# Determining Versatile Wavefunctions to Use in Variational Principle Calculations

## Document Type

Oral Presentation

## Event Website

https://source2022.sched.com/

## Start Date

18-5-2022

## End Date

18-5-2022

## Keywords

Quantum Mechanical System, Hamiltonian, Variational Principle

## Abstract

The energy states of a quantum mechanical system are one of the most important factors governing its physical characteristics and properties such as heat capacity, magnetization, and many others. At low temperature, there is a high probability the system will be in its “ground state” (the state with lowest energy). When working with quantum mechanical systems, it is generally impossible and/or impractical to calculate exact ground state energies. Rather, mathematical techniques, namely the variational principle, in which the expectation value of energy is calculated, are used to estimate the ground state energy. For a given Hamiltonian, representing the quantum mechanical system’s total energy, the expectation value is calculated using a normalized test wavefunction containing the expected characteristics of the system’s ground-state wavefunction. In this research project, three different test wavefunctions (Gaussian, Lorentzian, and Hyperbolic Secant functions) were used to estimate ground state energies for different quantum systems. Expectation values obtained using the variational principle provide an upper bound on ground state energy, so the wavefunction providing the lowest energy estimate is closest to the true ground state wavefunction. The goal of this project was to determine whether a particular test wavefunction worked well for a variety of quantum systems. The Gaussian test wavefunction generally gave the best results in terms of providing the lowest ground state energy of all test wavefunctions. This project concluded that the Gaussian test wavefunction offered reliable estimates for ground state energies for a variety of quantum systems.

## Recommended Citation

Goe, Zachary, "Determining Versatile Wavefunctions to Use in Variational Principle Calculations" (2022). *Symposium Of University Research and Creative Expression (SOURCE)*. 45.

https://digitalcommons.cwu.edu/source/2022/COTS/45

## Department/Program

Physics

## Additional Mentoring Department

Physics

Determining Versatile Wavefunctions to Use in Variational Principle Calculations

The energy states of a quantum mechanical system are one of the most important factors governing its physical characteristics and properties such as heat capacity, magnetization, and many others. At low temperature, there is a high probability the system will be in its “ground state” (the state with lowest energy). When working with quantum mechanical systems, it is generally impossible and/or impractical to calculate exact ground state energies. Rather, mathematical techniques, namely the variational principle, in which the expectation value of energy is calculated, are used to estimate the ground state energy. For a given Hamiltonian, representing the quantum mechanical system’s total energy, the expectation value is calculated using a normalized test wavefunction containing the expected characteristics of the system’s ground-state wavefunction. In this research project, three different test wavefunctions (Gaussian, Lorentzian, and Hyperbolic Secant functions) were used to estimate ground state energies for different quantum systems. Expectation values obtained using the variational principle provide an upper bound on ground state energy, so the wavefunction providing the lowest energy estimate is closest to the true ground state wavefunction. The goal of this project was to determine whether a particular test wavefunction worked well for a variety of quantum systems. The Gaussian test wavefunction generally gave the best results in terms of providing the lowest ground state energy of all test wavefunctions. This project concluded that the Gaussian test wavefunction offered reliable estimates for ground state energies for a variety of quantum systems.

https://digitalcommons.cwu.edu/source/2022/COTS/45

## Faculty Mentor(s)

Benjamin White