Harmonic Oscillator

Harmonic Oscillator

1. THE HARMONIC OSCILLATOR By- Manish Sahu M.Sc. Chemistry (Final) Sp.- Physical Chemistry

2. Introduction Definition History Force Frequency Energy Application Limitations Conclusion Reference SYNOPSIS

3. A Physical system in which some value oscillates above and below a mean value at one or more characteristic frequencies . Such system often arise when a contrary force result from displacement from a force neutral position and gets stronger in proportional to the amount of displacement , as in the force exerted by a spring that is stretched or compressed or by a vibrating string on a musical instrument . INTRODUCTION

4. A system executing harmonic motion is called a HARMONIC OSCILLATOR. It is a basic model for a vibrating diatomic molecule. DEFINITION

5. The Harmonic Oscillator played a leading role in the development of quantum mechanics. In 1900 , PLANCK made the bold assumption that atoms acted like oscillators with quantized energy when they emitted and absorbed radiation . In 1905 , EINSTEIN assumed that electromagnetic radiation acted like electromagnetic harmonic oscillators with quantized energy. HISTORY

6. In SIMPLE HARMONIC MOTION, the force acting on system at any instant , is directly proportional to the displacement . If the displacement of the system from a fixed point x, The linear restoring force is F , F α x F = - k x ………(1) Where , F = Restoring Force x = Displacement k = Force Constant FORCE OF HARMONIC OSCILLATOR

7. Now according Newton's 2 nd law of motion,If ‘m’ is the mass of a particle,than F = ma a = d 2 x /dt 2 F = m(d 2 x/dt 2 ) ………(2) Now equaling equation (1) & (2),then d 2 x/dt 2 + kx/m = 0 ……....(3) If x is the displacement Harmonic ,then x = Asin2 π ʋt ………..(4) where, A= amplitude of vibration , ʋ = frequency of vibration FREQUENCY OF HARMONIC OSCILLATOR

8. If we take differential form of eq n (4) dx/ dt = A (2 π ʋ) cos2 π ʋt Again differential… d 2 x/dt 2 = - A ( 4 π 2 ʋ 2 ) sin2 π ʋt d 2 x/dt 2 = - 4 π 2 ʋ 2 x Or d 2 x/dt 2 + 4 π 2 ʋ 2 x = 0 ………(5) Now equaling then eq n (3) & (5) , then, 4 π 2 ʋ 2 x = kx/m or in other words the frequency of the vibration in simple harmonic oscillator i.e. ……..(6) This equation is the frequency of LINEAR HARMONIC OSCILLATOR . ʋ =1/2 π√ k/m

9. If it is in classical treatment force is related to potential energy by the expression :- F = - ∂V/∂x where V = P.E. F = - kx - ∂V/∂x = - kx ∂V/∂x = kx ∂V = kx ∂x Now If we integrate these eq n ,then ∫ v dV = k ∫ x x dx V = k x 2 / 2 This equation is potential energy of LINEAR HARMONIC OSCILLATION . V = ½ kx 2 Fig:- P.E. Diagram from Linear Harmonic Oscillation POTENTIAL ENERGY

10. The problem of simple harmonic oscillator occurs frequency in physics because a mass at equilibrium under the influence of any conservative force,in the limit of small motions behaves as a SIMPLE HARMONIC OSCILLATOR . A conservative force is one i.e.,associated with a potential energy .The potential energy function of a HARMONIC OSCILLATOR is V(x) = ½ kx 2 APPLICATION

11. The HARMONIC OSCILLATOR is a great approximation of a molecular vibration,but has key limitations :- Due to equal spacing of energy ,all transitions occure at the same frequency.However experimentally many lines are often observed. The HARMONIC OSCILLATOR does not predict bond dissociation, you cannot break it no matter how much energy is introduced. LIMITATIONS

12. The HARMONIC OSCILLATOR is among the most important examples of explicitly solvable problems, whether in classical or Quantum Mechanics .It appears in every textbook in order to demonstrate some general principle by explicit calculation . CONCLUSION

13. Advanced physical chemistry by :- “Dr. J.N. Gurtu & A. Gurtu” Theoretical chemistry by :- “Samuel Glasstone” REFERENCE