## Quantum Numbers of an electron in an atom

Hello friends!  Once again I am back with a topic named Quantum numbers. In my last post I’ve discussed Bohr’s Correspondence principle. So let us start..

The study of the spectrum as suggested that an electron in atom exist in several discrete energy states called quantum states.

Each state is quantised. Each state of electron is described by four quantum numbers which are as :
i) Principle Quantum Number (n)
ii) Orbital Quantum Number (l)
iii) Magnetic Quantum Number(m)
iv) Spin quantum Number(s)

Also,In the solution of Schrodinger wave equation the three constants are used which are Principle quantum number(n), Orbital quantum number(l), magnetic quantum number(s). In addition to these, there is a fourth quantum number called spin quantum number (s), which represents the spin of the electron. These four quantum numbers completely specify an electron in an atom. Thus quantum numbers may be defined as :

“the set of four numbers which give a complete information about the electron in an atom i.e. it’s energy, shape, orientation & also it’s spin”.

Let us define these quantum numbers one by one. i.e.
i) Principle Quantum Number(n):

This quantum number was given by Bohr & denoted by ‘n’.  It determines the main the main energy level or shell in which an electron in an atom is present & also the energy associated with it. It also determines average distance of electron from nucleus in particular shell.
Starting from nucleus, the energy shells are denoted as K, L, M, N,… etc.  or as 1,2,3,4,…..etc.

The maximum number of electrons which a shell can accommodate is 2n^2.
The K shell(n=1) can have maximum of two electrons.
Similarly shell (n=2) have eight electrons.
& M (n=3) accommodate eighteen electrons.
It determines general size of electrons orbit & energy of the electron.
With the help of ‘n‘ we can determine energy of Hydrogen electron & other one electron species i.e. He+ & Li2+ ion. The mathematical expression for the same is :

ii) Azimuthal or Orbital Quantum Number :

All the electrons having the same value of principle quantum (n) do not have the same energy. Based on the energy of electrons, the shells are divided into sub-levels or sub-shells and these have been described with the help of ‘l‘. It was given by Sommerfeld.
The value of ‘l’ depends upon the value of shell(n).
“For a particular value of n, l can have all integer values ranging from 0 to (n-1).Each value of l represents one particular subshells”.

l = 0(s) ,1(p), 2(d) ,3(f) ,…….(n-1)
It determines the shape of electron orbit, & the orbital angular momentum of the electron.

It is also called Subsidary or angular quantum number.

iii) Magnetic quantum number(m):

When a source producing spectral line is placed in magnetic field, a spectral lines splits into a number of lines. This is called Zeeman effect.  In order to explain this effect, Linde proposed an electron because of it’s angular motion around the nucleus produces an electric field which in turn leads to magnetic field of the earth.
As a result, the electrons presents in subshells acquires some specific orientations in space called orbitals. These orientations are specified by quantum number called magnetic quantum number demoted by ‘m‘ . The values of m are linked to ‘l‘ therefore it is sometimes denoted by ‘ml‘.
The magnetic quantum numbers takesthe values from l to +l with 0.
It takes  m= 2l +1 values

e.g. for l =1 ,m = 2+1=3 values.
which are = -1 ,0 ,+1

for l=2, m = 4+1=5 values.
which are = -2 ,-1, 0 ,+1 ,+2.
It determines the orientation of electron orbit in an external magnetic field.
iv) Spin Quantum Number(s):

An electron along with charge & mass also spin around its axis and leads to quantum number called spin quantum number.
It is denoted by ‘s‘. It was given by Goudsmit
The electrons can spin either in the clockwise or anticlockwise direction, it can have two values i.e. +1/2 and -1/2. These two values are normally represented by two arrows pointing in the opposite direction. i.e. (spin up & spin down).
The spin quantum number leads to spin angular momentum which is denoted by ms.  This is given by the expression :
Inner Quantum Number:
In an orbit electron performs two type of motion:
i) Orbital
Due to orbital motion, electron has electron angular momentum

ii) Spin
Due to spin, electrons has spin angular momentum i.e.

Total angular momentum of electrons:

Thus the four quantum numbers specify completely the quantum state of an electron in atom & hence determines its energy & angular momentum.

When an electron jumps from one state to another some or all of these quantum number changes & hence energy and angular momentum of electron changes which results the spectrum.
I hope you understand all the quantum numbers. Still any query you can always ask in below comment box.
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