Quantum Mechanical Model

The wave mechanical model of atom comes from wave equation. The wave equation describes the electron in motion as a three dimensional wave.

This was given by Schrodinger

x, y, z are Cartesian coordinates m is mass of electron.

V is potential energy of electron

E is total energy of electron

h is Planck’s constant.

Ψ is wave function of electron.

This can be written as

               

               

is Laplacian operator.

Significance of Ψ and Ψ²:

•  Ψ is wave function. It gives the amplitude of the electron wave.
• Ψ may have positive (or) negative value depending upon values of coordinates.
• Ψ² is the probability of finding electron at a point in space is given by Ψ² (x,y,z)
• The wave function “Ψ” should satisfy certain conditions called boundary conditions. They are :

1. Ψ must be continuous
2. Ψ must be finite
3. Ψ must be single valued at any point.
4. The probability of finding the electron over the space from + ∞ to - ∞ must be equal to one.

Quantum numbers:

1. Principle Quantum number (n):-

·         It determines the size of the orbital.

·         It indicates the main energy level to which the electron belongs.

n = 1,2,3,4,5,…… are named as K,L,M,N,O.

·         The maximum number of electrons in K, L, M and N energy levels are 2,8,18 and 32.

2.  Azimuthal Quantum number:-

•  It is denoted by “l”
• It defines the three dimensional shape of the orbital.
• For a given values of n, l can have values ranging from 0 to n – 1
• If n = 1 , l = 0;
  If n = 2, l = 0, 1.
• It also denotes the number of sub shells in a principal shell.
  

  value of l	0	1	2	3	4
  subshell	s	p	d	f	g

  
  
• The number of subshells in a principal shell is equal to the value of n.
• Orbital angular momentum of electron is
  
  

      3. Magnetic Quantum Number:-

·         It is denoted by m (or) m_l

·         It gives the orientation of the orbital with respect to the co – ordinate axis.

·         For any sub – shell it will have 2l + 1 values of m_l
           = -l,-(l-1),-(l-2), -------0, ------ (l-2), (l-1), l

      
    4. Spin Quantum number:-

·         It is denoted by “s” or m_s

·         It denotes the magnetic property of the electron.

·         If  the electron spins in clockwise   the electron spins  in anti-clock wise.

·         It defines the direction of spin of the electron.

·         Spin angular momentum is