1. 講者： 許永昌 老師 1

2. Contents Green’s function Symmetry of Green’s Function Form of Green’s Functions Expansions: Spherical Polar Coordinate Legendre Polynomial Addition Theorem Circular Cylindrical Coordinate 2

3. Green’s Function ( 請預讀 P592~P594) L : linear Operator Green’s Function obeys: [z  L ]  G(r,r’)=  (r  r’) where z is a constant. Or [zI  L]G=I. G is the inverse Operator of [z  L]. Therefore, if [z  L (r 1 )]  y(r 1 )=f (r 1 ), we get one of the solution: Q: If  (r 1 ) obeys [z  L (r 1 )]   (r 1 )=0, is also be a solution? 3

4. Symmetry of Green’s function ( 請 預讀 P595~P596) 4

5. Form of Green’s Functions ( 請預讀 P596~P598) 1. It may not be possible to set  G/  n=0 over the entire surface. 1. Reason: 2. Green’s functions: 5

6. Form of Green’s Functions (continue) When r 1 ~r 2, the term  k 2 does not affect the behavior of G. The Gs show in P5 can be calculated by 1. Solving the differential equation. 2. Or 6

7. Spherical Polar Coordinate Expansion ( 請預讀 P598~P600) 1. We can define the Legendre polynomial obeys 2. Let’s find the expansion of G: 1. If and substitute it into  2 G=  (r 1 -r 2 ), we get (next page) 7

8. Spherical Polar Coordinate Expansion (continue) By Frobenius’ Method for r  r’ case,i.e.  (r-r’)=0, we get Therefore, we get We get Legendre Polynomial Addition Theorem: 8

9. Circular Cylindrical Coordinate Expansion ( 請預讀 P601~P603) We get 9

10. Circular Cylindrical Coordinate Expansion (continue) is a modified Bessel operator. See Sec. 11.5. Its eigenstates are I m (k  ) & K m (k  ), respectively. Owing to I m (0) is finite and K m (  )  0., C=-1 (Ex. 11.5.10) Perhaps you can get the value C from 10

11. Example: 9.7.1 ( 請預讀 P602~P603) Quantum Mechanical Scattering --- Neumann Series Solution. Hamiltonian: We get [E  H 0 ]|  >=V|  >, 11

12. Example 9.7.2 ( 請預讀 P603~P606) Quantum Mechanical Scattering  Green’s function Eqs. 9.204 ~9.213 : The author started from the Fourier Expansion of  r  G 0 V    and tried to find out  k  G 0 V    from the approximated Schrödinger Eq. Eqs. 9.214~9.218: Try to translate the representation of Green’s function from k-space to r-space. In this calculation the integrand has two poles and 12

13. Homework 9.7.2 (16.3.2e) 9.7.6 (16.7.6e) 9.7.16 (16.3.16e) 9.7.20 (16.7.20e) 13

14. Nouns 14