# Lecture 10: Wave Function/Equation, Orbitals, and Quantum Numbers

﻿$\Psi :$﻿ Wave function/equation atomic orbital

﻿$\Psi^{2} :$﻿ A measure of the intensity of electron density distribution

﻿$\int \Psi ^{2}dv:$﻿ Probability

﻿$\Psi$﻿ must by finite, single valued and continuous

﻿$\int$﻿ (Over all space) ﻿$\Psi ^{2}dv=1$﻿ (normalization)

1 - Finite probability of finding the electron (but don't know where)

﻿$\Psi = R(r)\Theta (\Theta )\Phi (\Phi )$﻿ Cartesian coordinates to Polar coordinate

### S Orbital

• Vertical
• Doesn't have an angle

### P Orbital

• Has an angle
• X, Y Z axis

Plug in Equation - ﻿$\Psi = nlm$﻿

• n - Defines Orbital size = 1,2,3 (Principle Quantum number)
• l - Defines Orbital shape = 0, 1, 2, 3 (Azimuthal Q, N) -n-1
• m - Defines Orbital orientation = -l.....0.....+l (Magnetic Q.N)
• s - Defines direction of ﻿$e^{-}$﻿spin = ﻿$+\frac{1}{2} or -\frac{1}{2}.$﻿ (Spin Q, N)

﻿$+\frac{1}{2}$﻿ - Clockwise

﻿$-\frac{1}{2}$﻿ - Counter clockwise

### Quantum Numbers

l = 0 - S orbital (vertical shape)

l = 1 - P orbital (dumbshell shape) ﻿$P_{x}, P_{y}, P_{z}$﻿

l = 2 - d orbital (double dumbshell shape)

l = 3 - f orbital

l = 1 m = -1, 0, +1

l = 2 m = -2, -1, 0. +1, +2

﻿$P_{x}, P_{y}, P_{z}$﻿ have the same size and energy - "Degenerate Orbital"