Approximation of the Quantum Anharmonic Oscillator: Investigating the Accuracy of Effective Models for the Double Well Potential

Sammanfattning

The report investigates how the quantum mechanical model of a particle in one dimension underthe influence of the double well potential V (x) = λ(x^2 − x_0^2)^2 can be approximated using aneffective model, where the system is approximated with a system of truncated dimension. Theinfinite dimensional space of energy eigenfunctions to the double well potential was estimatedby two finite dimensional spaces, ones which were spanned by the ground- and first excitedeigenstates of the harmonic oscillators corresponding to the two minima of the potential. Theenergy eigenvalues given by these models were then studied. Additionally, the effects on thepartition function, the internal energy and the heat capacity of the system were checked tofurther study the accuracy of the models. To determine the accuracy of the approximation ahigh order, highly resolved finite element method was used for comparison as stand in for anexact, analytical solution. All investigated quantities were found to be better approximated forlarger λ and lower temperatures, with the effective model of larger dimension achieving a betterapproximation. The result with respect to λ likely results from that, for low energy, the doublewell potential resembles two harmonic oscillator potentials more the higher the energy barrier,i.e. the higher the λ. The effective models being better approximations for lower temperaturescan be explained by the fact that at low temperature, the lowest energies have the strongestimpact on the behaviour of the system.