Physics 116C Helmholtz s and Laplace s Equations in Spherical Polar Coordinates: Spherical Harmonics and Spherical Bessel Functions

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1 Physics 116C Hemhotz s an Lapace s Equations in Spherica Poar Coorinates: Spherica Harmonics an Spherica Besse Functions Peter Young Date: October 28, 2013) I. HELMHOLTZ S EQUATION As iscusse in cass, when we sove the iffusion equation or wave equation by separating out the time epenence, u r,t) = F r)tt), 1) the part of the soution epening on spatia coorinates, F r), satisfies Hemhotz s equation 2 F k 2 F = 0, 2) where k 2 is a separation constant. In this hanout we wi fin the soution of this equation in spherica poar coorinates. The raia part of the soution of this equation is, unfortunatey, not iscusse in the book, which is the reason for this hanout. Note, if k = 0, Eq. 2) becomes Lapace s equation 2 F = 0. We sha iscuss expicity the soution for this important) case. A. Reminer of the Soution in Circuar Poars Reca that the soution of Hemhotz s equation in circuar poars two imensions) is Fr,θ) = J n kr)a kn cosnθb kn sinnθ) 2 imensions), 3) n=0 where J n kr) is a Besse function, an we have ignore the secon soution of Besse s equation, the Neumann function 1 N n kr), which iverges at the origin. For the specia case of k = 0 Lapace s equation) you showe in the homework that the soution for the raia part is Rr) = C n r n D n r n, 4) 1 Aternative names for the Neumann function are Besse function of the secon kin an Weber function.

2 2 for n = 0 the soution is C 0 D 0 nr). The r n soution in Eq. 4) arises as the imit of the J n kr) soution in Eq. 3) for k 0, whie the r n soution arises as the imit of the Neumann function N n x) soution of Hemhotz s equation not ispaye in Eq. 3) which ony incues the soution reguar at the origin). Since the soution of Hemhotz s equation in circuar poars two imensions) invoves Besse functions, you might expect that some sort of Besse functions wi aso be invove here in spherica poars three imensions). This is correct an in fact we wi see that the soution invoves spherica Besse functions. B. Separation of Variabes in Spherica Poars Now we set about fining the soution of Hemhotz s an Lapace s equation in spherica poars. In this coorinate system, Hemhotz s equation, Eq. 2), is 1 r 2 r 2 F ) 1 r r r 2 sinθ F ) 1 2 F sinθ θ θ r 2 sin 2 θ φ 2 k2 F = 0. 5) To sove Eq. 5), we use the stanar approach of separating the variabes, i.e. we write We then mutipy by r 2 /RΘΦ) which gives r 2R ) k 2 r 2 1 Rr r Θsinθ Fr,θ,φ) = Rr)Θθ)Φφ). 6) sinθ Θ ) θ θ 1 Φsin 2 θ 2 Φ = 0. 7) φ2 C. Anguar Part Mutipying Eq. 7) by sin 2 θ, the ast term, Φ 2 Φ/φ 2 ), ony invoves φ whereas the first two terms ony epen on r an θ), an so must be a constant which we ca m 2, i.e. The soution is ceary 2 Φ Φ φ 2 = m2. 8) Φφ) = e imφ, 9) with m an integer in orer that the soution is the same for φ an φ 2π). Substituting into Eq. 7) gives Rr r 2R ) k 2 r 2 1 r Θsinθ sinθ Θ ) m2 θ θ sin 2 θ = 0. 10)

3 3 The thir an fourth terms in Eq. 10) are ony a function of θ whereas the first two ony epen on r), an must therefore be a constant which, for reasons that wi be cear ater, we write as 1), i.e. sinθ Θ ) m2 Θsinθθ θ sin 2 θ = 1), 11) which can be written as sinθ Θ ) ) 1) m2 sinθθ θ sin 2 Θ = 0. 12) θ With the substitution x = cosθ, Eq. 12) becomes [ 1 x 2 ) Θx) ] ) 1) m2 x x 1 x 2 Θx) = 0. 13) Eq. 13) is the Associate Legenre equation, so the soution is Θx) = P m x) x = cosθ), 14) where the P m cosθ) are Associate Legenre Poynomias, an, as shown in the book, we nee = 0,1,2,, an m runs over integer vaues from to. If is not an integer one can show that the soution of Eq. 12) iverges for cosθ = 1 or 1 θ = 0 or π). Generay we require the soution to be finite in these imits, an this is the reason why we write the separation constant in Eq. 12) as 1) with an integer. For m = 0, P 0 x) P x), the Legenre poynomia. The functions Θ an Φ are often combine into a spherica harmonic, Y m θ,φ), where Y m θ,φ) = const. P m cosθ)e imφ, 15) where const. is a messy normaization constant, esigne to get the right han sie of Eq. 16) beow equa to unity when =,m = m. The spherica harmonics are orthogona an normaize, i.e. 2π 0 π φ 0 θ sinθ Y m θ,φ) Y m, δ m,m. 16) Note that since the spherica harmonics are compex we nee to take the compex conjugate of one of them in this orthogonaity-normaization reation.

4 4 The first few spherica harmonics are 1 Y0 0 θ,φ) = 4π, 3 Y1 1 θ,φ) = 8π sinθeiφ, 3 Y1 1 θ,φ) = 4π cosθ, 3 Y1 1 θ,φ) = 8π sinθe iφ. Spherica harmonics arise in many situations in physics in which there is spherica symmetry. An important exampe is the soution of the Schröinger equation in atomic physics. For the case of m = 0, i.e. no epenence on the azimutha ange φ, we have Φφ) = 1 an aso P m cosθ) = P cosθ), where the P x) are Legenre Poynomias. Hence Y 0 θ,φ) = const. P cosθ). 17) You wi reca from earier casses that the first three Legenre poynomias are P 0 x) = 1, P 1 x) = x, P 2 x) = 1 2 3x2 1). D. Raia Part We now focus on the raia equation, which, from Eqs. 10) an 12), is r 2R ) [ k 2 r 2 1) ] R = 0, 18) r r or equivaenty r 22 R r 2 2rR r [ k 2 r 2 1) ] R = 0. 19) It turns out to be usefu to efine a function Zr) by Substituting this into Eq. 19) we fin that Z satisfies Rr) = Zr). 20) kr) 1/2 r 22 Z r 2 rz r [ k 2 r 2 1/2) 2] Z = 0, 21)

5 5 which is Besse s equation of orer 1/2. The soutions are J 1/2 kr) an N 1/2 kr) which, together with the factor kr) 1/2 in Eq. 20), means that the soutions for Rr) are the spherica Besse an Neumann functions, j kr) an n kr) efine by j x) = π π 2x J 1/2x), n x) = 2x N 1/2x). 22) From now one we wi assume that the soution is finite at the origin, which rues out n kr), an so Rr) = j kr). 23) Hence, the genera soution of Hemhotz s equation, Eq. 2) which is reguar at the origin is Fr,θ,φ) = m= a m j kr)y m θ,φ), 24) where the coefficients a m wou be etermine by bounary conitions. In particuar one typicay ony nees certain iscrete vaues of k which are reate to zeros of the spherica Besse functions. Eq. 24) is the soution of Hemhotz s equation in spherica poars three imensions) an is to be compare with the soution in circuar poars two imensions) in Eq. 3). As iscusse earier in the course, the spherica Besse functions i.e. Besse functions of hafinteger orer, see Eq. 22)) are simper than Besse functions of integer orer, because they are are reate to trigonometric functions. For exampe, one has j 0 x) = sinx x, j 1x) = sinx x 2 cosx x, n 0x) = cosx x, n 1x) = sinx x cosx x 2. 25) Hence Eq. 24) is not quite as formiabe as it may seem. The ony situations consiere in etai in this course wi be those in which there is no epenence on the azimutha ange φ. In this case ony the m = 0 terms contribute. For these, Y 0 θ,φ) = const. P cosθ), see Eq. 17), an so Eq. 24) simpifies to Fr,θ) = a j kr)p cosθ) azimutha symmetry). 26) E. Exampe with azimutha symmetry: a pane wave As a specia case of Eq. 26), consier a pane wave traveing in the z irection the irection of the poar axis). We know that in cartesian coorinates the spatia part of the ampitue of the

6 6 wave is just expikz), which we can aso write as expikrcosθ). Since the ampitue of the wave satisfies Hemhotz s equation an there is no φ epenence), it must aso be given by Eq. 26) for the specifie vaue of k), i.e. e ikrcosθ = a j kr)p cosθ), 27) for some choice of the a. In fact one can show that a = 21)i, 28) which gives e ikrcosθ = i 21)j kr)p cosθ), 29) a resut which is very important in scattering theory in quantum mechanics. I have not been abe to fin a very simpe erivation of Eq. 28). II. LAPLACE S EQUATION Finay we consier the specia case of k = 0, i.e. Lapace s equation 2 F = 0. A. Separation of variabes Separating the variabes as above, the anguar part of the soution is sti a spherica harmonic Y m θ, φ). The ifference between the soution of Hemhotz s equation an Lapace s equation ies in the raia equation, which becomes r 22 R r 2 2rR 1)R = 0. r As for the anaogous case of circuar poars, we can see by inspection that the soution just has a singe power or r, i.e. Rr) r λ for some vaue of λ. To etermine λ we substitute r λ in to the equation, which gives λλ1) 1) = 0. Factoring gives λ )λ1) = 0, so the two soutions of are λ = an λ = 1).

7 7 If we speciaize to the case of azimutha symmetry for simpicity the genera soution is Fr,θ) = A r B ) r 1 P cosθ). 30) The r term correspons to the spherica Besse function j kr) in Eq. 26) in the imit k 0, whiether 1) termcorresponstothesphericaneumannfunctionn kr)notshownineq.26) which ony ispays the soution of Hemhotz s equation reguar at the origin) in the same imit. Eq. 30) escribes, for exampe, the eectrostatic potentia in regions of space where there is no charge. Couomb s aw, Fr) 1/r correspons to the case of = 0 remember that P 0 x) = 1). If the soution is vai in the region where r the A vanish since the potentia shou go to zero far away from any charges, an so Fr,θ) = B r 1P cosθ), 31) which is the mutipoe expansion that we iscusse in the earier part of the course.