Solving Two-Dimensional Schrodinger Equation for the Higgs Potential Using Numerical and Variational Methods

In this study, we solve the two-dimensional Schrödinger equation for the Higgs potential. We begin by reducing the problem to a one-dimensional differential equation using rotational symmetry. Then, through an asymptotic expansion and series solution methods, we simplify the problem further into another one-dimensional differential equation. Both equations are solved for bound states with identified initial conditions. To compute the eigenvalues numerically, we must provide the range of eigenstates and the number of terms in the series solution. Due to this, we apply the variational method as an additional approximation technique. By carefully selecting parameters, we achieve consistent results with both methods. We use the ground state and first excited wave functions obtained to verify the uncertainty relation. Finally, we attempted to analyze Higgs mode using the wave function obtained from the above procedures.