Introduction to Spherical Harmonics

Transcription

1 Introduction to Spherical Harmonics Lawrence Liu 3 June 4 Possibly useful information. Legendre polynomials. Rodrigues formula:. Generating function: d n P n x = x n n! dx n n. wx, t = xt t = P n xt n, usually t <. n= 3. Recurrence relations: n P n x n xp n x np n x =. P nx xp nx = n P n x. xp nx P n x = np n x. P nx P n x = n P n x. x P nx = np n x nxp n x. 4. Differential equation: [ x u nn u =. 5. Normalization: 6. Orthonormality: φ n x = n P nx. φ m xφ n x dx = δ mn.. Associated Laguerre polynomials. Rodrigues formula: L α nx = e x x α n! d n dx n e x x nα. When α =, these are the Laguerre polynomials; otherwise, they are called associated or generalized Laguerre polynomials.

2 . Generating function: 3. Recurrence relations: 4. Differential equation: wx, t = t α e xt t = L α nxt n, t <. n= n L α nx x α n L α nx n αl α n x =. x d dx Lα nx = nl α nx n αl α n x. L α n x L α nx = L α n x. d dx Lα nx = L α n x. xu α xu nu =. 5. Orthogonality: e } x {{ x α } L α mxl α nx dx = weight 6. Normalization: Γn α δ mn. n! φ α nx = e x/ x α/ n! Γn α Lα nx. 7. Orthonormality: φ α mxφ α nx dx = δ mn..3 Hypergeometric equation/function. Hypergeometric equation:. Solution: z zu [γ α β zu αβu =. u = c u c u ; α k β k u = F α, β; γ; z = z k, z < ; k!γ k k= u = z γ F γ α, γ β; γ; z, z <, arg z < π. Lemma. If uz is a solution to the hypergeometric differential equation with parameters α, β, and γ, then vz = u z is a solution to the hypergeometric equation with parameters α, β, and γ. Proof. Differentiate the hypergeometric equation: then write v = u : and we conclude z zu zu zu [γ α β zu α β u αβu = ; z zv [γ α β 3zv αβ α β v = ; }{{} αβ v = F α, β ; γ ; z.

3 .4 Gamma and beta functions.. 3. Γz = zγz. ΓzΓ z = π sin πz. z ΓzΓ z = π Γz Bx, y = t x t y dt. Bx, y = ΓxΓy Γx y..5 Fundamental constants. Planck s constant:. Electron mass: 3. Fundamental charge: 4. Coulomb constant: 5. Bohr radius: = J s. µ e = kg. e =.68 9 C. k = 4πɛ = N m C. a = 4πɛ µ e e = m. Legendre functions and associated Legendre functions. Legendre functions By taking..4 to be our definition for the Legendre polynomials, we can extend the definition and let n ν C, x z C. The equation can then be written [ z u νν u =, z u zu νν u =, which is called Legendre s equation. Make the substitution t = z. Then [ t 4 dt = dz = du dt = du dz ; d u dt t 3 du νν u =, dt

4 This has solution t t d u tdu νν u =. dt dt F ν, ν ; ; z P ν z; z <. Notice that when ν =,,,..., the function reduces to a polynomial; these turn out to be precisely the Legendre polynomials and will be shown later. We can also substitute t = z, u = z ν v. Then du dz = z ν dv dz νz ν v, d u dz = z ν d v dv νz ν dz dz ν νz 3 ν v, d v dz = t 3 d dt dt = z 3 dz, dv dz = t 3 t 3 dv dt dv dt, = 6t dv dt 4t3 d v dt. The equation becomes z u zu νν u =, z [ z ν d [ v dz dv νz ν dz ν νz 3 ν v z z ν dv dz νz ν v νν z ν v =, 5ν t t d v t dt The solution is so that t 3ν 3 dv 4t dt t t d v dt 5ν ν dv dt 3γ t ν νv [ t t { } 4tt d v [ 3 ν 5 νtdv ν ν v =, dt dt [ ν 3 ν 5 dv ν ν t dt v =. ν vt = F, ν ; ν 3 ; t, 4ν uz = ν z ν F, ν ; ν 3 ; z. dv dt ν ν t νv t ν νv =, We normalize this second solution: π Γν ν Q ν z Γ ν 3 F z ν, ν ; ν 3 ; z, z >, arg z < π, ν,,... so that the analytic continuation of Q ν has Q ν = π 3 tan νπ νγ ν Γ ν. The Legendre functions satisfy the same recurrences as the Legendre polynomials; this is proven later. 4

5 . Associated Legendre functions Consider the equation z u zu [νν m z u =, m N. Substitute u = z m v. Then u = z m v mz z m v, u = z m v mz z m v mz m z m v m z m v. The differential equation becomes z m v mz z m v m z z m v m z m v z z m v νν m z z m v =, z v mzv zv m z z v mv [ νν m z v =, z v m zv [mm νν v =, Substitute z = t, dv dz = dv dt. z v m zv m νm ν v =. tt d v t m dv m νm ν v =. dt dt This is the hypergeometric differential equation with parameters α = m ν, β = m ν, and γ = m expand the second term and reverse the sign of each term. Applying Lemma m times, we can write the solution vt as w m t, where Changing back into z, z z tt d w dt t dw νν w =. dt 4w z w νν w =, z w zw νν w =. This is Legendre s equation, so the solutions to the original equation, by going back through the substitutions, are d m u = z m dz m P νz Pν m z, u = z m d m dz m Q νz Q m ν z, the associated Legendre functions. Occasionally, these are called associated Legendre polynomials, but in general, they are not, since m odd gives the square root factor. Usually we are interested in real < x <, and these can be written: Pν m x = Pν m x ± i = e ± mπi x m d m dx m P νx = m x m d m dx m P νx; Q m ν x = m x m d m dx m Q νx, ν,,... 5

6 Notice also that for ν N, m > ν, P m ν z. The differential equation is invariant under a change of sign in m, so we might as well define, for m Z, P m ν z m d m dz P νz, m Q m ν z m d m dz Q νz. m Sometimes, a different convention is used for negative m, which should be kept in mind when encountering this expression. Furthermore, Pν m may be generalized to P ν µ, where µ C, but we will not be needing this, and integer m is the most commonly encountered situation. Also, notice that P m ν z = P m νz. This is easily proven since the hypergeometric function does not depend on the order of the first two arguments. 3 Problem: Laplace s equation We want to find those functions which are harmonic in spherical coordinates. We begin with Laplace s equation, written in spherical coordinates: u = r r u r r r sin θ u u sin θ θ θ r sin θ φ =. Separate the radial and angular components of the equation, by guessing that the solution looks like ur, θ, φ = RrY θ, φ. [N.B.: here we use the physics convention to spherical coordinates; that is, θ π, φ π. This is fairly standard for the spherical harmonics. u = r r = Y r r = R r r RY r r R r r R r R r sin θ θ r sin θ θ Y sin θ θ sin θ RY θ sin θ Y θ sin θ Y θ R Y r sin θ φ RY r sin θ φ Y Y sin θ φ =. Now that we have separated the radial and angular components, each must be equal to a constant, which, for convenience, we write as ll : d r dr = ll, R dr dr sin θ Y Y Y sin θ θ θ Y sin = ll. θ φ So far there is no loss of generality, since l could be any complex number at this point. Take the angular equation, and multiply by Y sin θ: sin θ θ sin θ Y Y θ φ = ll Y sin θ. [You may have seen this equation if you ve studied physics; it shows up in electrodynamics. Again, separate variables: Y θ, φ = ΘθΦφ. Φ sin θ d sin θ dθ ΘΦll sin θ Θ d Φ dθ dθ dφ =, 6

7 Θ sin θ d sin θ dθ ll sin θ d Φ dθ dθ Φ dφ =. The components are separated, so each must be a constant, which, without loss of generality, we write as m. Θ sin θ d sin θ dθ ll sin θ = m, dθ dθ The Φ equation is easy: d Φ Φ dφ = m. Φ = e imφ. [There are actually two solutions corresponding to ±m, but this is fine if we allow m to be negative or positive. There is also a possible constant in front, but we ll absorb that into Θ. Since Φφ = Φφ π, we must have that m is an integer. Now look at the Θ equation: sin θ d sin θ dθ [ ll sin θ m Θ =. dθ dθ Make a change of variables: Then: x [ x d dx x = cos θ, dx = sin θ dθ, df dx = df sin θ dθ, x df dx = df dθ, = d dθ x d dx. x x dθ dx x [ d x dθ dx dx [ d x dθ dx dx This is an equation we ve seen; the solution is Θ = AP m l [ ll x m Θ =, [ ll x m Θ =, [ll m x Θ =. x = AP m cos θ. The second solution, not of interest here because it is unbounded, is The spherical harmonics are then We can also normalize this so that π π Θ = BQ m l cos θ. Yl m θ, φ = APl m cos θe imφ. Y m l θ, φy m l θ, φ sin θ dθ dφ = δ ll δ mm. l 7

8 This determines A, which gives the normalized spherical harmonics: Y m l θ, φ = l l m! 4π l m! P l m cos θe imφ. Occasionally the normalized functions are defined to be the spherical harmonics; this usually happens in physics texts. Also, in other disciplines, several different normalization factors may appear. Now look at the radial equation: d r dr = ll R, dr dr r d R dr dr r ll R =. dr This is a Cauchy-Euler equation, and is easy to solve: R = Ar l Br l. If we want the solution to be bounded at the origin, then we pick B =. The set of solutions to Laplace s equation is then u l,m = A l r l B l r l Yl m θ, φ. The general solution is ur, θ, φ = l l= m= l 4 Application: hydrogen atoms 4. Brief introduction to quantum mechanics Al r l B l r l Yl m θ, φ. The central idea of quantum mechanics is wave-particle duality. The wave function of a system, Ψx, t, contains all the information about the system, and is in general complex. Born s interpretation of the wave function is that Ψ is a probability density; therefore, the wave function must be normalized: Ψ dx =. This is for a one-dimensional wave function. Schrödinger s equation describes the evolution of the wave function: i Ψ t = Ψ µ x V Ψ, where V, usually depending only on x, is a potential to be specified, µ is a mass, and is Planck s constant. By separation of variables, we obtain Ψ = ψxφt, where φt = e iet and ψ satisfies the time-independent Schrödinger equation: ψ µ x V ψ = Eψ. 8

9 4. Hydrogen wave functions It is easy to generalize the Schrödinger equation to R 3 : i Ψ t = µ Ψ V Ψ. The probability of finding a particle with wave function Ψr, t in the volume d 3 r = dx dy dz is Ψr, t d 3 r, so the normalization condition is Ψ d 3 r =. Again, by separation of variables, we can obtain a complete set of stationary states: and the ψ n satisfy Ψ n r, t = ψ n re ient, µ ψ V ψ = Eψ. The general solution is just a linear combination of these states. Now, if we rewrite the time-independent Schrödinger equation in spherical coordinates, [ µ r r ψ r r r sin θ ψ ψ sin θ θ θ r sin θ φ V ψ = Eψ. As usual, we begin by separating variables. Here we will split the solutions into a product of radial and angular components: ψr, θ, φ = RrY θ, φ. Substituting into the Schrödinger equation, [ Y d µ r r dr R dr dr r sin θ Divide by RY, multiply by µr / : d r dr µr R dr dr [V r E Y sin θ Y θ θ [ sin θ R Y r sin θ φ V RY = ERY. sin θ Y Y θ θ sin θ φ =. Since we have separated the angular and radial components, each is a constant, which we write ll : d r dr µr [V r E = ll, R dr dr Y [ sin θ Y Y sin θ θ θ sin θ φ = ll. The angular equation once again gives spherical harmonics: Y = Y m l θ, φ, l Z, m Z. [N. B.: In the following, e is used for both the elementary charge and the logarithmic base. The usage is fairly clear from context, and there should be no confusion as long as you are aware of it. For a hydrogen atom, the potential V is the Coulomb potential: e V r = 4πɛ r, 9

10 where e is the elementary charge, k = the differential equation becomes 4πɛ is the Coulomb constant. After changing variables, ur rrr, d u [V µ dr µ d u [ µ dr e 4πɛ r µ d u κ dr = ll r u = Eu, ll r u = Eu. Let κ µe which is real because E is negative for states of interest, and divide by E: [ u. Then let ρ κr, ρ µe πɛ κ : µe ll πɛ κ r κ r d [ u dρ = ρ ll ρ ρ u. Now consider the asymptotics. For large ρ the equation is approximately d u dρ = u. The general solution is u = Ae ρ Be ρ, but e ρ is not normalizable, so B =. For small ρ, the equation is approximately d u ll = dρ ρ u, which has solution Cρ l Dρ l, but ρ l is not normalizable, so D =. Factoring out the asymptotic behavior, we have, without loss of generality, Putting this into the differential equation, uρ = ρ l e ρ vρ. ρ d v l ρdv dρ dρ [ρ l v =. If we attempt a series solution for v, the asymptotic behavior is v c e ρ, which makes u not normalizable. Therefore, the series must terminate; that is, v must be a polynomial. Letting s = ρ, s d v l sdv ds ds ρ l v =. Writing ρ = n, n N it is easily shown that this is the case, we recognize the solution to be associated Laguerre polynomials [see..4: vρ = L l n l ρ. ρ determines E: µe 4 E n = 3π ɛ n. Going back through all the substitutions, we have the solution for ψ = ψ nlm : l [ ψ nlm = C e r r r na L l n l na na Yl m θ, φ.

11 Once C is determined to normalize the probability, we finally have the normalized hydrogen wave functions: 3 l [ n l! ψ nlm = na n[n l! 3 e r r r na L l n l Yl m θ, φ. na na They are all orthonormal: ψ nlm ψ n l m r sin θ dr dθ dφ = δ nn δ ll δ mm. Adding on the time dependence gives a complete set of stationary states; the general solution is a linear combination of these: Ψ nlm r, θ, φ, t = ψ nlm r, θ, φe ient. 5 Application: Helmholtz s equation Consider Helmholtz s equation, u k u =. Once again separate variables: u = RrΘθΦφ. We obtain: sin θ d dθ sin θ dθ dθ d r dr dr dr d Φ dφ = m Φ, [ ll sin θ m Θ =, [ k r νν R =. In the radial equation, substitute R = r v. The result is [ v ν r v k which has solution r v =, v = AJ ν kr BH kr, ν where J ν is Bessel s function of the first kind and H ν is the second Hankel function, H ν = J ν iy ν, Y ν being Bessel s function of the second kind. The particular solutions to Helmholtz s equation are then u = r [ AJ ν kr BH kr ν [CPν m cos θ DQ m ν cos θ e imφ. Usually, the situation of interest is rotationally symmetric and requires a bounded solution; this forces D =, m = : [ u = r AJ ν kr BH kr [CP ν ν cos θ. A solution can then be constructed from a superposition of these. 6 Properties of spherical harmonics Really, properties of Legendre/associated Legendre functions, but since the spherical harmonics are defined this way, the properties transfer somewhat.

12 6. Analytic continuation of P ν z Using π π sin ν θ dθ = and properties of the gamma function, we arrive at and write π π sin k φ dφ = Γ ν Γ ν, Rν >, k, k =,,,..., k! k ν k ν k z P ν z = k! k= = k π ν k ν k z π k= k k! sin k φ dφ = π k ν k ν k z dφ π k= k k! sin φ = π F ν, ν ; π ; z sin φ dφ, where uniform convergence justifies reversing the order of integration and summation. Using the identity F ν, ν, ν ν w w w w, w = w f ν w, w <, and some arguments from complex analysis, the analytic continuation of P ν z is P ν z = π z f ν sin φ dφ, argz < π, π f ν defined as above. Usually in applications we look for bounded solutions, so that Q ν is not as frequently used as P ν. Since the analytic continuation of Q ν is much more tedious, and we won t be using it here, the analytic continuation, without proof, is Q ν z = z ν z cosh ψ z cosh ψ cosh ψ 6. Some integral representations dψ, Rν >, argz < π. To derive these integral representations, the general technique is to assume z > through a substitution, then use analytic continuation to show that the formula is valid in an extended region. To illustrate this, begin with P ν z = π π f ν z sin φ dφ, argz < π, f ν defined as before. Let z = cosh α, α >, and in the integral, substitute sinh θ = sinh α sin φ.

13 The result is P ν cosh α = π α Similarly, by substituting z = cos θ, sin ψ = sin θ sin φ, P ν cos θ = π θ cosh ν θ dθ. cosh α cosh θ cos ν ψ dψ, cos ψ cos θ which gives an integral representation for < z <. By an analogous procedure, we can also derive an integral representation for Q ν z: Q ν cosh α = α e ν θ cosh θ cosh α dθ. Finally we state without proof the following for associated Legendre functions of the first kind, which may be useful in numerical computation of the spherical harmonics:. Pν m Γν m z m π z = m π Γ z ν m z m cos ψ sin m ψ dψ, Re z >, m =,,,... Γν m. 3. P m ν z = Γν m πγν π Pν m m Γν m cos θ = π Γ m Γν m sin θ m z z cos ψ ν cos mψ dψ, Re z >, m =,,,... θ cos ν φ dφ, < θ < π, m =,,,... cos φ cos θ m 6.3 Recurrence relations Using from above, and P ν cosh α = π Q ν cosh α = α α cosh ν θ dθ, cosh α cosh θ e ν θ cosh θ cosh α dθ, we can derive recurrence relations that relate the functions with varying integer degrees. For example: P ν cosh α P ν cosh α = 4 π = 4 π α α cosh ν θ cosh θ dθ cosh α cosh θ cosh α cosh ν θ dθ cosh α cosh θ π = cosh α P ν cosh α = cosh α P ν cosh α = cosh α P ν cosh α 4 ν π 4 ν π ν π α α α α cosh α cosh θ cosh ν dθ cosh α cosh θ d sinh ν θ sinh ν θ sinh θ dθ cosh α cosh θ cosh ν 3 θ cosh ν θ dθ cosh α cosh θ = cosh α P ν cosh α ν [P νcosh α P ν cosh α, 3

14 and going back to cosh α = z, we have ν P ν z ν zp ν z νp ν z =, analogous to..3. The other recurrence relations can be derived similarly. If we start from Q ν, it turns out that the exact same recurrence relations are obtained. Finally, we can prove that P ν reduces to the Legendre polynomials defined by the Rodrigues formula.. for ν N. Use ν = and ν = in the hypergeometric series definition of P ν to find that P z = and P z = z. Then since the same recurrence relations hold, the functions constructed by putting these into the recurrence must generate the exact same polynomials. The Legendre functions of the second kind are not polynomials, but using Q z = log z z, Q z = z log z z, which can be calculated from previously given formulas, the recurrence relations, and induction, it can be shown that Q n z = P nz log z z f n z, n =,,,... where f n is a polynomial of degree n, and f z. Without proof, we also give the following:. ν m P m ν ν zp m ν ν mp m ν =, m =,,,.... z dp m ν dz = νzp m ν ν mp m ν, m =,,,... These also hold exactly in the same way for Q m ν. 6.4 Relations to other special functions A value of interest is P ν. We begin computing this by using the hypergeometric series: P ν = F ν, ν ; ; = = Γ νγν sin νπ = π k= k= k= ν k ν k k k! Γk νγk ν k k! Γk νγk ν k k!, k 4

15 where the last step is the reflection formula. Replace one of the k! by Γk, and we have sin νπ P ν = π sin νπ = π sin νπ = π sin νπ = π k= k= Γk ν Γk νγν k k!γν } Γk {{ } Bk ν, ν Γk ν k k!γν t ν t ν k= t k ν t ν dt Γk ν k!γν t ν t ν t ν dt, k t dt where absolute convergence justifies switching the order of integration and summation. Another change of variables, t = s, gives P ν = ν sin νπ s ν s ν ds = ν sin νπ Γ νγ ν π π Γ }{{} ν, B ν, ν and finally, we have π P ν = Γ ν Γ ν. It also turns out that Legendre functions of degree ν = n, n N, can be written as elliptic integrals. First, for ν =, reduce, using a formula given previously, P cosh α = π cosh α K tanh α, where K is the elliptic integral of the first kind: Kk = π dφ k sin φ. Then, the recurrence relations can be used to obtain the other functions. Similarly, using Q cosh α = e α K e α, and the recurrence relations, we have the Legendre functions of the second kind, for half-integer degrees. 6.5 Other facts. Y m l θ, φ are eigenfunctions of the differential operator r = sin θ θ r Y m l sin θ θ sin θ = ll Y m l. φ ;. x m P n x dx = n Γm Γm n Γm n Γm n. 5

16 7 Pictures The pictures following the bibliography are from [. References [ M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. NIST, Maryland, 964. [ S. Brandt, H. D. Dahmen, The Picture Book of Quantum Mechanics. Springer, New York, 4th edition,. [3 D. J. Griffiths, An Introduction to Quantum Mechanics. Prentice Hall, New Jersey, nd edition, 5. [4 G. Kristensson, Second Order Differential Equations. Springer, New York,. [5 N. N. Lebedev, Special Functions & Their Applications. Dover Publications, USA, 97. 6

17 .3 Angular Momentum, Spherical Harmonics 93 [ Fig..3. The first ten Legendre polynomials P l u = l l! du u l. l d l

18 94. Wave Packet in Three Dimensions Fig..4. Graphs of the associated Legendre functions Pl m u, top, and of the absolute squares of the spherical harmonics Y lm ϑ,φ, bottom. Except for a normalization factor, the absolute squares of the spherical harmonics are the squares of the associated Legendre functions.

19 96. Wave Packet in Three Dimensions Fig..5. The spherical harmonics Y lm are complex functions of the polar angle ϑ, with ϑ π, and the azimuth φ,with φ<π. They can be visualized by showing their real and imaginary parts and their absolute square over the ϑ,φ plane. Such graphs are shown here for l = 3 and m =,,, 3. sphere. For all possible values l and m the functions Y lm are rotationally symmetric around the z axis. They can vanish for certain values of ϑ. These are called ϑ nodes if they occur for values of ϑ other than zero or π. It should be noted that Y ll does not have nodes, whereas Y lm possesses l m nodes.

20 .3 Angular Momentum, Spherical Harmonics 97 Fig..5. continued The Legendre polynomials possess the following orthonormality properties: P l up l udu = l δ ll. Here δ ll is the Kronecker symbol {, l = l δ ll =, l l.

21 98. Wave Packet in Three Dimensions Fig..6. Polar diagrams of the absolute squares of the spherical harmonics. The distance from the origin of the coordinate system to a point on the surface seen under the angles ϑ and φ is equal to Y lm ϑ,φ. Different scales are used for the individual parts of the figure.