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)