quantum view of Harmonic oscillator
Download as PPTX, PDF3 likes8,339 views
The harmonic oscillator played a key role in early quantum theory. Planck, Einstein, and others assumed that atoms, radiation, and solids behaved like quantum harmonic oscillators to explain phenomena like blackbody radiation and heat capacities. This led to the development of quantum theory for electromagnetic and mechanical oscillators. Solving the Schrödinger equation for the harmonic oscillator yields discrete energy levels and eigenfunctions that describe its quantized energy spectrum.
quantum view of Harmonic oscillator
- 1. Ahmed Haider Ahmed s.M.Sc. Physics – ASU
- 3. 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; and in 1907, Einstein assumed that the elastic vibrations of a solid behaved as a system of mechanical oscillators with quantized energy. These assumptions were invoked to account for black body radiation, the photoelectric effect and the temperature dependence of the specific heats of solids. Subsequently, quantum theory provided a fundamental description of both electromagnetic and mechanical harmonic oscillators.
- 4. So, the potential energy will be: kxdxxv Fdxxv kxF )( )( 2 2 1 )( kxxv Substituting in Schrödinger equation :
- 5. Schrödinger equation will be: )()()( 2 2 2 rErrv m In x direction )()( 2 1 2 2 2 22 xExkx xm )1()( 2 )( 2 2 22 2 xE m xx mk x
- 6. 2 24/1 22 2 2/1 2 2 4/1 2 2 2/1 2 Let, mk x mk x x mk x mk Putting this value in equation (1) we get:
- 8. The quantum energy spectrum consist of discrete values according to energy levels n() is the angular function for state n and En is the energy of the state n The problem now is to determine n , n
- 10. )3()()12()( nnn (3)with(2)equationcomparing )( :beillsolution wits 0)( 2 2 1 by en n (*)11 nn
- 13. This the quantized energy of harmonic oscillator nn nn n 0 1 1n 1 1 Ebut, 2 1 nEn n = 0,1,2,3,….
- 17. The three lowest energy eigenfunctions (n=0,1,2)
- 18. Introduction to Quantum Mechanics: A. C. Phillips, Physics and Astronomy Dept. University of Manchester. Lecture notes in quantum mechanics by Dr. Mahmoud Salah , Physics Dept., University of Minia, 2010 Quantum Mechanics 3rd EDITION : Eugen Merzbacher University of North Carolina at Chapel Hill.