1. Determinants and Matrices 行列式與矩陣 Chapters 17 & 18 Gialih Lin, Ph. D. Professor

2. 17.1 Concepts Linear algebra :matrix The concept of determinants has its origin in the solution of simultaneous linear equations. 聯立方程式 a 1 x+b 1 y=c 1 a 2 x+b 2 y=c 2 x=(c 1 b 1 -b 1 c 2 )/(a 1 b 2 -b 1 a 2 ) y=(a 1 c 2 -c 1 a 2 )/(a 1 b 2 -b 1 a 2 )

3. determinant The solution of the system can be written in the form x=D 1 /D y=D 2 /D

4. 17.2 Determinants of order 3 a 1 x+b 1 y+c 1 z=d 1 a 2 x+b 2 y+c 2 z=d 2 a 3 x+b 3 y+c 3 z=d 3 x=D 1 /D y=D 2 /D z=D 3 /D Gramer’s rule

5. Simultaneous equations, =1

6. Hermitian transformation Change the length of vector r but not its direction Hr = h  r h is a real number

7. 17.3 The general case Wave functions   = a 11  1 +a 12  2 +a 13  3 + ….+a 1n  n   = a 21  1 +a 22  2 +a 23  3 + ….+a 2n  n.. …..  n  = a n1  1 +a n2  2 +a n3  3 + ….+a nn  n

8. Quantum mechanics Any function, , can be expressed in terms of its components for a set of basis functions,  i, of unit length (column matrices or column vectors, see Chapter 18)  = a 1  1 +a 2  2 +a 3  3 + ….+a n  n  = b 1  1 +b 2  2 +b 3  3 + ….+b n  n

9. 17.4 The solution of linear equations Gramer’s rule D=0 no solution exists because the equations are inconsistent. The equations are said to be linearly dependent, and each equation can be expressed as a linear combination of the others.

10. Secular equations Shrödinger equation H  = E  In the form of matrix H is an nxn matrix for an n order wave function (  ) E is a constant called as eigenvalue (See chapter 18)

11. Hermitian transformation and eigenvalue (see Chapter 19) Change the norm of function , but keep the relative magnititudes of its components along its basis functions the same. (see Section 18.5 linear transformation) H  = h  h is a real number, and y is said to be an eigenfunction of operator H. (see chapter 19 The matrix eigenvalue problem) The operator H is said to be Hermitian when (as in quantum mechanics) the eigenvalue h is real.

12. Secular determinant The eigenvalue, h, of the Hermitian operator, H, are calculated by solving the secular determinant

13. 18 Matrices and linear transformations 18.1 concepts 3x3 (square) matrix Row Column Trace Tr A = a 1 +b 2 +c 3 A=

14. Vectors A matrix containing a single column only is called a column matrix or column vector; a matrix containing one row only is a row matrix or row vector. The elements of a vector are called components.

15. Quantum mechanics Any function, , can be expressed in terms of its components for a set of basis functions,  i, of unit length (column matrices or column vectors, see Chapter 18)  = a 1  1 +a 2  2 +a 3  3 + ….+a n  n  = b 1  1 +b 2  2 +b 3  3 + ….+b n  n

16. Matrix multiplication C=AB The number of columns of A=the number of rows of B. if A is mxn and B is nxp, the procdut C is mxp

17. Hermitian transformation Change the length of vector r but not its direction Hr = h  r h is a real number

18. Multiplication by a unit matrix If A is an mxn matrix and I m and I n are the unit matrices of orders m and n, respectively, then I m A= A =A I n

19. Simultaneous equations, =1

20. 18.5 Linear transformations x’ = Ax Simultaneous transformations Consecutive transformations Inverse transformations

21. Hermitian transformation Change the length of vector r but not its direction Hr = h  r h is a real number