#### Author ORCID Identifier

**0000-0001-9336-1355**

#### Standing

Undergraduate

#### Type of Proposal

Oral Research Presentation

#### Faculty

Faculty of Science

#### Faculty Sponsor

Gordon Drake

#### Proposal

Introduction:

The helium atom with two electrons serves as a model for many other three-body problems in atomic physics. Unlike hydrogen, it is the simplest system that cannot be solved exactly in the nonrelativistic limit; but it displays many of the complications found in multi-electron atoms. Thus, it has been researched extensively since the beginnings of quantum theory.

In this project, we compare the accuracy and speed of three methods of calculating eigenvalues of helium: Power Method, Tridiagonalization method, and Jacobi’s method by using David Bailey’s extended double quadruple precision (dq) module. This increases the numerical accuracy from 32 decimal digits in quadruple precision to 70 decimal digits in dq-precision - the machine epsilon is 7 x 10^{-70}.

Purpose:

- Study convergence without loss of accuracy or numerical stability.
- Compare our results with other values in previous calculations in the literature (with Dr. Drake’s and others' previous calculations).
- Compare the computation time for the three methods.

Methods:

1) Jacobi’s method: the slowest, but the most numerically stable (1 to 1.5 min)

The Jacobi method is used to find the complete set of eigenvalues of a symmetric matrix by repeated exact diagonalization of a 2 x 2 matrix formed by two diagonal elements and the matching largest off-diagonal element. This process is iterated until it converges.

2) Tri-diagonalization method (about 20 seconds). Tridiagonalization method reduces a Hermitian matrix to tridiagonal form that is then diagonalized to find the complete set of eigenvalues and eigenvectors.

3) Power method with inverse iteration (2-3 seconds): converges to the single eigenvalue that is closest to an initial guess and corresponding eigenvector.

Findings: In progress

#### Availability

March 29th, and April 1st

Study of Numerical Methods to Solve the Quantum Mechanical Three-Body Problem

Introduction:

The helium atom with two electrons serves as a model for many other three-body problems in atomic physics. Unlike hydrogen, it is the simplest system that cannot be solved exactly in the nonrelativistic limit; but it displays many of the complications found in multi-electron atoms. Thus, it has been researched extensively since the beginnings of quantum theory.

In this project, we compare the accuracy and speed of three methods of calculating eigenvalues of helium: Power Method, Tridiagonalization method, and Jacobi’s method by using David Bailey’s extended double quadruple precision (dq) module. This increases the numerical accuracy from 32 decimal digits in quadruple precision to 70 decimal digits in dq-precision - the machine epsilon is 7 x 10^{-70}.

Purpose:

- Study convergence without loss of accuracy or numerical stability.
- Compare our results with other values in previous calculations in the literature (with Dr. Drake’s and others' previous calculations).
- Compare the computation time for the three methods.

Methods:

1) Jacobi’s method: the slowest, but the most numerically stable (1 to 1.5 min)

The Jacobi method is used to find the complete set of eigenvalues of a symmetric matrix by repeated exact diagonalization of a 2 x 2 matrix formed by two diagonal elements and the matching largest off-diagonal element. This process is iterated until it converges.

2) Tri-diagonalization method (about 20 seconds). Tridiagonalization method reduces a Hermitian matrix to tridiagonal form that is then diagonalized to find the complete set of eigenvalues and eigenvectors.

3) Power method with inverse iteration (2-3 seconds): converges to the single eigenvalue that is closest to an initial guess and corresponding eigenvector.

Findings: In progress