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Quantum Number

Quantum mechanics is the branch of science which deals with the behaviour and character of matter and energy interaction with the atoms of matter. One of the successes of quantum mechanics is quantum numbers. The four quantum numbers were first used in the one dimensional model of Bohr. These numbers are related to structure of atoms and it in mathematical form. These numbers show the electronic distribution in the atom. All numbers give the information about the size of the orbit, distribution of electrons, magnetic property, orbital size, shape, and orientation in space.

The model of Schrodinger used these numbers in three dimensional spaces. So, the three coordinates are taken from Schrodinger's wave equations. These three coordinates are the principal number, angular number, and magnetic numbers. Now these numbers are studied in their four parts that are principal, angular, magnetic, and spin quantum number. Here we are discussing about all the quantum number and rules for finding these numbers. Let’s discuss that.

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Quantum Number Definition

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Quantum Numbers explains values of conserved quantities in the quantum system dynamics. Quantum numbers are discrete sets of integers or half-integers. This is different from classical mechanics where the values can vary in continuous range. Quantum numbers describe properties of orbital and of electrons in orbital.

Quantum numbers specifies main energy level or distance from the nucleus, shape of orbital, orientation of orbital, spin of electron. Any quantum system can have one or more quantum numbers and it is therefore difficult to list all possible Quantum Numbers.

Four Quantum Numbers

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To describe an electron completely inside an atom requires four quantum numbers:
  1. Principal quantum number (n) – it specifies the energy level.
  2. Azimuthal quantum number (ℓ) – it describes the shape of orbital.
  3. Magnetic quantum number (m ℓ) – which specifies the orientation of orbital in 3 D space.
  4. Spin quantum number (ms) -- it describes the spin of electron in an atom.

Principle Quantum Number

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Principal quantum number is used to specify main energy level occupied by an electron in an atom. Its value is greater than one. It never holds a zero value. n always holds integral values. Higher the value of n larger is the energy of electron. Higher the value of n, closer is the corresponding energy levels.

For an atom its value ranges from 1 to the energy level containing outermost electron. It holds all positive whole number integer values 1, 2, 3…. These values are denoted by K, L M, N, …. energy levels. Therefore for Quantum Number n = 1, K energy level (closest to the nucleus); for n = 2, L energy level; for n = 3, M energy level; for n = 4, N energy level and so on. It also determines size of an orbital. Maximum number of electrons in nth main level is 2n2.

Azimuthal Quantum Number 

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Azimuthal Quantum Number is the second quantum number also known as angular momentum quantum number or Angular Quantum Number. Azimuthal quantum number describes the shape of the orbital. It is denoted by l. The following table describe the shape of different sub level.

Sub shell     s       
 Value of l    0   1   2     3 
   Shape  Spherical  Dumb bell   Double-dumb bell   Complex
 Structure s Orbital
p Orbital
d Orbital
f Orbital

Magnetic quantum number determines the position of an orbital or electron cloud around the nucleus in x, y and z directions. It splits the sub - levels into orbital. It is denoted by the symbol ml. Its values ranges from - l to + l. It holds only integer values. Each orbital holds maximum of 2 electrons.

For example for l = 3, ml = -3, -2, -1, 0, +1, +2,+3, Hence for any value of l total number of orbital possible is 2 l + 1. We can also say that for some l and ml holds 2l+1 values.
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Spin Quantum Number parameterizes the angular momentum of a given particle. Spin quantum number is denoted by 's' or sometimes ms (magnetic spin quantum number). It is the fourth quantum number among all quantum numbers defined for complete description of electron. Spin quantum number for different particles can hold different values i.e. integer or half integer values. For fermions (like electron, proton etc.) and delta baryons spin quantum number holds half integer values. Like electron has spin quantum number s = + ½ or – ½. For bosons ‘s’ holds integer values.

For a particle, ‘s’ can hold only two values, one positive and other negative. Spin quantum number basically shows the spin or rotation of a particle around its axes. For example for electron + ½ shows anticlockwise rotation of electron while – ½ shows clockwise rotation.

In any orbital only two electrons at maximum can exist and there spin is always opposite to each other. For example if one electron has positive half spin then the second filled electron will have negative half spin.
The first experiment which showed the presence of spin of electron or in which electron spin was observed was Stern–Gerlach experiment.
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Quantum Number Rules

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The following set of rules is defined for quantum numbers:
  1. No two electrons in an atom can have same set of four quantum numbers or we can say same quantum state. This rule is known as Pauli’s exclusion principle.
  2. No two electrons in the same orbital can have same spin quantum number. This rule is known as Hund’s rule.
  3. Principal quantum number can never hold a value equal to zero.
  4. Magnetic quantum number (m) and azimuthal quantum number (ℓ) holds only integer values.
  5. Principal quantum number ranges from 1 to $\infty$.
  6. Angular momentum and magnetic quantum numbers are derived from principal quantum number in following rules. 
    Azimuthal quantum number in an integer from 0 to n – 1. For example for n = 4, ℓ = 0, 1, 2 and 3.
    •  Magnetic quantum number holds integer value from - ℓ to + ℓ. For example for ℓ = 3, m ℓ = -3, -2 , -1, 0 , + 1, + 2 and +3.
    • Spin quantum number (m s) can either be +1/2 or – ½ for an electron.

Quantum Number Chart 

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Following table shoes Quantum Number Chart where values of quantum numbers ℓ, mand ms are shown according to given value of n by using Quantum Number Rules mentioned above.

Quantum Number Chart
In the quantum mechanics, the spin is one of the important characteristics of the elementary particles, like electrons. The spin gives the important part of the quantum state of the particle. The direction of spin of the particle determines several things, like degree of freedom, angular momentum, spin quantum number, etc. → Read More
More topics in Quantum Number
Electron Spin Spin Quantum Number
Principal Quantum Number Azimuthal Quantum Number
Magnetic Quantum Number
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