f(k) e p 2 (k) e iax 2 (k a) r 2 e a x a a 2 + k 2 e a2 x 1 2 H(x) ik p (k) 4 r 3 cos Y 2 = 4

Transcription

1 Fouie tansfom pais: f(x) 1 f(k) e p 2 (k) p e iax 2 (k a) 2 e a x a a 2 + k 2 e a2 x 1 2, a > 0 a p k2 /4a2 e 2 1 H(x) ik p (k) The fist few Y m Y 0 0 = Y 0 1 = Y ±1 1 = l : 1 Y2 0 = 4 3 ±1 cos Y 2 = sin e±i Y 2 ±2 = 5 16 (3 cos2 1) 15 sin cos e±i sin2 e ±2i Di eential opeatos in spheical symmety: f() = df d ˆ [ˆf()] = 1 2 d d 2 f() = 1 2 d d 2 f() apple 2 df d 2

2

3 1. (20 points) When we pefomed sepaation of vaiables on Laplace s equation in spheical coodinates, we found the geneal solution (,, )= 1X lx l=0 m= l Alm l + B lm (l+1) Y m l (, ). The Y m l satisfy the following othogonality condition: Z Yl m Yl m0 d = 0 ll 0 mm 0. A specially designed sphee of adius R applies a spatially vaying electic potential at its inne suface, (R,, )= V (, ). Expess (,, )inside the sphee, < R,in tems of Y m l and V. 3

4 2. (20 points) A sti wie obeys the following wave 2 y 2 2 v 2 R y =0. Fo a wie of infinite length, solve the following initial value poblem by Fouie tansfom. Will you tansfom with espect to x, t, oboth? y(x, 0) t=0 =0 You may expess the solution in the Fouie domain. 5

5 3. (20 points) A Hemitian patial di eential opeato D on domain S, with homogeneous bounday conditions, has known eigenfunctions and eigenvalues, with two quantum numbes n and k: D nk() =k n 2 +1 nk(). The eigenfunctions obey the following othogonality elation: Z ml() nk() () d 2 = mn lk. (a) Solve the inhomogeneous poblem S Du() =f(), in tems of a nk, whee f() = X n,k a nk nk(). (b) If you did not know the coe cients a nk, how would you calculate them fom f()? 7

6 4. (20 points) Use the method of images to wite a Geen s function fo the Laplacian opeato, 2,onx>0, x=0 =0;! 0asx!1. Then, use you Geen s function to build a solution to Laplace s eqation with the following x=0 = V (y), whee V is a constant. The following integal may help. 1 Z dx p x2 ± a =ln x + p x 2 ± a The limit on the integal is ticky. It equies that you discad a diveging tem (call it an abitay potential o set) and theeby give up the idea that! 0 as x!1. 9

7 5. (20 points) Recall ou Fouie di action fomula, D(x, y, z) = k iz eik F x 0,y 0 [A(x0,y 0 ) I (x 0,y 0 )]. (2) This assumes Diichlet bounday conditions, D = A I, in an apetue defined by the dimensionless function A. The physical wavenumbe is k = 2 /. The fequencies conjugate to sceen coodinates x 0,y 0 ae poducts of k with angles of popagation into the fa field: = kx, = ky. (3) (a) Find the Fouie tansfom of the top-hat function a (x), which is one fo a/2 < x < a/2, and zeo elsewhee. (b) Use this esult to calculate the di acted wave D fom adiation of wavenumbe k at nomal incidence to a ectangula slit, a/2 < x < a/2 and b/2 <y<b/2. 11

8 6. (20 points) Considethedi eentialequation H 00 n(x) 2xH 0 n(x)+2nh n (x) =0, (4) on the open inteval 1 <x<1. Thefisttwo(nomalized)eigenfunctionsae H 0 (x) = 1 p ; H 1 (x) = x p. (5) The eigenvalues ae n =2n. (a) Put equation (4) in Stum-Liouville fom. 2 What is this equation called? (b) We have specified no bounday conditions, yet it is possible to conclude that the opeato is Hemitian. Why? (c) Conside the inhomogeneous equation, u 00 2xu 0 = F (x). (6) Suppose we happen to have F in tems of the eigenfunctions, F (x) = X n c n H n (x). In tems of the coe (d) Solve equation (6) with F (x) =5x. cients c n and eigenfunctions H n,whatisthesolutionu(x)? 2 As usual, this can be done by multiplying the equation by some f(x). 14

9

10

11

12

13

14