Masters Thesis

Similarity solutions of the two dimensional stationary Schrodinger equation

We have investigated a similarity analysis for a two-dimensional stationary Schrodinger equation that is based on the group theoretic method introduced by Lie. Working in the complex plane, and using a family of one-parameter transformations, forming a group, we have found a requirement on the "energy function" for the model problem to have Lie group solutions. This requirement reduces the parameter coefficients for the matrix in the invariance condition to be a diagonal matrix. Furthermore, analyzing its characteristic system, we have shown there are subclasses that one can obtain solutions in closed form. Several of such solutions obtained are illustrated graphically. In addition, we used a predetermined form in the extended similarity form in the real plane, we obtained several closed form solutions from a reduced ODE. Some of these solutions are presented graphically.

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