## Table of Contents

# Quantum Number Definition

Quantum numbers are often used to define the trajectory and the movement of an electron in an atom.

Quantum numbers are defined as the set of numbers that are used to determine the position and energy of the electron in an atom.

## Types of Quantum Number

There are basically four quantum numbers;

i. Principal Quantum Number

ii. Azimuthal Quantum Number or Orbital angular momentum quantum number

iii. Magnetic Quantum Number

iv. Spin Quantum Number

Number | Symbol | Possible Value |

Principal Quantum Number | n | 1,2,3,4,… |

Azimuthal Quantum Number | l | 0,1,2,3,….(n-1) |

Magnetic Quantum Number | m_{1} | -l,….,-1,0,1,…l |

Spin Quantum Number | m_{s} | +1/2,-1/2 |

### i. Principal Quantum Number

• Principal quantum numbers are represented by the symbol ‘n’.The Principal Quantum Number denotes the principal energy level or shell in which an electron revolves around the positively charged nucleus.

• The principal quantum number are described in whole numbers with a value that is equal to one or greater than one (n = 1,2,3,4,5…..) The value of n=1 signifies the innermost electron shell of an atom, which further corresponds to the lowest energy state (or also known as the ground state) of an electron.

• Thus, it can be observed here that a principal quantum number (n) cannot be zero nor it can have a negative value.

• When a given electron absorbs energy, then it jumps from one principle shell to a higher shell, thus causing an increase in the value of n. Correspondingly, when electrons emit or produce energy, then they rapidly jump back into lower shells thus causing a decrease in value of n.

• The principal quantum number helps us in determining the energy shell of an electron. It was first considered to differentiate between different energy levels in the Bohr model of the atom but remains valid to the modern atomic orbital theory.

The energies of the numerous principal shells will follow the sequence as given below;

K < L < M < N < ….

1 < 2 < 3 < 4 < …..

#### ii. Azimuthal Quantum Number

• The azimuthal (or orbital angular momentum) quantum number determines the shape of a given orbital. It is represented by the symbol ‘l’ and its value is always equal to the total number of angular nodes present in the orbital.

• A value of the azimuthal quantum number designates an s, p, d, or f subshell which differ in shape. This value of l or Azimuthal Quantum number depends on the value of the principal quantum number.

• For instance, if n =3, the azimuthal quantum number or l can take the following values which are– 0,1, and 2.

• When l = 0, the subsequent subshell is an ‘s’ subshell. Likewise, when l =1 and l = 2, the resultant subshells are ‘p’ and ‘d’ subshells. Consequently, when n = 3, the three subshells that an atom can have are 3s, 3p, and 3d.

• Another example for the same includes when n = 5, the possible values of l or azimuthal quantum number are 0, 1, 2, 3, and 4. If l = 3, then this represents that there are a total of three angular nodes present in the atom. Angular momentum quantum number can have a set of positive values ranging from zero to (n − 1).

For the 1st Shell or K, n =1, thus the value of l will be 0

For the 2nd Shell or L, n = 2, thus the value of l will be 0 and 1

For the 3rd Shell or M, n = 3, thus the value of l will be 0, 1 and 2

For the 4th shell or N, n = 4, thus the value of l will be 0, 1, 2 and 3

##### iii. Magnetic Quantum Number

The magnetic quantum number determines the total number of orbitals present in a subshell and the orientation of these orbitals. It is represented by the symbol ‘m_{1}’.

The value of the magnetic quantum number depends on the value of the azimuthal quantum number.

For a known value of l, the value of ml ranges from -l to +l.

For instance, if n = 3 and l = 2 in an atom, then the possible values magnetic quantum number can have are – -2, -1, 0, +1,and +2.

##### iv. Electron Spin Quantum Number

• The electron spin quantum number does no depend on the values of n, l, and ml. The electron spin quantum number helps in determining the direction in which the electron is spinning, and is represented by the symbol ms.

• The electron spin quantum number can have two values which are +½ and -½.

• The positive value of ms indicates an upward spin on the electron which is also termed as ‘spin up’ and is represented by the symbol ↑. The negative value of ms indicates downward spin, or it is also termed as ‘spin down’, which is represented by the symbol ↓.

• The value of ms can be summarised to ±½. and an orbital of an atom cannot hold more than two electrons.

Quantum numbers | Meaning and Possible Values of quantum numbers |

Principal quantum number or n | Electron shell, n can be greater than or equal to1 |

Azimuthal quantum number or l | Subshells (s=0, p=1, etc.) , For a given value of n it can have values ranging from 0 to (n-1) |

Magnetic quantum number or m_{1} | Total number and orientation of orbitals, l≥m_{1}≥-l |

Electron spin quantum number or m_{s} | Represents the direction of electron spin, ms = ±½ |

**Note:** Hund’s rules states that two electrons of the same atom cannot have exactly the same set of quantum numbers.

##### Quantum Number Examples

**Q1.** What are the Possible ml values for l = 3?

The value of the magnetic quantum number commonly ranges from -1 to 1,

Thus, the possible values of ml or magnetic quantum number when l = 3 are: -3, -2, -1, 0, 1, 2, and 3.

**Q2.** An electron is present in its 2p orbitals. What are the possible values of n, l, and ml for this electron?

For the 2p orbital,

• Principal quantum number (n) = 2

• Azimuthal quantum number (l) = 1

• Magnetic quantum number (ml) = – 1 , 0 ,+ 1

##### Quantum Number Citations

- Polyad quantum numbers and multiple resonances in anharmonic vibrational studies of polyatomic molecules. J Chem Phys . 2013 Nov 14;139(18):184101.
- Sorting Photons by Radial Quantum Number. Phys Rev Lett . 2017 Dec 29;119(26):263602.
- The splitting of atomic orbitals with a common principal quantum number revisited: np vs. ns. J Chem Phys . 2012 Apr 14;136(14):144112.

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