Harmonic Oscillator Harmonic Oscillator W. Udo Schröder, 2004

Harmonic Vibrations of N-Body Systems Separation of overall (center-of-mass “com”), rot and vib motion: z x Superposition of Nvib independent vibrational “normal modes”, normal coordinates {qj} 3 Normal modes of linear molecules symmetric stretch (infrared) No interaction with other degrees of freedom i≠j asymmetric stretch Harmonic Oscillator Generic example: Diatomic molecule (H-Cl),.. Discrete bound (stationary) states at low excitations Dissociation at high excitations bend (2 x) Classical vibrational frequency cj= Hooke’s constant Dissociation W. Udo Schröder, 2004

Morse Potential and Harmonic Approximation Equilibrium bond length Taylor expansion about Dissociation Harmonic Oscillator W. Udo Schröder, 2004

Stationary Schrödinger Equation Harmonic Oscillator Reset energy scale  U(q0)=0 x := displacement from equilibrium m:= reduced mass Stationary Schrödinger Equation x= D Symmetric potential  Harmonic Oscillator Energy eigen states y(x) : spatially either symmetric or asymmetric W. Udo Schröder, 2004

Harmonic Oscillator: Asymptotic Behavior Asymptotic boundary conditions (y(x) in classically forbidden region)? x2 k2/l  k2« l2x2 Harmonic Oscillator solved app. by Gaussian decay of wf in forbidden region, faster for heavy particles (m) and steep potential (c) . For HCL: l ~ 84 Å-2 W. Udo Schröder, 2004

Harmonic Oscillator Eigen Value Equation Trial: =exact solution for Harmonic Oscillator Found one energy eigen value and eigen function (= ground state of qu. oscillator) W. Udo Schröder, 2004

Harmonic Oscillator: Hermite’s Differential Equ. All asymptotic: Trial function Harmonic Oscillator Hermite’s DEqu. Asymptotic behavior: H(x) not exponential  H = finite power series W. Udo Schröder, 2004

Solving with Series Method Hermite’s DEqu. Asymptotic behavior  H(x) not exponential  H= finite power series Discrete EV spectrum, count states n=,0, 1,… ai from differential equ. Harmonic Oscillator all =0 W. Udo Schröder, 2004

Energy Spectrum of Harmonic Oscillator Recursion relations for ai terminate at i = n i = even or i = odd. Harmonic Oscillator w Energy eigen values of qu. harmonic oscillator W. Udo Schröder, 2004

Harmonic Oscillator Level Scheme Equidistant level scheme x= D E0 E1 E2 E3 n=0 n=1 n=2 n=3 unbounded, n   HCl Zero-point energy +1: absorption -1: emission Elm. E1 transitions: D n = ± 1 D Harmonic Oscillator Absorption and emission in infra-red spectral region W. Udo Schröder, 2004

Rot-Vib Spectra of Molecules Assume independent rot and vib  neglect centrifugal stretching L r m1 z f m2 q Very different energy scales Selection rules for elm transitions: J=0 J=1 J=2 J=3 J=4 n = 2 J=0 J=1 J=2 J=3 J=4 n = 1 Harmonic Oscillator J=0 J=1 J=2 J=3 J=4 n = 0 rotational band vibrational band Vibrational transition change in   change in angular momentum W. Udo Schröder, 2004

Rot-Vib Spectra of Molecules Assume independent rot and vib neglect centrifugal stretching L r m1 z f m2 q J2 J1 Rot-vib absorption/emission branches Harmonic Oscillator W. Udo Schröder, 2004

HOsc-Wave Functions: Hermite’s Polynomials Definite parity p=+1 , p=-1 Harmonic Oscillator W. Udo Schröder, 2004

Oscillator Eigenfunctions Unnormalized: n=4 n=3 n=2 n=1 Normalization n=0 x Harmonic Oscillator All have definite parity, spatial symmetry. n = # of nodes W. Udo Schröder, 2004

Calculating Expectation Values Mean square deviation from equilibrium bond length HCl MathCad rendition: Deviation from equilibrium bond length Harmonic Oscillator W. Udo Schröder, 2004

End of Section Harmonic Oscillator W. Udo Schröder, 2004

Harmonic Oscillator: General Solution W. Udo Schröder, 2004

Drawing Elements L r z f q z a L=4 a L r R z f L r R z f q L r R z f z +1 +2 +3 +4 m -1 -2 -3 -4 a L r R m1 z f m2 L r R m1 z f m2 q L r R m1 z f m2 z x y L r R f m z Harmonic Oscillator W. Udo Schröder, 2004