape surrounded by a torus.

The shape of the orbital is related to the angular momentum of the electron. We can find a relationship between the azimuthal number and this quantity:

$L=\sqrt{ (l(l+1)}\frac{h}{2\pi} ​$

If your electron lies in the $d$ shell, you will find it in the central block of the periodic table, with most metals. In this case, add $1$ to the period to see the principal quantum number. If your electron is in the $f$ shell, you can find it in the two rows of dangerous-and-if-not-artificial elements at the bottom of the table. In this case, add $2$ to the period to find $n$.

## How to find the magnetic quantum number

After dealing with energy and shape, we can define the orientation of the orbitals. This distinction again creates a subdivision of the energy level. The associated quantum number is $m$. Following the same pattern we met when we learned how to find the azimuthal quantum number, we find multiple values of $m$ for each value of $l$, according to the formula:

$m = -l,-(l-1), ..., 0, ..., l-1,l$
• For $n=1$ and $l=0$, $m$ has a single value: $m=0$. A sphere is always the same in every orientation!
• For $n=2$ and $l=1$, $m=-1,0,+1$. In this case, there are three possible orientations: the bilobated orbitals follow the axis, and we find three possible shells: $p_x$, $p_y$, and $p_z$.
• For the value $l=2$ the number of magnetic numbers is $5$: $m=-2,-1,0,1,2$. There are four possible orientations for the cloverleaf (one for each axis and one for the $x-y$ plan) and an extra orientation for the weird bilobated-and-doughnut-shaped orbital.

You can find $7$ orbitals for $l=3$, the $f$ orbitals, but again, the complexity makes the description not on the scope.

## Complete the quantum number calculator: you spin me round

Electrons have an additional property: spin. The spin is a measure of the intrinsic rotation of the particle and has an associated quantum number, $s$.

$s$ can assume values $1/2$ and $-1/2$, and we add it to the $n-l-m$ set of quantum numbers we calculated before. This last number creates an additional division in two sublevels in each orbital. Thus, the final tally of electrons in an atomic shell is:

• $l=0$: $2$ electrons;
• $l-1$: $6$ electrons;
• $l=2$: $10$ electrons; and
• $l=3$: $14$ electrons.

## How to find your quantum numbers: calculate the quantum numbers for an energetic shell

How to calculate the quantum numbers with our tools? Insert the value of $n$: as you know, it will help you find the other quantum numbers. We will tell you:

• The values of all the quantum numbers;
• The number of orbitals;
• The maximum number of electrons; and
• A table with all the possible combinations of the quantum numbers.

Now take a periodic table, and try to assign to every element the quantum numbers corresponding to the outermost electron in the cloud.

Discover quantum mechanics with Calctools and other calculators: the two photon absorption calculator, the photoelectric effect calculator, and many more!

Davide Borchia
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