OVERSTANDING REMOVES MISUNDERSTANDING

OVERSTANDING REMOVES MISUNDERSTANDING

OVERSTANDING REMOVES MISUNDERSTANDING

GUARANTEED WEALTH; HEALTH; WISDOM AND IS "toe-curling" RELATIONSHIPS ARE YOURS IF YOU OVER-STAND!

Simple harmonic motion Atoms connected via chemical bonds are equivalent to masses connected by springs. We can describe these using Hooke’s law (Q is a displacement of an atom away from eqn position) From Newton’s second law Where m is the reduced mass Thus Find a general solution where F = -kQ F = m d2 Q dt2 m d2 Q dt2 +kQ = 0 Q(t)= Acos(wvibt) wvib = k m;  Insert potential into time independent Schrodinger equation: To find quantized solutions V(x) = 1 2 kx2 k = d2 V dx2 æ è ç ö ø ÷ From classical to quantum V(x) = V(0) + x dV dx æ è ç ö ø ÷+ 1 2 x2 d2 V dx2 æ è ç ö ø ÷ + 1 3! x3 d3 V dx3 æ è ç ö ø ÷ + ... If two nuclei are slightly displaced from equilibrium positions (x = R - Re), can express their potential energy in a Taylor series: Not interested in absolute potential, so set V(0) = 0. At equilibrium, dV/dx = 0 (a potential minimum). Providing displacement is small, third order term can be neglected. We can therefore write: EY(x) = - 2 2m Ñ2 +V(x) é ë ê ù û úY(x) En = wvib (n+ 1 2 ) his creates a ladder of vibrational modes This is well-known case of a harmonic oscillator. The energy of a quantum-mechanical harmonic oscillator is quantized and limited to the values. Selection rules dictate that harmonic Oscillator transitions are only allowed for Dn = ± 1 En = (n+ 1 2 ) wvib 0 1 2 3 4 5 6 7 8 Displacement (x) Energy Potential energy V wvib V(x) = 1 2 kx, Molecules have many different vibrational modes O C O O C O Asymmetric stretch mode O C O Bending mode CO O Symmetric stretch mode (100)