Quantum Number Calculator

Created by Purnima Singh, PhD
Reviewed by Steven Wooding
Last updated: Jun 05, 2023

Use the quantum number calculator to find out the possible values of quantum numbers for a given electron shell or subshell.

Did you know that every electron inside an atom has a unique address? And that no two electrons in an atom can share the exact same address? Quantum numbers serve as an address for the electrons and hence are fundamental to understanding the quantum mechanical model of the atom.

Please continue reading to understand what a quantum number is and how to determine the allowed values of quantum numbers (i.e., angular momentum, magnetic and spin quantum numbers) for a shell using our quantum number calculator. We will also take a look at the famous Pauli's exclusion principle.

What is a quantum number?

Quantum numbers are a set of values that we use to describe each electron in an atom. These numbers tell us about the electron's energy, angular momentum, spin, and magnetic moment.

In a sense, the quantum numbers give us the electron's address (or most probable location) inside an atom.

There are four quantum numbers that we need to get a complete description of an electron within an atom:

  • Principal quantum number;
  • Angular momentum quantum number;
  • Magnetic quantum number; and
  • Spin quantum number.

In the following sections, we will try to explain what each of these quantum numbers represents.

Principal quantum number

The principal quantum number tells us about the main energy shell or state of the electron. It is denoted by nn. The allowed values for nn are:


When the electron is in the shell n=1n=1 (also called the first shell or K shell), it has the lowest energy and is closest to the nucleus.

As nn increases, the average distance of the electron from the nucleus increases, along with the size of the atom. Furthermore, the energy of the electron in an atom increases with increasing nn. You may read more about this in our hydrogen energy levels calculator.

๐Ÿ’ก Although nn can take any positive integer value, the highest shell occupied by the heaviest known element (Oganesson, Z=118Z=118) is the n=7n=7.

Angular momentum quantum number

The angular momentum quantum number or the azimuthal quantum number determines the orbital angular momentum of the electron. It is denoted by ll, and it can take any integer value from zero to nโˆ’1n-1, i.e.,

l=0,1,2,...,nโˆ’1l = 0, 1, 2, ..., n-1

For a given ll, the magnitude of orbital angular momentum of the electron LL is given by:

L=(lโ€‰(l+1)โ€‰h2ฯ€\qquad L = \sqrt{(l\,(l+1)} \, \frac{h}{2\pi}

The angular momentum quantum number also gives us the number of subshells or sublevels in a given shell, as well as the shape of the subshell. For example, if n=2n=2, then the allowed values for ll are 0,10, 1. This means that there are two subshells in the shell n=2n=2:

  • l=0l=0 or s subshells (or s orbitals); and
  • l=1l=1 or p subshells (or p orbitals).

The following table lists the name of orbitals corresponding to different values of ll.

Value of l







Name of the orbital







๐Ÿค” No known elements in their ground state configuration have electrons in g or h subshells.

Magnetic quantum number

The magnetic quantum number gives information about the relative spatial orientation of an orbital. It is denoted by mlm_l. For a given ll, the possible values of mlm_l can range from โˆ’l-l through 00 to +l+l, i.e.,

ml=โˆ’l,โˆ’(lโˆ’1),...,0,...,(lโˆ’1),lm_l = -l, -(l-1),..., 0,..., (l-1), l

For example, for l=1l=1 (i.e., p-orbital), the allowed values of magnetic quantum number are ml=โˆ’1,0,+1m_l =-1, 0, +1 corresponding to the three possible orientations, px, py, and pz.

Spin quantum number

In addition to revolving around the nucleus, the electron in an atom is also spinning around its own axis. The spin quantum number describes the spin orientation of the electron and is denoted by msm_s.

Since there are only two possible ways an electron can spin; clockwise or counter-clockwise, the spin quantum number can take only two values; โˆ’1/2-1/2 or +1/2+1/2, i.e.,

s=ยฑ1/2s = \pm 1/2

For a given ss, the magnitude of spin angular momentum of the electron SS is given by:

S=(sโ€‰(s+1)โ€‰h2ฯ€\qquad S = \sqrt{(s \, (s+1)} \, \frac{h}{2\pi}

To understand how the orbital and spin motion of electrons influence the magnetic properties of an atom, check out our magnetic moment calculator.

In the next section, we will learn how to use the quantum number calculator.

How to use the quantum number calculator?

Now that we know what quantum numbers are, let's see how we can use the quantum number calculator to find out all possible values of quantum numbers for the n=3n=3 shell.

  1. Using the drop-down menu, choose principal quantum number.

  2. Enter the value n=3n=3.

  3. The result section of the calculator will display a summary showing the possible values of angular momentum quantum number (0,1,2)(0,1,2) and spin quantum numbers (โˆ’1/2,+1/2)(-1/2,+1/2).

  4. You can also see the number of orbitals (9)(9) and the maximum number of electrons (18)(18) allowed in the shell.

  5. Finally, the tool will display a table of the allowed combinations of quantum numbers.

Where do quantum numbers appear too? Discover it in our effective nuclear charge calculator! We also have a dedicated atom calculator to calculate the various atomic properties.


What does the principal quantum number determine?

The principal quantum number describes the main energy level or electron shell of an atom. It also determines the size and energy of an orbital as well as the size of the atom.

Which characteristic is given by the angular momentum quantum number?

The angular momentum quantum number characterizes the following:

  • The number of subshells in a given shell;
  • The shape of orbitals; and
  • The energy of electrons in a multi-electron atom.

Which quantum number describes the shape of an orbital?

The angular momentum or the azimuthal quantum number describes the shape of an orbital. For example, the l=0 orbitals (also known as s-orbitals) are spherical in shape, and the l=1 orbitals (p-orbitals) are dumbbell-shaped.

Which quantum number describes the orientation of an orbital?

The magnetic quantum number describes the spatial orientation of an orbital.

How do I find magnetic quantum number?

For a given value of the azimuthal quantum number, l, the magnetic quantum number has (2l+1) possible values. Hence, to find the magnetic quantum number for a given l, determine all possible values from -l through 0 to +l, i.e., -l, -(l-1), ..., 0, ..., (l-1), l.

How do I find angular momentum quantum number?

To determine the allowed angular momentum (or azimuthal) quantum number for a given electron, follow these instructions:

  1. Find out the principal quantum number n or the main electron shell.

  2. The allowed set of values for angular momentum quantum number will be from 0 to n-1, i.e., 0, 1, 2, ..., n-1.

What is Pauli's exclusion principle?

Pauli's exclusion principle is a rule that determines the restriction on the number of electrons that can occupy a specific atomic orbital. According to this principle: No two electrons in an atom can have the same values for the four quantum numbers.

For a given atomic orbital (e.g., 1s, 2s, ...), the value of the principal, angular momentum, and magnetic quantum numbers are the same. So it can accommodate two electrons, provided the electrons have opposite spin.

Purnima Singh, PhD
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