When displaced from equilibrium this system is a basic model for the simple harmonic oscillator (SHO) - neglecting anharmonicity of course. 1

When displaced from equilibrium this system is a basic model for the simple harmonic oscillator (SHO) - neglecting anharmonicity of course. The reason it is so fundamental is that it applies to all dynamics involving displacements from equilibrium from the vibrations of the CO2 molecule responsible for the Greenhouse effect…… 2

ω1 =1388 cm-1 symmetric stretch ω2 = 667 cm-1 bend ω3 = 2349 cm-1 asymmetric stretch 3

ω3 = 2349 cm-1 active ω1 =1388 cm-1 inactive ω2 = 667 cm-1 active

2800 2500 2200 1900 1600 1300 1000 700 cm-1

…to the lateral oscillations of the Millennium Bridge 6

G(r) G(v) = ω (v + ½) v = 3 3½ ω 2½ ω v = 2 1½ ω v = 1 v = 0 ½ ω re r – re 

It is worth noting that in Heisenberg’s first paper on Quantum Mechanics we find his first test is on the energy of the harmonic oscillator: 8

It is worth noting that in Heisenberg’s first paper on Quantum Mechanics we find his first test is on the energy of the harmonic oscillator: W = nhωo/2π (22) 9

It is worth noting that in Heisenberg’s first paper on Quantum Mechanics we find his first test is on the energy of the harmonic oscillator: W = nhωo/2π (22) W = (n + ½)hωo/2π (23) 10

It is worth noting that in Heisenberg’s first paper on Quantum Mechanics we find his first test is on the energy of the harmonic oscillator: W = nhωo/2π (22) W = (n + ½)hωo/2π (23) “According to this idea, therefore, even with the harmonic oscillator the energy cannot be represented by classical mechanics, by (22) but has the form (23)” ie zero point energy 11

Translation in “Wave Mechanics” It is worth noting that in Heisenberg’s first paper on Quantum Mechanics we find his first test is on the energy of the harmonic oscillator: W = nhωo/2π (22) W = (n + ½)hωo/2π (23) “According to this idea, therefore, even with the harmonic oscillator the energy cannot be represented by classical mechanics, by (22) but has the form (23)” ie zero point energy Translation in “Wave Mechanics” Gunther Ludwig P 180 Pergamon 1968 12

13

symmetric stretch of a homonuclear diatomic molecule such as H2