講者： 許永昌 老師 1

Contents 2

Why do we need the Dirac Delta function? (Example) For a charge q, it builds an electric field We get  E=0 except at r=0. It means that for a single charge, its charge density is because of the equation  E=  /  0. How about the information of q? 3

Why do we need the Dirac Delta function? We still need a third relation:  (r)d  =q. Therefore, we can introduce a function obeys Therefore, for a single charge q, its charge density can be written as  =q  (r) and  E=  /  0. In general, Dirac delta function is defined in 1D 4

Kronecker delta Kronecker delta: Dirac Delta function: Usage: ê i  ê j =  ij. =  (r i -r j ) (u(r)=, 補充而已，量力會講 ) 5

The approximations 6

The functions of approximations 7 Code:plot_dirac_delta_sequences.mplot_dirac_delta_sequences.m Therefore,  (x) is an even function.

Change of the variable Q:  (g(x))=? A: We need to think from If g(x i )=0, we get Therefore, 8 The interval is changed, so that we need to add an absolute to dg/dx.

Change of the variable 9

The derivative of Dirac Delta function 10

Dirac Delta function in 3D In Cartesian Coordinate: In Spherical Coordinate: 11

Introductions of Green’s function Reference: Green’s functions in Quantum Physics, E.N. Economou, 3 rd edition. [z  L (r)]G(r,r’;z)=  (r-r’). L (r) : a Linear operator. E.g.  2. z: a constant. E.g. z=0. One of its usage: If [z  L (r)]u(r)=f(r), and there is no  (r) obeys [z  L (r)]  (r)=0, we get u(r)=  G(r,r’;z)f(r’)dr’. If u(r) describes physically the response of a system to a source f(r), then G(r,r’;z) describes the response of the same system to a unit point source located at r’. 12

Examples P91 Evaluate 13

Summary 14

Homework e (1.15.3) e (1.15.8) e ( ) e ( ) e ( ) Please Write a summary of Chapter 1, Vector Analysis. 15

Nouns 16