2. Chapter Overview Classical HO and Hooke’s Law HO Energy
Diatomic Molecules & HO
HO & Morse Potential
Quantum Mechanical HO
HO & IR
HO Wave functions
Even/Odd functions

4. Classical HO Hooke’s Law Newton Applied to Hooks Law
consider a mass connected to a wall by a spring where:
l0 is the eq length and
l is the length after the spring
has been stretched or
compressed
***Demonstration
Classical HO
Newton Applied to Hooks Law
Getting the Wavefunction
A = amplitude
= frequency (rad/s)
 = /2 (1/s)

6. Energy Time Potential Energy, V(x) Kinetic Energy, T(x) or K(x)
Total Energy, E(x) = K(x) + V(x)
This means that energy is conserved and total energy is constant

8. HO Applied to Diatomics
The Model
Finding our DiffEq Solution
a spring between 2 atoms
1. Subtract 2 equation & manipulate
Equations of motion for each mass:
HO Applied to Diatomics
2. Substitute with x = x2 – x1 – l0 with l0 = 0
Center of Mass (CoM), M
3. Recall CoM
4. Use CoM to simplify
Suggest CoM moves uniformly in time
Momentum is constant
Motion depends on relative separation of masses or
Same as that of the one mass system

10. HO & Morse Potential Meet the Real Potential
For l < l0 have a sharp rise due to repulsion
For l = l0 we have the equilibrium bond length
As we pull the atoms apart the bond breaks and the energy will tail off
HO Potential
Real Potential
HO & Morse Potential
Comparison between HO & Real Pot:
Good for many molecules at 298K
Fits well close to the minimum
We will use a Taylor Expansion about l = l0

11. HO & Morse Potential Taylor Expansion Applied to HO Setting V(l0) = 0
Our potential now becomes:
This will do 2 things:
Allow us to match the real potential curve’s minimum
Force this to be set to zero through
Only small displacements from l0 are allowed
This rule eliminates higher order terms which will be discussed later in the semester

12. HO & Morse Potential Meet the Morse Potential
Looks very much like the real potential
 Shown here is H2’s
HO & Morse Potential

14. SE for 1D QHO The Math QHO Energy QHO Energy Levels
We know the energy must be quantized
Solving this system is beyond us so we will skip the wavefunction for now and skip to E
SE for 1D QHO
QHO Energy Levels
h is the E difference between all levels
v = 0 is ½h above the well bottom
Called the zero point energy
It is a direct results of HUP
(KE & PE cannot be zero simultaneously)

16. HO & Morse Potential Harmonic Oscillator Models IR
The absorption frequency is on the order of IR
HO & Morse Potential
obs must satisfy E = h obs to be absorbed/emitted
The only allowed transitions are v = 1
We will address this more in Chapter 13

17. SKIP (at least for now)

19. All about QHO  What  looks like First few 
Normalization
Constant
All about QHO 
Gaussian Distribution (exponential term)
The Hermite Polynomial, Hv
For ξ = α½x
H0(ξ) 1
H1(ξ)
2(ξ)
H2(ξ)
4(ξ)2 - 2
H3(ξ)
8(ξ)3 - 12(ξ)
H4(ξ)
16(ξ)4 - 48(ξ)2 + 12
H5(ξ)
32(ξ) (ξ) (ξ)

20. Proof that these s are eigenfunctions of the Hamiltonian
All about QHO 

21. All about QHO  These s are normalized These s are orthogonal
You get to prove this in your homework
The two combined give orthonormal
All about QHO 
Use of Even/Odd – it will come in handy
For odd: integration over all space produces zero
For even: integration over all space can be performed by doubling the 0   portion

22. SKIP (at least for now)