Project Title: Approximation Methods for the One-Dimensional Time-Independent Schrodinger Equation with Power-Law Potential
Student: Khanh Le ’16
Mentor: Dr. Barbara Andereck
The Schrodinger equation is the corner stone of quatum physics. The equation describes the behavior of small particles in terms of the wave function Psi(x,t). The solution of the Schrodinger equation can be expressed as a superposition of solutions of the time-independent Schrodinger equation psi(x), namely the stationary states.
In the research project, we are interested in the one-dimensional time-independent Schrodinger equation. In particular, we look at the equation with the class of potential of the form V(x)=a|x|^N, namely the power-law potential. For specific value of N, the solutions of the equation have played an important role in our understanding of quantum physics. The solutions in the case N=2 can be used as a good approximation of the solution of generic potential near the stable equilibrium. For N= -1, the solutions are related to the radial solution of the hydrogen atom.
We investigate three different approximations of the solutions of the one-dimensional time-independent Schrodinger equation, namely the Verlet integration, the Numerov matrix method and the WKB approximation. The first two methods are numerical approximation for the Schrodinger equation. For those two methods, we develop programs in Mathematica to give approximations for the energy eigenvalues E_n and the wavefunction psi_n(x). From these two approximate solutions, we conjecture a dependency between the energy eigenvalues E_n of each energy state and the quantum number n which we later derive using the WKB approximation. We also re-discover the peculiar behavior of the potential well with N= -2 in which we no longer have quantization of energy due to the strong attraction of the potential.