Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
A detailed explanation of the harmonic oscillator model in quantum mechanics, including the classical mechanics background, the time-independent schrödinger equation solutions, and the physical significance of the zero-point energy. It covers various systems such as one-dimensional and three-dimensional boxes, rectangular wells, and particles on a ring or sphere. The document also includes the harmonic oscillator wavefunctions, probability density, and the correspondence principle.
Typology: Study notes
1 / 9
This page cannot be seen from the preview
Don't miss anything!
Solving the Schrödinger Equation Exactly for Simple Model Systems Harmonic Oscillator Model of Particle Vibrations (2) Vibrational Motion (1) Translational Motion One particle in free space, 1D-box, 3D-box, and 1D-rectangular well Particle on a ring, and sphere (3) Rotational Motion
2 Harmonic Oscillator Model of Vibrational Motion: Classical Mechanics F kx dx dV F Force for harmonic oscillations is given by: This force corresponds to a potential energy of: 2 2 1 V kx See Engel & Reid CH. 18.6 for details of the classical harmonic oscillator. The harmonic oscillator model is for one particle of mass m moving in a parabolic potential. x x 1 t x 2 t x 1 x 2 (^) equilib. This is easily extended to two particles with reduced mass m = (m 1 m 2 )/(m 1 + m 2 ).
The separation between adjacent energy levels in the harmonic oscillator is given by v 1 v E E for all values of v. Plot V(x) with respect to x : Energy of the Ground Vibrational State: 2 2 1 0 0 ^ ^ E E 0 = v This is the zero-point energy. Harmonic Oscillator (continued)
What is the physical significance of the zero-point energy? Qualitatively, why does this occur? Harmonic Oscillator (continued)
7 Ground and First Excited State Wavefunctions & Probability Density: 2 2 ( ) ( ) v v v y x N H y e
, , v 0 , 1 , 2 , 4 1 2 mk x y 2 2 2 2 2 ( ) ( ) ( 1 ) 0 0 0 0
y x x N H y e N e Ground State ( v = 0 ): 2 2 2 0 2 0
x x N e 2 2 2 2 ( ) ( ) ( 2 ) 1 1 1 1 y y x N H y e N y e
Ground State ( v = 1 ): 2 2 2 2 2 1 (^22) 1 2 1 ( ) ( 4 ) 4
x e x x N y e N y What do these look like (plot versus x)? Harmonic Oscillator (continued)
The Wavefunctions of the Harmonic Oscillator ^ *y
