# 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

$$\frac{{{\partial }^{2}}\psi }{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}\psi }{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}\psi }{\partial {{z}^{2}}}+\frac{8{{\pi }^{2}}m}{{{h}^{2}}}\left( E-V \right)\psi =0$$

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

$${{\nabla }^{2}}\psi +\frac{8{{\pi }^{2}}m}{{{h}^{2}}}\left( E-v \right)\psi =0$$
$${{\nabla }^{2}}=\left( \frac{{{\partial }^{2}}}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{y}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{z}^{2}}} \right)$$

$$\nabla$$  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 :
• Ψ must be continuous
• Ψ must be finite
• Ψ must be single valued at any point.
• 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 $$\sqrt{l(l+1)}\times \frac{h}{2\pi }=\sqrt{l(l+1)}\times \hbar$$

1. Magnetic Quantum Number:

• It is denoted by m (or) ml
• 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 ml.
ml = – l, -(l – 1), -(l – 2), … 0, … (l – 2), (l -1), l

2. Spin Quantum number:

• It is denoted by “s” or ms
• It denotes the magnetic property of the electron.
• If s = + ½ the electron spins in clockwise
s = – ½   the electron spins in anti-clock wise.
• It defines the direction of spin of the electron.
• Spin angular momentum is
$$\sqrt{s(s+1)}\times \frac{h}{2\pi }=\sqrt{s(s+1)}\times \hbar =\frac{\sqrt{3}}{2}\times \hbar$$   (As s = ½)