Applications of the Variation Method

Abstract

The variation method is applied to two examples selected for illustration of fundamental principles of the method along with ease of calculation. The first example applies the linear version of the variation method to the particle in a box model, using a basis with explicit parity symmetry, Phik(t) = N (1-t2)tk, where t = 2x/L -1 and N is the normalization constant. Convergence of ground and excited state energies and wavefunctions with increasing length of the expansion basis is shown. The second example employs an exponential variational wavefunction to describe the harmonic oscillator model, using the nonlinear variation method. The ground state of even parity is a special case, due to the quantum requirement that the derivative of the wavefunction be continuous everywhere; a (fixed) linear combination of two exponential functions Phi(x) = exp(-Y|x|) - exp(-s|x|)/s is used to enforce this. In this variational wavefunction Y is the variational parameter, and s is any positive constant. The numerical por...