8. Quantum numbers..

We know that any atom is composed of many shells named  K, L, M, N..etc. All the shells contains subshells i.e. s, p, d, f, ...etc, and each subshell composed of various orbitals..i.e. 

S, px , py , pz

dz2,

dxz

,dyz

,dxy

,dx2−y2

--Simply this is a  process of naming of these shells, subshells ,  orbitals and their spin orientation by a numbers and these numbers are called Quantum number.
-- It is found by the solution of Schrodinger wave equation for hydrogen atom.

There are four types of Quantum number

1. principal q.no. (n)-  which reprasent main energy level or shells in which the electron is present.        
                    n= 1, 2, 3, 4, ....
2. Orbital angular momentum quantum number or Azimuthal  q. no. (l)- It described the orbital motion of the electron around the nucleus which is described by the orbital angular momentum of the electron.thus it is called orbital angular momentum quantum number.     
                                 l = n-1
 3. magnetic q. no.  (m)- Since the magnetic or electrical field generated by the angular momentum of the electron interact with external magnetic or electric field.The electron orient or revolve themselves in a specific regions of space around the nucleus called orbitals.The number of  orbitals in a given sub energy level (l) within a principal energy level (n) is given by the number represented by m ,called magnetic quantum number.
Possible values of m is
                                     m = 2l +1
  
4. Spin q. no.  (s)-- It arises by the spnning of electron around the nucleus as well as around its own axis. We will discuss it below. 
    

Shells	K, L, M, N..	Represented as   n
Subshells	S, p, d, f…	Represented as   l
Orbitals	S, px , py , pz dz2, dxz   ,dyz ,dxy ,dx2−y2 ,fz3 ,fxz2 ,fyz2 ,fxyz ,fz(x2−y2) ,fx(x2−3y2) ,fy(3x2−y2. …	Represented as  m
Spin	Up  or Down	Represented as  s

Example:-- for electronic configuration  given below  the value of n , l, and no. of electron  is 1, 0, 2 respectively.
    

--Possible values for n, l, m, s                             

n	1, 2, 3, 4…
l	0 to n-1
m	+ ι , 0, - ι
S	+1/2, or   -1/2

-- Quantum number presents the position of electron and it also indicates the distance of electron from the nucleus..
If n = 1
    l = 0   
then the position of electron = 1s
If  n = 2
    l = 1
then position of electron = 2p
 If  n =3
    l = 2
then position of electron = 3d
-- Picture given below shows the distance of orbitals from nucleus and the energy as well..                                     

 --If we know the principal quantum number we can determine the radius , velocity , and energy of the electron .

Spin quantum number-
                      Fourth q. no. doesn't follow from wave mechanical treatment. It arises from the spectral evidence that electron in its motion around the nucleus also rotates or spin about its own axis. Because of this rotation the electron has magnetic moment called spin magnetic moment which can be either up or down spin .
    
-Spin angular momentum is characterised by a Q . number S
                       S = 1/2,   either +1/2 or -1/2.
 Hydrogen spectra fine structure is observed as a doublet corresponding to two possibilities for the Z-component of the angular momentum , where for  any given direction Z the value of Spin  angular momentum S is ..
                   

                                               Sz =   ± 1/2 ћ

--Its solution give to possible  Z- component for electron  spin up and spin down

--When atom have even no. of electron in each orbitals , orientation of one electron will be opposite to other , or each electron will be opposite in orientation to that of its immediate neighbour.

-In the early year of Quantum mechanics atomic spectra external field can't be predicted with just n, l, m.

-Unlenbeck , Goudsmit and Kronig introduce an idea of self rotation of the electron , which would naturally give rise to an angular momentum vector in addition to the one associate with orbital rotation (l , m).

-- Electron Spin magnetic moment   µs = -( e/2m) g s

                where e= charge  and      g  =  Lande - g - factor.

Thus , these four quantum numbers play an important role to determine the value of radius , energy level (n), sub energy level (l), the orientation of  orbital(m) and  the direction of spin.

In other word " Quantum numbers serve as an address for an electron.