QUANTUM MECHANICS
TIME DEPENDENT SCHRODINGER EQUATION
FINITE DIFFERENCE TIME DEVELOPMENT METHOD WAVEPACKET CONFINED TO A PARABOLIC POTENTIAL WELL Ian
Cooper matlabvisualphysics@gmail.com DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS qm007.py This
article will consider a Gaussian wavepacket for an electron confined within a
parabolic (harmonic) potential well. The motion of the electron corresponds
to a quantum harmonic oscillator as shown in figure 1. Note the changes that
occur in the shape of the probability density function.
Fig.
1. Animation of the wavepacket in a
harmonic well. The red plots show the initial wavefunction (real and
imaginary parts) and the normalized probability density function. Figure 2
shows the time-dependence of the expectation values of position <x> and
the kinetic energy <K>,
potential energy <U>,
and the total energy <E>.
Fig.2. Time-dependence of the expectation values
of position <x>,
kinetic energy <K>,
potential energy <U>,
and total energy <E>. The
expectation values vary sinusoidally for the position <x>,
the kinetic energy <K>,
potential energy <U>.
The total energy <E>
remains constant since zero external forces act on the system. The harmonic motion is similar to a
classical particle, expect we never know the exact position of the electron
because of the uncertainty principle, but the motion of the wavepacket is
deterministic. |
