Open Access Senior Thesis
Bachelor of Arts
@2020 Clara G Chilton
In this paper, I will look at a mass-spring system that can be described by a Hamiltonian. In most systems described by a Hamiltonian, the energy levels will be quantized, and the system will be able to jump between them. However, many methods of finding these jumps aren’t well-suited to numerical analysis. I’ll use a Markovian approximation (The Liouville von Neuman Equation), which allows me to use only the last time step to find the current one. Using this, I will analyze the system to find the time evolution of the probability density matrix – whose diagonal shows the probability of the object being in each energy state at a specific time. I will then repeat this process with an added dissipation energy added to the Hamiltonian, which makes it decay over time into its ground state. From this, and using Fortran coding and Fourier Analysis, I will find the most probable number of energy levels that the object will jump between when said dissipation is applied. The result of this is that I will see what information is lost in this approximation by comparing it with a different method of computing it that Vidar previously modeled.
Chilton, Clara, "Decay and Dissipation: Finding Energy Level Jumps in a Harmonic Oscillator System using Fortran and Fourier Analysis" (2020). Scripps Senior Theses. 1521.