The Mathematics for Chemists (I) (Fall Term, 2004) (Fall Term, 2005) (Fall Term, 2006) Department of Chemistry National Sun Yat-sen University 化學數學（一）

Review: General Vector Determinant Matrix 1st Order LODE 2nd Order LODE PDE Orthogonal Expansion Dot, cross, orthonormal basis, gradient, convergence, curl (rot) Order-lowering, row and column operations, Triangular form, linear inhomogeneous equations Linear homogeneous equations Transpose, cofactor, adjoint, inverse Symmetric,orthogonal,Hermitian,unitary Trace, eigenvalue, eigenvector, diagonalization, Canonical form of quadratic forms Constant coefficient inhomogeneous Variable coefficient inhomogeneous Constant coefficient + special inhomogeneous Variable coefficient + homogeneous: (Associated) Legendre, Hermite,(Associated) Laguerr Separation of variables: PDE  ODE General expansion, Fourier series, FT

Base Vectors axiaxi azkazk ayjayj Orthogonal basis: Nonorthogonal basis: a bc x1 x2x3

Vector Spaces … Orthonormal basis General orthogonal basis:

Vector Algebra

Vector Calculus

Major Theorems in Integration L S V S ba

The Solutions of Linear Equations … Cramer’s Rule:

Properties of Determinants 1. Transpose: 2. Multiplication by a scalar: 3. Zero row or column: 4. Addition rule: 5. Interchange of rows/columns: antisymmetry: 6. Two equal rows/columns: 7. Proportional rows/columns: 8. Additions of rows/columns: 9. Differentiation:

Reduction to Triangular Form

Square Matrix: Transpose and Inverse symmetric matrix If

Orthogonal Matrices: General Cases Again, Kronecker symbol.

Orthogonal Matrices: General Cases

Matrix Multiplication

Other Useful Matrices in Chemistry 1. Hermitian matrices: 2. Unitary matrices: A.For real matrices, Hermitian means symmetric. B.All physical observables are Hermitian matrices. For real matrices, unitary means orthogonal. (Hermitian matrix is the complex extension of symmetric matrix) (Unitary matrix is the complex extension of orthogonal matrix)

Matrix Algebra The associative law: The distributive law: The (non-)commutative law: Commutator:

The Determinant and Trace of a (Square ) Matrix Product

The Matrix Eigenvalue Problem and Secular Equations The condition for the existence of nontrivial solution: (secular determinant)

Matrix Diagonalization Diagonalization of a square matrix is essentially the same as finding the eigenvalues and their respective eigenvectors.

Application: Quadratic Forms

General Quadratic Forms

Solving First Order ODE Separable Equations: First-order linear equations: + initial conditions

Reduction to Separable Form: Homogeneous Equations For n=0: Example:

First-Order Linear Equations: The inhomogeneous Case

Three Cases

The determination of the coefficient(s) in y p is obtained by substituting it back to the inhomogeneous equation. However, if y p is already in y h then the general solution should be: where the choice of c(x): If the characteristic equation of the corresponding homogeneous equation has two (real or complex) roots, then c(x) =x, or else, c(x)=x 2. If r(x) is the sum of terms given in above table, the total y p (x) is the sum of respective y p of all terms. [This leads to a method of series expansion for general r(x) ]

Second-Order ODE: Special Cases of Variable Coefficients It’s hard or impossible to obtain the solution of a general second-order ODE Inhomogeneous, linear, variable coefficients:

The Legendre Equation

The Associated Legendre Functions Under conditions: The particular solutions are associated Legendre functions:

The Hermite Equation

The Laguerre Equation n: real number Laguerre polynomials: Recurrence relation:

Associated Laguerre Functions The associated Laguerre equation It’s solution is associated Laguerre polynomials: they arise in the radial part of the wavefunctions of hydrogen atom in the form of associated Laguerre functions: which satisfy: and are orthogonal with respect to the weight function x 2 in the interval [0,∞]:

Hydrogen-Like Atoms Normalization factor Laguerre polynomails

Separation of Variables: Turn PDE into ODE

A 2D problem reduced to two 1D problems!

Orthonormal Expansion

Fourier Series f(x) l-l 0 x +A -A

Example: Fourier Series f(x) l-l 0 x +A -A

Fourier Transform Pairs if exists.

Example y a/π 0 FT