1. Harmonic Oscillator
Harmonic Oscillator
W. Udo Schröder, 2004

2. 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

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

4. 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

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
W. Udo Schröder, 2004

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)
W. Udo Schröder, 2004

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

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

14. 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

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

16. End of Section
Harmonic Oscillator
W. Udo Schröder, 2004

17. Harmonic Oscillator: General Solution
W. Udo Schröder, 2004

18. 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