Solutions of Laplace s equation in 3 Motivation The general form of Laplace s equation is: Ψ = 0 ; it contains the laplacian, an nothing else. This section will examine the form of the solutions of Laplaces equation in cartesian coorinates an in cylinrical an spherical polar coorinates. Of course it is nice to know how to solve Laplace s equation in these coorinate systems, particularly recalling that the choice of coorinate system is generally etermine by the symmetry of the bounary conitions. But there is a much more important reason why these solutions are of interest. Similar parts of ifferent partial ifferential equations separate off in the same way. Thus, for example, let s take the (time inepenent) Schröinger equation with a spherically symmetric potential. Now the angular part of the equation is all in the angular part of the laplacian. So the angular part of this Schröinger equation is equivalent to the angular part of Laplace s equation in spherical polar coorinates. Then the angular part of the solution to the Schröinger equation with a spherically symmetric potential will be exactly the same as the angular part of the solution to Laplace s equation in spherical polar coorinates: the spherical harmonics we shall iscover below. Of course the raial part of the solution will be ifferent because here the potential will have an effect. So although we are here examining solutions to Laplace s equation, the solutions we shall fin will have relevance to other equations which involve the laplacian. Rectangular Cartesian Coorinates In rectangular cartesian coorinates Laplace s equation takes the form + + Ψ Ψ Ψ = 0. x y z The solution by the separation of variables metho is accomplishe in a number of steps. Step 1: Write the fiel variable as a proct of functions of the inepenent variables. Ψ xyz,, = XxYyZz. Step : Substitute the proct solution into the partial ifferential equation. The erivatives are now total erivatives. X1x6 Y y Zz Y 0 y Z z X x Zz XxYy + + =. x y z Step 3: Divie through by the proct expression for the solution. PH130 Mathematical Methos 1
1 X x 1 Y y 1 Zz + + = 0. X x x Y y y Z z z Now the first term is a function of the inepenent variable x only, the secon term a function of the inepenent variable y only an the last term a function only of the inepenent variable z. Since x, y, an z are inepenent of each other, each of the three terms in the equation must be constant an their sum equal to zero. We will set the first term equal to a, the secon term equal to b, an the thir term equal to 1a+ b6. There may well be restrictions on the allowe values of the separation constants from the bounary conitions on the system. We have the three orinary ifferential equations X x ax x = 0 x Y y by 0 y = y Zz + a b Z z 0 1 + 6 =. z These are three Simple Harmonic Oscillator equations an their solutions are ± ax X x = const e Y y Zz = const e ay ± i a+ bz = const e. The solution, corresponing to particular values of a an b is then ax ay i a bz Ψ ab xyz,, const e e e. = ± ± ± + But since the original equation is linear, any linear combination of possible solutions is also a solution. So we may write the general form of the solution as ± ax ± ay ± i a+ bz Ψ1xyz,, 6= Cab e e e. ab The constants C ab remain to be etermine from the bounary conitions that the particular solution must satisfy. In two imensions (let us say there is no epenence on the z coorinate), we may put a+ b=0 an we obtain solutions of the form ax i ay Ψ1xy, 6= const e ± e ±. ± PH130 Mathematical Methos
Cylinrical Polar Coorinates In cylinrical polar coorinates Laplace s equation takes the form Ψ + + + Ψ 1 1 Ψ Ψ = 0. z We procee by the three stanar steps for solution by the separation of variables metho. Step 1: Write the fiel variable as a proct of functions of the inepenent variables. Ψ,,z = R Φ Z z. Step : Substitute the proct solution into the partial ifferential equation. The erivatives are now total erivatives. R16 1 R 1 Zz Zz 0 Φ Zz R Zz R Φ + Φ + + Φ =. z Step 3: Divie through by the proct expression for the solution. 1 R16 1 1 R16 1 1 Φ16 1 Zz + + + = 0. R R Φ Zz z Now the last term is a function of z only, an so it must be a constant. Let s set this constant equal to a. Then we have an orinary ifferential equation for Z: Zz az z 0 =, z a familiar SHO equation. We are left with an equation in an : 1 R 1 1 R 1 1 Φ + + + a = 0. R R Φ This may be separate by multiplying through by, giving 1 R16 1 R16 1 Φ16 + + a + = 0. R R Φ The last term may now be separate. For convenience we shall use the separation constant n ; we shall justify this choice later. Separating off the last term gives the orinary ifferential equation for Φas Φ + n Φ = 0, a straightforwar SHM equation, together with the ODE for the raial function, which we write in stanar form as: PH130 Mathematical Methos 3
R 1 R n + + a R = 0. This is Bessel s equation of orer n. We have previously encountere Bessel s equation of orer zero when we stuie the (circularly symmetric) vibrations of a circular rum. Now we see how the equation arises in the general case. Allowe values for n The solution to the SHM equation for Φis, to within an arbitrary constant Φ= e in. Now there is an important property of angular variables: as the angle increases the path trace out returns again an again to the same point. But the fiel variable, which we are trying to solve for, must have a given value at a given point; it must be single-value. The physical requirement of single-valueness translates into the mathematical requirement that Φ = Φ1 + π6. So if Φ= e in is a solution to the equation, then so is in π Φ= e e in. An for this to be the case, we must have e inπ = 1, which is only true when n is a positive or negative integer. Previously we have foun that restrictions on the allowe values of the separation constants are impose by bounary conitions. In this case restrictions on the allowe values of the separation constant are impose by the requirements of single-valueness of functions of angular variables. This is a secon cause of quantisation. It follows from this iscussion that the raial solutions of cylinrical problems will involve Bessel functions of integer orer n. Spherical Polar Coorinates In spherical polar coorinates Laplace s equation takes the form Ψ Ψ r r 1 1 Ψ sinϑ r sinϑ ϑ ϑ sin ϑ + + = 0. We procee by the three stanar steps for solution by the separation of variables metho. Step 1: Write the fiel variable as a proct of functions of the inepenent variables. Ψ r, ϑ, = R r Θ ϑφ. Step : Substitute the proct solution into the partial ifferential equation. The erivatives are now total erivatives. PH130 Mathematical Methos 4
r r Rr r Rr Θ ϑ RrΘ ϑ Φ Θ ϑ Φ + sinϑ Φ + = sinϑ ϑ ϑ sin ϑ 0 Step 3: Divie through by the proct expression for the solution. 1 1 1 0 Rr r r Rr Θ ϑ + r Φ sinϑ + = Θ ϑ sinϑ ϑ ϑ Φ sin ϑ Now the first term is epenent only on r, thus it must be constant an we choose ll+1 as the separation constant. This choice must be justifie later. This gives us the orinary ifferential equation for the Rr function: 1 Rr r r Rr = ll 1 + 6 r or r r Rr ll 1 + Rr = 0. r This equation has solutions of the form l l R r = Ar + Br 1 +. Because the solutions of this equation are quite simple, particularly since the solutions can be expresse in terms of the elementary functions, this equation has no special name. However for completeness, we shall refer to it as the spherical R equation. Now we have separate off the spherical R equation, the remainer of the Laplace equation is sinϑ Θ ll+ + 1 ϑ Φ sin ϑ sinϑ + = 0, Θ1ϑ6 ϑ ϑ Φ16 where we have tiie things up so that the last term is a function only of. The last term is thus now separable, an similar to the cylinrical case, we shall use m. The justification for this is the same as in the cylinrical case: the single-valueness of the solutions requires that m be a positive or negative integer. The equation for Φis then simply an SHM equation Φ + m Φ 0 =. : When the Φpart is separate off we are left with the equation for Θ ϑ ll 1 + m + 7 sin = 0 ϑ ϑ sin ϑ sin ϑ Θ ϑ Θ. ϑ ϑ This equation is conventionally transforme into one of the stanar equations through change of the inepenent variable ϑ to u by:. PH130 Mathematical Methos 5
u = cosϑ. Now = sinϑ ϑ so that = = sin ϑ. ϑ ϑ Substituting this into the equation gives ll + m + Θ 1 sin ϑ 7 Θ sin ϑ sin ϑ u u = 0 an eliminating ϑ in favour of u: Θ u 3ll 1 + 1 u 7 m8 u + 1 u7 1 u = 0 u 7 Θ where now Θ is regare as a function of u. Conventionally this is written (not quite in stanar form) as u u m 1 u 7 Θ Θ u + ll+ 1 u = 0 u u Θ 1 u. This is the Associate Legenre equation. In the particular case that m = 0 we get the simpler equation u u 1 u 7 Θ Θ u + ll+ 1 u = 0 Θ known as Legenre s equation. You shoul recognise that the m = 0 case is when we are consiering cylinrically symmetric solutions. Summary of special equations We are now in a position to list the various orinary ifferential equations we have iscovere in looking at solutions of Laplace s equation in ifferent coorinate systems. 1 The SHM equation. Zz + nzz 0 = z Here the inepenent variable z can be a linear or an angular variable. In the linear case bounary conitions will usually restrict the allowe values of the separation constant n. In the angular case the requirement that the solution be single-value restricts the allowe values of n to integers. The solutions of the SHM equation are sines an cosines (equivalently complex exponentials). PH130 Mathematical Methos 6
Bessel s equation R 1 R n + + a R 0 = Recall that n is restricte to integer values, following from the SHM equation for Φ The solutions to Bessel s equation are the Bessel functions Jn r an Yn. r These are known to Mathematica as BesselJ[n, r] an BesselY[n, r]. 3 The spherical R equation r r Rr ll 1 + Rr = 0. r This equation has simple solutions: l l R1r6 = Ar + Br 1 +; we on t nee any special functions. Although we have not shown this, the separation constant l is restricte to integer variables by the requirement that solutions to the Θ equation are a single-value function of its angular argument ϑ. 4 Legenre s equation u u 1 u 7 Θ Θ u + ll+ 1 u = 0 Θ. Expresse as functions of u = cosϑ, the solutions of Legenre s equation are polynomials, known as Legenre polynomials, Pu l. Since u = cosϑ, it follows that the variable u ranges between 1< u < 1an this explains the interval for Legenre polynomials. Mathematica knows the Legenre polynomials as LegenreP[l, u]. 5 The associate Legenre equation u u m 1 u 7 Θ Θ u + ll+ 1 u = 0 u u Θ 1 u. When expresse as functions of u = cosϑ, the solutions of the associate Legenre equation are polynomials, known as the associate Legenre polynomials, P m l 1u6. Mathematica knows the associate Legenre polynomials as LegenreP[l, m, u]. Spherical Harmonics The angular part of the solutions of Laplace s equation (an any other equation involving which has spherical symmetry) is containe in the proct of the azimuthal function Φ an the polar function Θ. ϑ The azimuthal function Φwill comprise a complex exponential e im an the polar function will be a solution of the associate. PH130 Mathematical Methos 7
Legenre equation P m l 1cosϑ6. Thus a solution corresponing to given values of the separation constants l an m will be m m Yl 1ϑ, 6 Pl 1cosϑ6e to within an arbitrary factor. Since these parts always go together in this way for spherical problems the Y m l ϑ, given their own name; they are calle spherical harmonics. By convention the assocoate Legenre polynomials an the spherical harmonic have their own (ifferent) normalisation conventions. Mathematica knows the spherical harmonics as SphericalHarmonicY[l, m, ϑ, ] im are PH130 Mathematical Methos 8