# 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"

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