Computer Science & Engineering Quantum ChemistryList Of Experiments

Eigen Systems: Geometrical Interpretation

Numerical computation in Quantum Mechanics (and quantum chemistry, in particular) uses the techniques and language of Linear Algebra, by usage of vectors, matrices and various operations on them.

General Motivation: The wave function of a state of the system is typically represented by a vector ( for the coefficients) on a basis. The Operators are naturally represented as matrices over this basis. Since all properties are expectation values of the a particular operators, we see that we are required to multiply a vector with a matrix. Thus understanding matrix operations (on a vector, in particular) becomes paramount for understanding the numerical methodology of quantum mechanics.

Application of variational theorem (in the matrix formulation) would require a a solution to a set of linear equations; they are of type \( {\bf A} \vec x = \vec b \), where matrix \( {\bf A }\) and vector \(\vec b\) are known, and a solution for vector \( \vec x \) has to be found. Also, the Schrodinger equation \(\hat H \psi = E \psi \) is also naturally converted to a matrix equation on a basis set \( {\bf H} \vec c = E \vec c \), which is an Eigen System. Understanding the solutions for such Eigen systems are the main goal of this experiment.