Quantum Numbers

Quantum numbers can be used to describe the trajectory and the movement of an electron in an atom. The quantum numbers of all the electrons in a given atom, when combined, must comply with the Schrodinger equation.

What are Quantum Numbers?

The values of the conserved quantities of a quantum system are given by quantum numbers. The definition of the quantum numbers that are related to electrons can be written as – a group of numerical values which provide solutions that are acceptable by the Schrodinger wave equation for hydrogen atoms.

Quantum Numbers

Four quantum numbers can be used to completely describe all the attributes of a given electron belonging to an atom, these are:

  • Principal quantum number, denoted by n.
  • Orbital angular momentum quantum number (or azimuthal quantum number), denoted by l.
  • Magnetic quantum number, denoted by ml.
  • The electron spin quantum number, denoted by ms.

A brief description of each of these numbers in the set of four quantum numbers that describe the unique quantum state of an electron in atomic physics can be found in the next subsection.

Types of Quantum Numbers

When the characteristics of an electron must be described in compliance with the Schrodinger wave equation, a total of four quantum numbers are used. Each of these values is described below.

Principal Quantum Number

  • Principal quantum numbers are denoted by n and they designate the principal electron shell of the atom. Since the most probable distance between the nucleus and the electrons are described by it, a larger value of the principal quantum number implies a greater distance between the electron and the nucleus, which in turn implies a greater atomic size.
  • The value of the principal quantum number can be any integer with a positive value that is equal to or greater than one. The value n=1 denotes the innermost electron shell of an atom, which corresponds to the lowest energy state (or the ground state) of an electron.
  • Thus, it can be understood that the principal quantum number, n, cannot have a negative value or be equal to zero because it is not possible for an atom to have a negative value or no value for a principal shell.
  • When a given electron is infused with energy (excited state), it can be observed that the electron jumps from one principle shell to a higher shell, causing an increase in the value of n. Similarly, when an electron loses energy, they jump back into lower shells and the value of n also decreases.
  • The increase in the value of n for an electron is called absorption, emphasizing the photons or energy being absorbed by the electron. Similarly, the decrease in the value of n for an electron is called emission, where the electrons emit their energy.

Azimuthal Quantum Number (or the Orbital Angular Momentum Quantum Number)

  • The azimuthal or orbital angular momentum quantum number describes the shape of a given orbital. It is denoted by the symbol l, and its value is equal to the total number of angular nodes in the orbital.
  • A value of the azimuthal quantum number can indicate either an s, p, d, or f subshell which vary in shapes. This value depends on and is capped by the value of the principal quantum number, i.e. the value of the azimuthal quantum number ranges between 0 and (n-1).
  • In an example where the value of n is 5, the possible values of l are 0, 1, 2, 3, and 4. If l = 3, then there are a total of three angular nodes in the atom.

Magnetic Quantum Number

The total number of orbitals in a subshell and the orientation of these orbitals are determined by the magnetic quantum number. It is denoted by ml. This number yields the projection of the angular momentum corresponding to the orbital along a given axis.

Electron Orbitals as per Quantum Numbers

The value of the magnetic quantum number is dependant on the value of the azimuthal (or orbital angular momentum) quantum number. For a given value of l, the value of ml ranges between the interval -l to +l. Therefore, it indirectly depends on the value of n.

For example, if n = 4 and l = 3 in an atom, the possible values of the magnetic quantum number are -3, -2, -1, 0, +1, +2, and +3.

Electron Spin Quantum Number

  • The electron spin quantum number is independent of the values of n, l, and ml. The value of this number gives insight into the direction in which the electron is spinning, and is denoted by the symbol ms.
  • The value of ms offers insight into the direction in which the electron is spinning. The possible values of the electron spin quantum number are +½ and -½.
  • The positive value of ms implies an upward spin on the electron which is also called ‘spin up’ and is denoted by the symbol ↑.  If ms has a negative value, the electron in question is said to have a downward spin, or a ‘spin down’, which is given by the symbol ↓.
  • The value of the electron spin quantum number determines whether the atom in question has the ability to produce a magnetic field. The value of ms can be generalized to ±½.


In order to simplify the details of the four different quantum numbers that are related to atomic physics, a tabular column detailing their names, symbols, meanings, and possible values is provided below.

Name and Symbol Meaning and Possible Values
Principal quantum number, n Electron shell, n ≥ 1
Azimuthal quantum number, l Subshells (s=0, p=1, etc.) , (n-1) ≥ l ≥ 0
Magnetic quantum number, ml Total number and orientation of orbitals, l≥ml≥-l
Electron spin quantum number, ms The direction of electron spin, ms = ±½

It is important to note that it is impossible for two electrons of the same atom to have exactly the same quantum state or exactly the same values of the set of quantum numbers, as per Hund’s rules.

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Practise This Question

Which of the following sets of quantum numbers is correct for an electron in 4f orbital?

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