Post on 19-Jan-2016
V
6
5
4
3
2
1
0
V
6
5
4
3
2
1
0
The wavefunctions of the Simple Harmonic Oscillator are
Orthonormal
Orthonormal wavefunctions
∫+ ∞
- ∞ψv” ψv’ dτ = δv” v’
Orthonormal wavefunctions
∫+ ∞
- ∞ψv” ψv’ dτ = δv” v’
Orthonormal wavefunctions
Kronecker delta δij = 1 when i = j and 0 when i ≠ j
+ +
∫+ ∞
- ∞ψv=0 ψv =0 dτ = 0
Orthonormal wavefunctions
← – ∞ + ∞ →
V
6
5
4
3
2
1
0
+–
∫+ ∞
- ∞ψv=1 ψv =0 dτ = 0
Orthonormal wavefunctions
← – ∞ + ∞ →
V
6
5
4
3
2
1
0
+ +–
∫+ ∞
- ∞ψv=2 ψv =0 dτ = 0
Orthogonal wavefunctions
← – ∞ + ∞ →
Anharmonic oscillator wavefunctions and probabilities
Anharmonic oscillator wavefunctions and probabilities
Anharmonic oscillator wavefunctions and probabilities
Anharmonic oscillator wavefunctions and probabilities
Anharmonic oscillator wavefunctions and probabilities
V
6
5
4
3
2
1
0
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
↑y
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
↑y
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
↑y
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
↑y
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
↑y
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
↑y
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
↑y
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
↑y
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
x →
↑y
-8 -6 -4 -2 0 2 4 6 8
12
10
8
6
4
2
0
y = x2
http://en.wikipedia.org/wiki/File:Simple_harmonic_oscillator.gif
H + H
E(r)
r
0
Rotational levels
Continuum wavefunction of a dissociating state
http://131.104.156.23/Lectures/CHEM_207/uv-vis.htm
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html
H + H
r
E(r)
H2
http://jchemed.chem.wisc.edu/JCEDLib/SymMath/collection/article.php?id=29
http://www.pci.tu-bs.de/aggericke/PC3e_osv/Kap_III/Molekuelschwingungen.htm