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

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)

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

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

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

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

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

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)

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

SKIP (at least for now)

All about QHO  What  looks like First few  Normalization Constant All about QHO  Gaussian Distribution (exponential term) The Hermite Polynomial, Hv For ξ = α½x https://commons.wikimedia.org/wiki/File:Gaussian_distribution.svg 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)5 - 160(ξ)3 + 120(ξ)

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

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

SKIP (at least for now)