3 Mathematics of the Poisson Equation

Transcription

1 3 Mthemtics of the Poisson Eqution 3. Green functions nd the Poisson eqution () The Dirichlet Green function stisfies the Poisson eqution with delt-function chrge 2 G D (r, r o ) = δ 3 (r r o ) (3.) nd vnishes on the boundry. It is the potentil t r due to point chrge (with unit chrge) t r o in the presence of grounded (Φ = 0) boundries The simplest free spce green function is just the point chrge solution G o = (3.2) 4π r r o In two dimensions the Green function is which is the potentil from line of chrge with chrge density λ = G o = 2π log r r o (3.3) (b) With Dirichlet boundry conditions the Lplcin opertor is self-djoint. The dirichlet Green function is symmetric G D (r, r 0 ) = G D (r 0, r). This is known s the Green Reciprocity Theorem, nd ppers in mny clever wys. The intuitive wy to understnd this is tht for grounded boundry b.c. 2 is rel self djoint opertor (i.e. rel symmetric mtrix). Now 2 G D (r, r 0 ) = δ 3 (r r 0 ), so in functionl sense G D (r, r 0 ) is the inverse mtrix of 2. The inverse of rel symmetric mtrix is lso rel nd symmetric. If the Lplce eqution with Dirichlet b.c. is discretized for numericl work, these sttements become explicitly rigorous. (c) The Poisson eqution or the boundry vlue problem of the Lplce eqution cn be solved once the Dirichlet Green function is known. Φ(r) = d 3 x o G D (r, r o )ρ(r o ) ds o n o ro G D (r, r o )Φ(r o ) (3.4) V where n o is the outwrd directed norml. The first term is volume integrl nd is the contribution of the interior chrges on the potentil. The second term is surfce integrl, nd is the contribution of the boundry vlue to the interior. (d) A useful technique to find Green function is imge chrges. You should know the imge chrge green functions i) A plne in D nd 2D (clss) ii) A sphere (homework) iii) A cylinder (homework + recittion) 7 V

2 8 CHAPTER 3. MATHEMATICS OF THE POISSON EQUATION (e) The Green function cn lwys be written in the form G(r, r o ) = G o (r, r o ) +Φ ind (r, r o ) (3.5) 4π r ro where the induced potentil, Φ ind (r, r o ), is regulr nd stisfies the homogeneous eqution 2 Φ ind = 0. The force of point chrge q nd the grounded boundries (i.e. between the chrge q nd the induced chrges on the grounded surfces) is entirely due to the induced potentil F = q 2 r Φ ind (r, r 0 ) (3.6) r=r0 Using the green reciprocity theoem Φ ind (r, r 0 ) = Φ ind (r 0, r), we cn write 2 F = r0 U int (r 0 ) (3.8) where U int (r o ) = 2 q2 Φ ind (r o, r o ) = 2 q2 lim (G(r, r o ) G o (r, r o )) (3.9) r r o (f) Finding the Green function by seprtion of vribles This is best illustrted by exmple. Pick two dimensions of surfce (sy θ, φ). The method is motivted by the fct tht δ 3 (r r o ) cn be written s sum δ 3 (r r o ) = r 2 δ(r r o)δ(cos θ cos θ o )δ(φ φ o ) = r 2 δ(r r o) lm Y lm (θ, φ)y lm(θ o, φ o ) (3.0) Thus the green function is cn lso be written s G(r, r o ) = l=0 l g lm (r, r o )Y lm (θ, φ)y lm(θ o, φ o ) (3.) leding to n eqution for g lm (r, r o ) [ l(l + ) r 2 r r2 + r r 2 g lm (r, r o ) = r 2 δ(r r o) (3.2) This remining eqution in D is then solved for the green function following the strtegy outlined in Sect. 3.2 (see Eq. (3.37)). This depends on the conditions boundry conditions. Similr expressions cn be derived in other coordintes. (g) For free spce, the two solutions to Eq. (3.2) re y out (r) = /r l+ nd y in (r) = r l, p(r) = r 2 nd p(r)w (r) = 2l +. Then the free spce Green fcn cn be written 4π r r o = l=0 l [Y lm (θ, φ)y lm(θ o, φ o ) 2l + r< l r> l+ (3.3) Some useful identities cn be derived from Eq. (3.3): The green function is the potentil for unit chrge q =. The induced chrges re proportionl to q. The electro-sttic field from the induced chrges is E ind (r, r 0 ) = q rφ ind (r, r o) while the force on q is F = qe ind (r 0, r 0 ). 2 We use [ rφ ind (r, r 0 ) = r=r0 2 rφ ind (r, r 0 ) + rφ ind (r 0, r) = r=r 2 r Φ 0 ind(r 0, r 0 ) (3.7) 0

3 3.2. SOLVING THE LAPLACE EQUATION BY SEPARATION 9 i) The generting function of Legendre Polynomils is found by setting r o = ẑ nd r < with Y l0 = (2l + )/4πP l (cos θ) + r2 2r cos θ = r l P l (cos θ) (3.4) ii) The sphericl hrmonic ddition theorem which we find by writing by setting r o = nd r < nd using / r r o = / + r 2 2rˆr ˆr o P l (ˆr ˆr o ) = 4π 2l + m= l where ˆr ˆr o is the cosine of the ngle between the two vectors. iii) The shell structure reltion which you find by setting ˆr = ˆr o = 4π 2l + m= l l=0 Y lm (θ, φ)y lm(θ o φ o ) (3.5) Y lm (θ, φ)y lm(θ, φ) (3.6) This reltion is wht is responsible for shell structure in the periodic tble (h) Similr expnsion exists in other coordintes. e.g. in cylindricl coords y out (ρ) = K m (κρ) nd y in (ρ) = I m (κρ), leding to 4π r r o = 2π m= 3.2 Solving the Lplce Eqution by Seprtion dk [ e im(φ φo) e ik(z zo) I m (kρ < )K m (kρ > ) (3.7) 2π A summry of seprtion of vribles in different coordinte systems is given in Appendix D. The most importnt cse is sphericl nd crtesin coordintes. Solving the Lplce eqution We use technique of seprtion of vribles in different coordinte systems. The technique of seprtion of vribles is best illustrted by exmple. For instnce consider potentil in squre geometry. The b ϕ = 0 on sides z y x ϕ o (x, y) specified on bottom Figure 3.: A rectngle illustrting seprtion of vrs potentil Φ(x, y, z) is specified t z = 0 to be Φ o (x, y) nd zero on the remining boundries () We look for solutions of the seprted form Φ = Z(z) to surf X(x)Y (y) to surf (3.8)

4 0 CHAPTER 3. MATHEMATICS OF THE POISSON EQUATION Substituting this into the lplce eqution, nd seprting vribles gives two equtions for X, Y (the prllel directions) nd one eqution for the perpendiculr eqution [ d2 dx 2 k2 n X(x) =0, (3.9) [ d2 dy 2 k2 m Y (x) =0. (3.20) [ d2 dz 2 + k2 z Z(z) =0, (3.2) where k 2 z = k 2 n + k 2 m. The signs of k x, k y, k z re chosen for lter convenience, becuse it will be impossible to stisfy the BC for k 2 x < 0 or k 2 y < 0. The first step is lwys to seprte vribles nd write down the generl solutions to the seprted equtions X(x) =A cos(k n x) + B sin(k n x) (3.22) Y (y) =A cos(k m y) + B sin(k m y) (3.23) Z(z) =Ae kzz + Be kzz (3.24) (b) It is best to nlyze the prllel equtions first which re ll of the form of Sturm Louiville eigenvlue eqution (see below). These determine the (eigen) functions X(x), Y (y) nd the eigenvlues (or seprtion constnts) k x nd k y. The generl solution for X(x) is X(x) = A cos k x x + B sin k x x, (3.25) nd we re specifying boundry conditions t x = 0 nd x =. In order to stisfy the boundry condition X(0) = X() = 0, we must hve A = 0 nd k = nπ/, leding to X(x) = B sin(k n ) k n = nπ n =, 2,.... (3.26) Similrly Y (y) = B sin(k m ) k m = mπ m =, 2,... (3.27) Thus the prllel directions determine both the functions nd the seprtion constnts. The complete eigen functions re ( nπx ) ( mπy ) ψ nm (x, y) = sin sin b n =... m =... (c) Finlly we return to the perpendiculr direction, Eq. (3.2). This eqution does not usully constrin the seprtion constnts. The generl solution is Z(z) = Ae kzz + Be kzz (3.28) with k z = k 2 n + k 2 m. With Z(z) specified The generl solution then is liner combintion [ Anm e γnmz + B nm e +γnmz ψ nm (x, y) (3.29) n= m=

5 3.2. SOLVING THE LAPLACE EQUATION BY SEPARATION Solving the seprted equtions: After seprting vribles, ll of the equtions we wil study cn be written in Sturm Louiville form: [ d dx p(x) d dx + q(x) y(x) = λr(x)y(x) (3.30) where p(x) nd r(x) re postive definite fcns. Here we record some generl properties of these equtions. () Given two independent solutions to the differentil eqution y (x) nd y 2 (x) The wronskin times p(x) is constnt. p(x) [y (x)y 2(x) y 2 (x)y (x) = const (3.3) wronskin(x) This usully mounts to sttement of Guss Lw. (b) If homogeneous boundry conditions re specified t two endpoints, x = nd x = b, then the problem becomes n eigenvlue eqution. Exmples of the eigenfunctions we need re given in Appendix C. In this cse only certin vlues of λ = λ n re llowed nd the functions re uniquely determined up to normliztion [ d dx p(x) d dx + q(x) ψ n (x) = λ n r(x)ψ n (x) (3.32) The prllel equtions will hve this form (see Eq. (3.9)), nd notice how the boundry conditions t x = 0 nd x = fixed the vlue of k n (see Eq. (3.25) nd Eq. (3.26)). i) The resulting eigenfunctions re complete 3 nd orthogonl with respect to the weight r(x) ψ n, ψ m = b dx r(x)ψ n(x)ψ m (x) = C n δ nm (3.33) where nd b re the endpoints where the boundry conditions re specified. The eigen-functions re usully not normlized. ii) Completeness mens tht ny function f(x) stisfying the boundry conditions, cn be expnded in the set f(x) = f n ψ n (x), (3.34) C n n where the C n re the normliztion constnts of the eigen-functions (Eq. (3.33)), nd f n is the inner product between ψ n (x) nd f(x) nd f(x) f n = ψ n, f = b dx r(x)ψ n(x)f(x) (3.35) iii) One cn esily show by substuting Eq. (3.35) into Eq. (3.34) tht completeness implies (c) Solving the seprted equtions with δ function source terms n ψ n (x)ψ n (x ) C n = r(x) δ(x x ) (3.36) We will lso need to know the green function of the one dimensionl eqution [ d dx p(x) d dx + q(x) g(x, x o ) = δ(x x o ) (3.37) 3 See Morse nd Freshbch

6 2 CHAPTER 3. MATHEMATICS OF THE POISSON EQUATION The Green function for such D equtions is bsed on knowing two homogeneous solutions y out (x) nd y in (x), where y out (x) stisfies the boundry conditions for x > x o, nd y in (x) stisfies the boundry conditions for x < x o. The Green function is continuous but hs discontinuous derivtives. outside nd inside it tkes the form: Since we know the solutions G(x, x o ) =C [y out (x)y in (x o )θ(x x o ) + y in (x)y out (x o )θ(x o x) (3.38) Cy out (x > )y in (x < ) (3.39) where C is constnt determined by integrting the eqution, Eq. (3.37), cross the delt function. In the second line we use the common (but somewht confusing nottion) which mkes the second line men the sme s the first line. x > the greter of x nd x o (3.40) x < the smller of x nd x o (3.4) Integrting from x = x o ɛ to x = x o + ɛ we find the jump condition which enters in mny problems: p(x) dg + p(x) dg dx dx =, (3.42) xo ɛ which cn be used to find C. xo+ɛ (d) In fct the jump condition will lwys involve the Wronskin of the two solutions. Eq. (3.38) into Eq. (3.42) we see tht C = /(p(x o )W (x o )) G(x, x o ) = [y out(x)y in (x o )θ(x x o ) + y in (x)y out (x o )θ(x o x) p(x o )W (x o ) y out(x > )y in (x < ) p(x o )W (x o ) Substituting (3.43) (3.44) where W (x o ) = y out (x o )y in (x o) y in (x o )y out(x o ) is the Wronskin. p(x o )W (x o ) is constnt nd is independent of x o. Note tht the denomintor