化學數學（一）

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

2. Review: General Dot, cross, orthonormal basis, gradient, convergence, curl (rot) Order-lowering, row and column operations, Triangular form, linear inhomogeneous equations • Vector • Determinant • Matrix • 1st Order LODE • 2nd Order LODE • PDE • Orthogonal Expansion 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

3. azk ayj axi c x3 b x2 a x1 Base Vectors Orthogonal basis: Nonorthogonal basis:

4. Vector Spaces … Orthonormal basis General orthogonal basis:

5. Vector Algebra

6. Vector Calculus

7. S b a L S V Major Theorems in Integration

8. … The Solutions of Linear Equations Cramer’s Rule:

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

10. Reduction to Triangular Form

11. Square Matrix: Transpose and Inverse symmetric matrix If

12. Orthogonal Matrices: General Cases Again, Kronecker symbol.

13. Orthogonal Matrices: General Cases

14. Matrix Multiplication

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

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

17. The Determinant and Trace of a (Square ) Matrix Product

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

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

20. Application: Quadratic Forms

21. General Quadratic Forms

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

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

24. First-Order Linear Equations:The inhomogeneous Case

25. Three Cases

26. The determination of the coefficient(s) in yp is obtained by substituting it back to the inhomogeneous equation. However, if yp is already in yhthen 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)=x2 . If r(x) is the sum of terms given in above table, the total yp(x) is the sum of respectiveypof all terms. [This leads to a method of series expansion for general r(x) ]

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

28. The Legendre Equation

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

30. The Hermite Equation

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

32. 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 functionx2 in the interval [0,∞]:

33. Hydrogen-Like Atoms Laguerre polynomails Normalization factor

34. Separation of Variables: Turn PDE into ODE

35. Separation of Variables: Turn PDE into ODE A 2D problem reduced to two 1D problems!

36. Orthonormal Expansion

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

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

39. Fourier Transform Pairs if exists.

40. a/π y 0 Example FT