Computational Physics II
• Assignment 5 - Time Evolution of QM Forced Harmonic Oscillator

Due: April 2

Code

• The first thing that needs to be done is to modify the program ot allow for time-dependent potentials. This is relatively easy using the given program. All that needed to be done was to modify the imaginary part of the diagonal which holds the potential elements. For each loop of the time evolution, the potential is produced for all values of x and the current time. This was an easy modification to the potential function, allowing a second parameter.
• The program was also modified to allowed command line arguments alpha and omega that defined the total potential.
• In order to start the project an eigenstate of the normal harmonic oscillator needed to be found. The Crank-Nicholson algorithm that was given to us already had a gaussian solution with the proper width to satisfy this requirement. Here is a plot of the probability distribution function (PDF) for all time that shows that is in fact an eigenstate:

2. Finding Resonance

• Now we can play with the parameters alpha and omega to find a resonating driving frequency. Since alpha is just the amplitude of the driving force, I set that to zero for all runs. I then modified omega from 1 to 3 in steps of one. I found that resonance occured near an omega value of two. Here are the three plots:

Below Resonance

At Resonance

Above Resonance

• I increased the time so that 10 full units would pass making the time evolution more apparent.
• As you can see, the PDF below resonance begins to fade along with the PDF above resonance. At resonance, we see that the PDF steadily increases as the time increases.