Introduction to Quantum Mechanics:
Project 1 Quantum 1D Oscillator
Instructor: Lizhi Ouyang
(Due on March 2, 2007)
The 1-Dimension time-independent Schrödinger Equation can be written as
2 2

  V ( x)  E
2m x 2
Consider a particle moving in 1-Dimension potential field V(x) that can be expanded into the
following taylor series:
V ( x)  v 0  v1 x  v 2 x 2  v 3 x 3  v 4 x 4  v 5 x 5  
When only v2 is nontrivial, the Schrödinger equation reduces to a 1D quantum harmonic oscillator
problem. In this project, you are setting out to find the eigenvalues and wavefunctions to the
Schrödinger equation for the following cases:
(1) Harmonic Oscillators
V = 1/2 K x2
(2) Simplest anharmonic oscillator
V = 1/2 K x2 + 1/24 M x4
(3) Double well Oscillator
V = -1/2K x2 + 1/24 M x4
Using either numeric methods or in special cases, analytical approaches.