Harmonic Oscillator (9) use pib to think through 01 VI 9 Particle in box; Stubby box; Properties of going to finite potential w/f penetrate walls, w/f oscillate, # nodes increase with n, E n -levels less separation with n, for E>V 0, spacing collapse, E n continuous Consider one ball (mass) on a spring Hook s law states that restoring force F = -k (x x e ) = -kq F d q = Δx = x - x e displacement x e F = - V/ q [recall: d(kq )/dq = kq] V = 1 / kq k = force constant F u h m x T = h = m q here dq = dx Hψ = h m d + (1/) kq dq ψ = Eψ T V a) like a box with sloped sides soft potential - expect penetrate b) still a well - expect oscillator c) must be integrable expect damped, i.e. ψ(x) 0 at q = ± 9
VI 30 Trial: ψ ~ ƒ(q)e αq Fail: q(+) OK, blows up q(-) side Solution: ψ ~ ƒ(q)e αq works both side - (α must positive) ƒ(q) polynomial form works f 0 ~const - orthog. f 1 ~q (odd-even) orthog. if ƒ (q) ~ q -const Result Atkins sect. confusing see House Ch. 6 not to derive, just get idea Wave function (from Engel ) modify variable to simplify: ψ υ (y) = H υ (y) e -y / υ = 0, 1,, (quantize) recursion formula: H υ (y) = (-1) υ e y d υ /dy υ (e -y ) y = α 1/ q α =π mk h= mk h non-classical: see wavefunction penetrate potential [Engel - difference classical: sin q & turning point at V(q m )] Hermite polynomials (H υ ): ex: H 0 = 1 note: odd - even progression H 1 = y alternate exponents H = 4y number solutions H 3 = 8y 3 1y exponential damping 30
thus: ψ υ = A υ H υ (y)e -y / VI 31 y=α 1/ q A υ =( υ υ!) -1/ (α/π) 1/4 Homework: insert ψ υ into Schrödinger equation to get: E υ = (υ + 1/) hω ω = k m = (υ + 1/) hν ν = ω/π note: even energy spacing: ΔE = hν zero point energy: 1/ hν heavier mass ΔE 0 classical weaker force constant ΔE 0 Shapes: wave functions probabilities (House) Model problems from Web Site: 31
3 VI 3
VI 33 Probabilities: low υ high in middle; high υ high at edge This fits classical, turnaround points slower motion A. Solutions for υ=0-4 B. ψ(x)*ψ(x), probabilities for υ = 0,4,8 compared to classical result (---) Plots of ψ ψ for υ = 0 4 and for n = 1, from Engel To describe two masses on a spring (relate to molecules) need change variable q = (x - x 1 ) (x 0 - x 0 1) = r r eq (3-D representation) r = x x 1 relative 1-D position in this case: μ = m 1 m /(m 1 + m ) reduced mass into Η -mass harmonic oscillator: Ηψ = [(-h /μ) d /dq + 1 / kq ]ψ υ = E υ ψ υ and get E υ = (υ+ 1 / )hω = (υ+ 1 / )hν ω = k μ ν = 1 / k π μ 33
Use to model vibration of a diatomic molecule low υ VI 34 harmonic (ideal) spacing regular E-levels collapse in real molec. anharmonic anharmonic probability distributions, even υ multiatom 3n - 6 relative coord complex but separable Two-dimensional Harmonic oscillator: H = T + V T = -h /m ( / x + / y ) V = V(x, y) expand about x=0,y=0 f(x)=σ n [1/n!]d n f/dx n x0 (x-x 0 ) n = V(0, 0) + V/ x 0 x + V/ y 0 y + ½ V/ x 0 x + ½ V/ y 0 y + ½ V/ x y 0 xy +1/6 3 V/ x 3 0 x 3 etc. more complex potential many terms, Taylor expansion not separated -- cross-terms like V/ x y mix variables V(0, 0) = 0 arbitrary constant just shift E, ignore V/ x 0 = V/ y 0 = 0 evaluate derivative at min. choose x = 0, y = 0 as the minimum same as choosing q = 0 as x e -x 34
VI 35 Then V = ½( V/ x ) 0 x +½ ( V/ y ) 0 y +½ ( V/ x y) 0 xy = ½ k x x + ½ k y y + ½ k xy xy + where k x = ( V/ x ) 0 etc. force constant so form just like harmonic oscillator, if neglect high order terms, like x 3 solvable if can separate variables do change of variable x, y q 1, q where q 1, q chosen so that potential is not coupled, mixed coord. V(q 1, q ) = ½ k 1 q 1 + ½ k q q n normal coordinates call this potential diagonalized use matrix approach can do to arbitrary accuracy also works for n-dimensions: (3n - 6) vibration Basis for vibrational spectroscopy IR and Raman We will discuss further in spectroscopy section at end 35