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Harmonic Oscillator

This work explores the intricate phenomena of harmonic oscillation, focusing on the dissociation processes and vibrational modes within N-body systems. Detailed discussions include the separation of overall motion into center-of-mass, rotation, and vibrational modes, as well as an analysis of symmetric and asymmetric stretches in diatomic molecules. We delve into the classical frequency, the variational principles of the Schrödinger equation, and the implications of Morse potential in harmonic approximation. Numerical techniques, such as Hermite polynomials for wave function analysis, will be presented, alongside the examination of energy eigenstates and transitions.

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Harmonic Oscillator

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  1. Harmonic Oscillator Harmonic Oscillator

  2. z x Dissociation Harmonic Vibrations of N-Body Systems Separation of overall (center-of-mass “com”), rot and vib motion: Superposition of Nvibindependent 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 Generic example: Diatomic molecule (H-Cl),..Discrete bound (stationary) states at low excitations Dissociation at high excitations Harmonic Oscillator bend (2 x) Classical vibrational frequencycj= Hooke’s constant

  3. Dissociation Taylor expansion about Morse Potential and Harmonic Approximation Equilibrium bond length Harmonic Oscillator

  4. Harmonic Oscillator Reset energy scale  U(q0)=0x := 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

  5. 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

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

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

  8. all =0 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

  9. w Energy Spectrum of Harmonic Oscillator Recursion relations for aiterminate at i = n i = even or i = odd. Harmonic Oscillator Energy eigen values of qu. harmonic oscillator

  10. E3 n=3 E2 n=2 E1 n=1 E0 x= D n=0 Harmonic Oscillator Level Scheme Equidistant level scheme 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

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

  12. z r m2 L q f m1 Rot-Vib Spectra of Molecules Assume independent rot and vibneglect centrifugal stretching J2 J1 Rot-vib absorption/emission branches Harmonic Oscillator

  13. HOsc-Wave Functions: Hermite’s Polynomials Definite parity p=+1 ,p=-1 Harmonic Oscillator

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

  15. Calculating Expectation Values Mean square deviation from equilibrium bond length HCl MathCad rendition: Deviation from equilibrium bond length Harmonic Oscillator

  16. End of Section Harmonic Oscillator

  17. Harmonic Oscillator: General Solution Harmonic Oscillator

  18. z z z z z z y r m2 m2 m2 m2 q L R q f f f f f x r r r r m m1 m1 m1 m1 R R R L L L L z m +4 +3 a +2 +1 0 -1 -2 -3 -4 Drawing Elements L=4 a Harmonic Oscillator

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