Schroedinger's equation in atomic units (h = 2
The ground-state wave function (coordinate space) and energy for an oscillator with reduced mass
The first thing we want to illustrate is that tunneling occurs in the simple harmonic oscillator. The classical turning point is that position at which the total energy is equal to the potential energy. In other words, classically the kinetic energy is zero and the oscillator's direction is going to reverse. For the ground state the classical turning point is,
has solution(s)
From the quantum mechanical perspective the oscillator is not vibrating; it is in a stationary state. To the extent that the oscillator's wave function extends beyond the classical turning point, tunneling is occurring. The calculation below shows that the probability that tunneling occurs is independent of the values of k and
A Fourier transform of the coordinate wave function provides its counter part in momentum space.
The uncertainty principle can now be illustrated by comparing the coordinate and momentum wave functions for a variety values of k and
For k = 2 and
Force constant:
Effective mass:
Classical turning point:
Force constant:
Effective mass:
Classical turning point:
The uncertainties in position and momentum are calculated as shown below because <x> = <p> = 0.
A summary of the four cases considered is provided in the table below.
Prepared by:
Frank Rioux
Department of Chemistry
CSB|SJU