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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.
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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