aa r X i v : . [ m a t h - ph ] A ug Angular decomposition of tensor products of a vector
Gregory S. Adkins
Franklin & Marshall College, Lancaster, Pennsylvania 17604 USA ∗ (Dated: October 13, 2018)The tensor product of L copies of a single vector, such as p i · · · p i L , can be analyzed in terms ofangular momentum. When p i · · · p i L is decomposed into a sum of components ( p i · · · p i L ) Lℓ , eachcharacterized by angular momentum ℓ , the components are in general complicated functions of the p i vectors, especially so for large ℓ . We obtain a compact expression for ( p i · · · p i L ) Lℓ explicitlyin terms of the p i valid for all L and ℓ . We use this decomposition to perform three-dimensionalFourier transforms of functions like p n ˆ p i · · · ˆ p i L that are useful in describing particle interactions. I. INTRODUCTION
Three-dimensional Fourier transforms of the general form I n ; i ··· i L ( ~r ) = Z d p (2 π ) e i~p · ~r p n ˆ p i · · · ˆ p i L (1)(where ˆ p = ~p/p ) have a wide variety of uses. For example, I − ( ~r ) = Z d p (2 π ) e i~p · ~r p = 14 πr (2)is the Fourier representation of the Coulomb potential. Two related transforms that occur in the study of fermion-fermion interactions [1–3] are I − i ( ~r ) = Z d p (2 π ) e i~p · ~r p i p = i ˆ x i πr , (3a) I ij ( ~r ) = Z d p (2 π ) e i~p · ~r p i p j p = 13 δ ij δ ( ~r ) − πr (cid:18) ˆ x i ˆ x j − δ ij (cid:19) . (3b)(In our notation the position vector has components ~r = ( x , x , x ) = ( x, y, z ), and ˆ r = ~r/r with componentsˆ x i = x i /r is the associated unit vector.) The structure of three dimensional Fourier transforms such as (1) isorganized by angular momentum. One sees that both the original function 1 /p and the transform 1 / (4 πr ) of (2) arescalars under rotation. The original function p i /p of (3a) is a vector with ℓ = 1 because the components ˆ p i can beexpressed linearly in terms of spherical harmonics Y m (ˆ p ) with ℓ = 1. The transform i ˆ x i / (4 πr ) of (3a) also has ℓ = 1as ˆ x i can be expressed linearly in terms of Y m (ˆ r ). The original function in (3b), ˆ p i ˆ p j , is a combination of ℓ = 0 and ℓ = 2: ˆ p i ˆ p j = (ˆ p i ˆ p j ) + (ˆ p i ˆ p j ) = (cid:18) δ ij (cid:19) + (cid:18) ˆ p i ˆ p j − δ ij (cid:19) (4)where (ˆ p i ˆ p j ) = δ ij is the ℓ = 0 component and (ˆ p i ˆ p j ) = ˆ p i ˆ p j − δ ij is the ℓ = 2 component. (In general we willwrite ( p i · · · p i L ) Lℓ for the component of p i · · · p i L of angular momentum ℓ .) We know that (ˆ p i ˆ p j ) has ℓ = 2 becauseit can be expressed linearly in terms of Y m (ˆ p ): (ˆ p i ˆ p j ) = P m C mij Y m (ˆ p ). It is apparent that the ℓ = 0 and ℓ = 2components of ˆ p i ˆ p j behaves differently under the Fourier transform, acquiring different radial factors. It is generallytrue that in transforms like (1) it is useful to decompose ˆ p i · · · ˆ p i L into components of definite ℓ and deal with eachcomponent separately.The purpose of this work is to show how the decomposition of p i · · · p i L can be done and to give explicit expressionsfor the components of ( p i · · · p i L ) Lℓ having various values of angular momentum ℓ . This is a generalization of (4) toan arbitrary number of vectors L . Our derivations are presented in terms of unit vectors because the relation( p i · · · p i L ) Lℓ = p L (ˆ p i · · · ˆ p i L ) Lℓ (5) ∗ [email protected] allows us to immediately obtain the general case.This work is organized as follows. In II we will review the method for performing three-dimensional Fouriertransforms like (1) making use of angular decomposition. In III we obtain the general expression for the componentof ˆ p i · · · ˆ p i L of angular momentum ℓ . Finally, in IV, we give some examples and applications of our results.More general studies of the relation between Cartesian and spherical components of tensors have been done, [4–8]but the results of those studies are not in a form useful for our purposes here. II. THREE-DIMENSIONAL FOURIER TRANSFORMS USING ANGULAR DECOMPOSITION
A systematic procedure exists for the evaluation of transforms such as (1) based on the decomposition of ˆ p i · · · ˆ p i L into components of definite angular momentum. [9] Our purpose in this section is to review this procedure. We beginby noting that any function of angles, such as ˆ p i · · · ˆ p i L , can be written in terms of spherical harmonics:ˆ p i · · · ˆ p i L = X ℓ = L ℓ X m = − ℓ C ℓmi ··· i L Y mℓ (ˆ p ) (6)for some constants C ℓmi ··· i L . The values of ℓ that enter this sum are ℓ = L , L −
2, etc., down to 1 or 0 dependingon whether L is odd or even. There are no values of ℓ greater than L because ˆ p has ℓ = 1 and the combination of L objects having ℓ = 1 can lead to angular momentum L at the most. Matching the parity ( − L of ˆ p i · · · ˆ p i L tothe parity ( − ℓ of Y mℓ (ˆ p ) gives the requirement that only odd or only even values of ℓ can contribute. We define(ˆ p i · · · ˆ p i L ) Lℓ to be the component of ˆ p i · · · ˆ p i L of angular momentum ℓ (ˆ p i · · · ˆ p i L ) Lℓ ≡ ℓ X m = − ℓ C ℓmi ··· i L Y mℓ (ˆ p ) , (7)so that ˆ p i · · · ˆ p i L = X ℓ = L (ˆ p i · · · ˆ p i L ) Lℓ . (8)It follows that any transform of the form given in (1) can be expressed as a linear combination of transforms like I nℓm ( ~r ) = Z d p (2 π ) e i~p · ~r p n Y mℓ (ˆ p ) . (9)It is convenient to express the exponential in (9) as a Rayleigh expansion: [10–12] e i~p · ~r = ∞ X ℓ =0 i ℓ (2 ℓ + 1) j ℓ ( pr ) P ℓ (ˆ p · ˆ r ) (10)where the j ℓ ( x ) are spherical Bessel functions [13] and the P ℓ ( x ) are Legendre polynomials. We substitute (10) into(9) and use the addition theorem of spherical harmonics [14] P ℓ (ˆ p · ˆ r ) = 4 π ℓ + 1 ℓ X m = − ℓ Y mℓ (ˆ r ) Y m ∗ ℓ (ˆ p ) (11)to factor the angular dependence present in P ℓ (ˆ p · ˆ r ) into parts involving the angles of ˆ p and ˆ r separately. We integrateover the angles of ˆ p using orthogonality Z d Ω p Y m ∗ ℓ (ˆ p ) Y m ′ ℓ ′ (ˆ p ) = δ ℓℓ ′ δ mm ′ , (12)where d Ω p = dθ p sin θ p dφ p is the element of solid angle for ˆ p , to write the transform as I nℓm ( ~r ) = i ℓ π Y mℓ (ˆ r ) Z ∞ dp p n +2 j ℓ ( pr ) . (13)The integral R nℓ ( r ) ≡ Z ∞ dp p n +2 j ℓ ( pr ) (14)converges for − ( ℓ + 3) < n < − n and ℓ are real, as here), and has the value R nℓ ( r ) = χ nℓ /r n +3 where [15] χ nℓ = 2 n +1 √ π Γ (cid:0) ℓ +3+ n (cid:1) Γ (cid:0) ℓ − n (cid:1) . (15)We can extend the useful range of n by generalizing (14) to R nℓ ( r ) = lim λ → + Z ∞ dp e − λp p n +2 j ℓ ( pr ) , (16)which is also given by R nℓ ( r ) = χ nℓ /r n +3 for all n in the larger range − ( ℓ + 3) < n < ℓ . When n = ℓ the integralcontains a delta function : [9] R ℓℓ ( r ) = lim λ → + Z ∞ dp e − λp p ℓ +2 j ℓ ( pr ) = 2 π (2 ℓ + 1)!! r ℓ δ ( ~r ) . (17)We always integrate over the spherical angles before doing the radial integration as part of the definition of thesepossibly singular integrals. (Non-spherical regularization alternatives have been considered by Hnizdo. [16] Moregeneral results for Fourier transforms of the form (9) have been obtained by Samko. [17]) All in all, we see that theinitial transform (1) can be written as I n ; i ··· i L ( ~r ) = Z d p (2 π ) e i~p · ~r p n ˆ p i · · · ˆ p i L = X ℓ = L i ℓ π R nℓ ( r ) (ˆ x i · · · ˆ x i L ) Lℓ , (18)where (ˆ x i · · · ˆ x i L ) Lℓ is defined in terms of Y mℓ (ˆ r ) just as (ˆ p i · · · ˆ p i L ) Lℓ is in terms of Y mℓ (ˆ p ). It follows that if we canarrive at a useful expression for (ˆ p i · · · ˆ p i L ) Lℓ , then we will be able to perform Fourier transforms of the form shownin (1) in a systematic way. III. ANGULAR DECOMPOSITION OF ˆ p i · · · ˆ p i L Our goal in this section is to obtain an explicit and useful expression for the component (ˆ p i · · · ˆ p i L ) Lℓ of angularmomentum ℓ . We will discuss both a constructive method most useful for low values of L and a general result validfor all L . Both approaches make use of the explicit solution for the constants C ℓmi ··· i L in (7): C ℓmi ··· i L = Z d Ω p Y m ∗ ℓ (ˆ p ) ˆ p i · · · ˆ p i L , (19)obtained through use of the orthogonality of the spherical harmonics. From this it is easy to see that the constants C ℓmi ··· i L , and thus the components (ˆ p i · · · ˆ p i L ) Lℓ , are completely symmetric in all indices. The constructive methodalso uses the tracelessness of the maximum angular momentum component (ˆ p i · · · ˆ p i L ) LL , which follows from thetracelessness of C Lmi ··· i L , which is a consequence of the fact that an object composed of L − C Lmi ··· i L δ i L − i L = Z d Ω p Y m ∗ L (ˆ p ) ˆ p i · · · ˆ p i L − = 0 . (20)The most convenient way to obtain (ˆ p i · · · ˆ p i L ) Lℓ for small values of L is by straightforward construction. Weillustrate the constructive approach with a number of examples. The procedure starts with the maximum angular When ℓ is not an odd integer we use (2 ℓ + 1)!! ≡ ℓ +1 Γ( ℓ + 3 / / Γ(1 / momentum component (ˆ p i · · · ˆ p i L ) LL , which is completely symmetric and traceless. This maximum angular momentumcomponent can be written as ˆ p i · · · ˆ p i L plus a linear combination of symmetric terms involving fewer momentum factors(but still with the same parity) added in with unknown coefficients. The condition of tracelessness determines thecoefficients.As a first example of the constructive approach, consider the case L = 3. The maximum angular momentumcomponent is (ˆ p i ˆ p j ˆ p k ) = ˆ p i ˆ p j ˆ p k −
15 (ˆ p i δ jk + ˆ p j δ ki + ˆ p k δ ij ) , (21)where the − / p i ˆ p j ˆ p k ) , is thedifference ˆ p i ˆ p j ˆ p k − (ˆ p i ˆ p j ˆ p k ) : (ˆ p i ˆ p j ˆ p k ) = 15 (ˆ p i δ jk + ˆ p j δ ki + ˆ p k δ ij ) . (22)It is clear that (ˆ p i ˆ p j ˆ p k ) has ℓ = 1 because each of its terms is linear in ˆ p .As a second example of explicit construction, we consider the term with L = 4. The term with maximal angularmomentum is(ˆ p i ˆ p i ˆ p i ˆ p i ) = ˆ p i ˆ p i ˆ p i ˆ p i −
17 (ˆ p i ˆ p i δ i i + perms) + 135 ( δ i i δ i i + perms) , (23)where the coefficients − / /
35 were obtained by applying the tracelessness condition. We are only writingone representative permutation of indices–the others are represented by “+ perms” and an indication of how manyindependent permutations in all there are. We identify the ℓ = 2 component (ˆ p i ˆ p i ˆ p i ˆ p i ) by isolating the term in thedifference ˆ p i ˆ p i ˆ p i ˆ p i − (ˆ p i ˆ p i ˆ p i ˆ p i ) that is quadratic in ˆ p and subtracting the appropriate momentum-independentterms so that each part of (ˆ p i ˆ p i ˆ p i ˆ p i ) has ℓ = 2:(ˆ p i ˆ p i ˆ p i ˆ p i ) = 17 (cid:16) (ˆ p i ˆ p i ) δ i i + perms (cid:17) . (24)The ℓ = 0 component is the remainder:(ˆ p i ˆ p i ˆ p i ˆ p i ) = 115 ( δ i i δ i i + perms) . (25)We have also constructed the L = 5 decomposition and just give the results:(ˆ p i ˆ p i ˆ p i ˆ p i ˆ p i ) = ˆ p i ˆ p i ˆ p i ˆ p i ˆ p i −
19 (ˆ p i ˆ p i ˆ p i δ i i + perms)
10 terms + 163 (ˆ p i δ i i δ i i + perms)
15 terms , (26a)(ˆ p i ˆ p i ˆ p i ˆ p i ˆ p i ) = 19 (cid:0) (ˆ p i ˆ p i ˆ p i ) δ i i + perms (cid:1)
10 terms , (26b)(ˆ p i ˆ p i ˆ p i ˆ p i ˆ p i ) = 135 (cid:0) ˆ p i δ i i δ i i + perms (cid:1)
15 terms . (26c)The basis of our general construction of (ˆ p i · · · ˆ p i L ) Lℓ is an inductive argument using a recursion relation givinga component with angular momentum ℓ in terms of components with lower values of ℓ . We will propose a generalexpression for (ˆ p i · · · ˆ p i L ) Lℓ and show that it satisfies both the recursion relation and the appropriate initial values.We can obtain a useful expression for (ˆ p i · · · ˆ p i L ) Lℓ by using (19) in (7) along with the addition theorem for sphericalharmonics: (ˆ p i · · · ˆ p i L ) Lℓ = Z d Ω p ′ ˆ p ′ i · · · ˆ p ′ i L ℓ X m = − ℓ Y m ∗ ℓ (ˆ p ′ ) Y mℓ (ˆ p )= (2 ℓ + 1) Z d Ω p ′ π ˆ p ′ i · · · ˆ p ′ i L P ℓ (ˆ p ′ · ˆ p ) . (27)It can be seen from this expression both that (ˆ p i · · · ˆ p i L ) Lℓ = 0 for ℓ > L , and that (ˆ p i · · · ˆ p i L ) Lℓ = 0 unless L − ℓ iseven (by use of a parity argument). For any particular value of ℓ we could write out P ℓ (ˆ p ′ · ˆ p ) as a polynomial of order ℓ and perform the angular integral using [18] Z d Ω4 π ˆ x i · · · ˆ x i N = δ N, even ( N + 1)!! (cid:0) δ i i · · · δ i N − i N + perms (cid:1) ( N − . (28)It is easy to perform the integral in (27) for ℓ = 0 and ℓ = 1 where P (ˆ p ′ · ˆ p ) = 1 and P (ˆ p ′ · ˆ p ) = ˆ p ′ · ˆ p = ˆ p ′ j ˆ p j (withan implied sum over j from 1 to 3). One finds that(ˆ p i · · · ˆ p i L ) L = Z d Ω p ′ π ˆ p ′ i · · · ˆ p ′ i L = δ L, even ( L + 1)!! (cid:0) δ i i · · · δ i L − i L + perms (cid:1) ( L − , (29a)(ˆ p i · · · ˆ p i L ) L = 3 Z d Ω p ′ π ˆ p ′ i · · · ˆ p ′ i L ˆ p ′ j ˆ p j = 3 (ˆ p i · · · ˆ p i L ˆ p j ) L +10 ˆ p j = 3 δ L, odd ( L + 2)!! (cid:16) ˆ p i (cid:0) δ i i · · · δ i L − i L + perms (cid:1) ( L − + perms (cid:17) L terms . (29b)These results will serve as initial values for the inductive argument. The recursion relation for (ˆ p i · · · ˆ p i L ) Lℓ is(ˆ p i · · · ˆ p i L ) Lℓ = 2 ℓ + 1 ℓ (cid:26) (ˆ p i · · · ˆ p i L ˆ p j ) L +1 ℓ − ˆ p j − ℓ − ℓ − p i · · · ˆ p i L ) Lℓ − (cid:27) , (30)which follows from (27) and the recursion relation for Legendre polynomials ℓP ℓ ( x ) = (2 ℓ − xP ℓ − ( x ) − ( ℓ − P ℓ − ( x ) . (31)In order to write down a general expression for (ˆ p i · · · ˆ p i L ) Lℓ and perform the inductive proof of its correctness itwill be useful to introduce a little notation. First, we will use the usual summation symbol to represent a ‘sum overpermutations’ instead of the ‘+ perms’ notation used up until now. Specifically, we will write X ( L − (cid:16) δ i i · · · δ i L − i L (cid:17) (32)for the sum over the ( L − L − L ! permutations in a sum over permutations of A i · · · A Li L , but only one in a sum over permutationsof S i ··· i L if S i ··· i L is completely symmetric. Also, we define the new symbol X L,ℓi ··· i L ≡ X ( Lℓ ) (cid:16) ˆ p i · · · ˆ p i ℓ X ( L − ℓ − (cid:16) δ i ℓ +1 i ℓ +2 · · · δ i L − i L (cid:17)(cid:17) (33)to represent the completely symmetric object with L indices formed of ℓ momentum unit vectors and ( L − ℓ ) / (cid:0) Lℓ (cid:1) notation represents the combinatoric factor L ! ℓ !( L − ℓ )! for the number of ways to pick ℓ indices out of a collection of L indices. We can represent X L,ℓi ··· i L slightly more compactly as X L,ℓi ··· i L = X ( Lℓ ) ( L − ℓ − (cid:16) ˆ p i · · · ˆ p i ℓ δ i ℓ +1 i ℓ +2 · · · δ i L − i L (cid:17) . (34)We note that the X L,ℓi ··· i L symbol only makes sense when L and ℓ are either both even or both odd–we define it to bezero otherwise. We also define X L,ℓi ··· i L to be zero if L or ℓ is negative or if ℓ is greater than L . With the new notationwe can write the results of (29) as (ˆ p i · · · ˆ p i L ) L = 1( L + 1)!! X L, i ··· i L , (35a)(ˆ p i · · · ˆ p i L ) L = 3( L + 2)!! X L, i ··· i L . (35b)Identities among X L,ℓi ··· i L quantities can often be found by simple counting. For instance, consider the followingidentity for the symmetrized product of two X s: X ( L + NL ) (cid:16) X L,ℓi ··· i L X N,ni L +1 ··· i L + N (cid:17) = κX L + N,ℓ + ni ··· i L + N . (36)Both sides of (36) involve the same set of L + N indices, both are symmetric in all indices, and both have exactly ℓ + n momentum unit vectors in each term, so the two sides of (36) are proportional. Since all terms enter with thesame sign and there are no cancellations, the constant of proportionality κ can be found simply by counting the totalnumber of terms on each side. On the left there are (cid:16) L + NL (cid:17) (cid:16) Lℓ (cid:17) ( L − ℓ − (cid:16) Nn (cid:17) ( N − n − (cid:16) L + Nℓ + n (cid:17) ( L + N − ℓ − n − κ is the ratio: κ = (cid:18) L + N − ℓ − nL − ℓ (cid:19) (cid:18) ℓ + nℓ (cid:19) ( L − ℓ − N − n − L + N − ℓ − n − . (37)Two additional identities that will be useful to us involve the contraction X L,ℓi ··· i L ˆ p i L of an X with ˆ p and thecontraction X L,ℓi ··· i L δ i L − i L of two indices of an X . For the first identity, we note that X L,ℓi ··· i L = X ( L − ℓ ) (cid:16) ˆ p i · · · ˆ p i ℓ X ( L − ℓ − (cid:16) δ i ℓ +1 i ℓ +2 · · · δ i L − i L (cid:17)(cid:17) + X ( L − ℓ − ) (cid:16) ˆ p i · · · ˆ p i ℓ − ˆ p i L X ( L − ℓ − (cid:16) δ i ℓ i ℓ +1 · · · δ i L − i L − (cid:17)(cid:17) , (38)where in the first term it is understood that index i L is definitely on a δ , while in the second term index i L is attachedto a ˆ p . Contraction with ˆ p i L then gives two corresponding terms: ˆ p i L times the first term of (38) has L − ℓ + 1 factors of ˆ p , while ˆ p i L times the second term of (38) has L − ℓ − p , so that X L,ℓi ··· i L ˆ p i L = αX L − ,ℓ +1 i ··· i L − + βX L − ,ℓ − i ··· i L − for some constants α and β . Counting terms allows us to identity the constantsto be α = ℓ + 1 and β = 1, so that X L,ℓi ··· i L ˆ p i L = ( ℓ + 1) X L − ,ℓ +1 i ··· i L − + X L − ,ℓ − i ··· i L − . (39)For the second identity we write X L,ℓi ··· i L as X L,ℓi ··· i L = X ( L − ℓ ) ˆ p i · · · ˆ p i ℓ n X ( L − ℓ − (cid:16) δ i ℓ +1 i ℓ +2 · · · δ i L − i L (cid:17) + X ( L − ℓ − L − ℓ − (cid:16) δ i ℓ +1 i ℓ +2 · · · δ i L − i L − δ i L − i L (cid:17)o! + X ( L − ℓ − ) (cid:16) ˆ p i · · · ˆ p i ℓ − ˆ p i L − X ( L − ℓ − (cid:16) δ i ℓ i ℓ +1 · · · δ i L − i L (cid:17)(cid:17) + X ( L − ℓ − ) (cid:16) ˆ p i · · · ˆ p i ℓ − ˆ p i L X ( L − ℓ − (cid:16) δ i ℓ i ℓ +1 · · · δ i L − i L − (cid:17)(cid:17) + X ( L − ℓ − ) (cid:16) ˆ p i · · · ˆ p i ℓ − ˆ p i L − ˆ p i L X ( L − ℓ − (cid:16) δ i ℓ − i ℓ · · · δ i L − i L − (cid:17)(cid:17) , (40)where it is understood that in the first two terms the indices i L − and i L are definitely on δ s, on the same δ in thefirst and on different δ s in the second; in the third term i L − is on a ˆ p while i L is on a δ ; in the fourth i L is on aˆ p and i L − on a δ ; and in the last term both i L − and i L are on ˆ p s. Contraction of i L − with i L gives rise to twostructures: X L,ℓi ··· i L δ i L − i L = ρX L − ,ℓi ··· i L − + σX L − ,ℓ − i ··· i L − , with the first four terms of (40) contributing to ρ and only thelast contributing to σ . Again, we use term counting to identify values for ρ and σ , finding ρ = L + ℓ + 1 and σ = 1.(The four contributions to ρ are, in order, 3, L − ℓ − ℓ and ℓ .) The final form of the contraction identity is X L,ℓi ··· i L δ i L − i L = ( L + ℓ + 1) X L − ,ℓi ··· i L − + X L − ,ℓ − i ··· i L − . (41)We propose the following form for the general decomposition formula(ˆ p i · · · ˆ p i L ) Lℓ = (2 ℓ + 1)( L − ℓ − L − ℓ )!( L + ℓ + 1)!! X n = ℓ ( − ℓ − n ( L − n )!( ℓ + n − ℓ − n − ℓ − n )!( L − n − X L,ni ··· i L , (42)where the sum is over n = ℓ , ℓ −
2, etc., ending with 1 or 0 depending on whether ℓ is odd or even. We proceed to verifythe correctness of this formula by: ( i ), showing that it is consistent with the initial values of (35a), (35b); and ( ii ),verifying that it satisfies the recursion relation (30). Verification of consistency with the initial values is immediateby substituting ℓ = 0 and ℓ = 1 into (42) and noting that in each case there is only one term in the sum and that itagrees with (35a), (35b). For step ( ii ), verification that (42) satisfies the recursion relation (30), we substitute (42)into the right hand side of (30) and obtain two terms. The first of these is ℓ + 1 ℓ (2 ℓ − L − ℓ + 1)!!( L − ℓ + 2)!( L + ℓ + 1)!! X n ′ = ℓ − ( − ℓ − n ′− ( L − n ′ + 1)!( ℓ + n ′ − ℓ − n ′ − ℓ − n ′ − L − n ′ )!! (cid:16) ( n ′ + 1) X L,n ′ +1 i ··· i L + X L,n ′ − i ··· i L (cid:17) (43) and the second is − ℓ + 1 ℓ ℓ − ℓ − ℓ − L − ℓ + 1)!!( L − ℓ + 2)!( L + ℓ − X n = ℓ − ( − ℓ − n − ( L − n )!( ℓ + n − ℓ − n − ℓ − n − L − n − X L,ni ··· i L . (44) We shift the summation index in the first part of (43) according to n ′ → n − n ′ → n + 1, sothat all terms are proportional to X L,ni ··· i L . We add the two parts of (43) to (44) and, after some algebraic simplification,find that the sum is equal to (42). Thus (42) satisfies the recursion relation and by induction is correct for all valuesof ℓ . It is useful to find an expression for the component of ˆ p i · · · ˆ p i L having maximal angular momentum. With ℓ → L in the general decomposition formula (42) we find that(ˆ p i · · · ˆ p i L ) LL = X n = L ( − L − n ( L + n − L − X L,ni ··· i L . (45)This expression is traceless on all pairs of indices as required by (20) and as can be confirmed by applying the traceidentity (41) to (45).Using (45), the product identity (36), and expression (35a) for (ˆ p i · · · ˆ p i L ) L , it is easy to see that (ˆ p i · · · ˆ p i L ) Lℓ canbe written in an alternative, and illuminating, form:(ˆ p i · · · ˆ p i L ) Lℓ = (2 ℓ + 1)!!( L − ℓ + 1)!!( L + ℓ + 1)!! X Lℓ ! (ˆ p i · · · ˆ p i ℓ ) ℓℓ (cid:0) ˆ p i ℓ +1 · · · ˆ p i L (cid:1) L − ℓ . (46)This form displays clearly the fact that every sub-part of (ˆ p i · · · ˆ p i L ) Lℓ has angular momentum ℓ among some subsetof momenta unit vectors and angular momentum zero among the rest. This behavior is illustrated by the examplesshown in (24), (26b), and (26c) (since (ˆ p i ) = ˆ p i ). Expressions (45) for (ˆ p i · · · ˆ p i L ) LL and (46) for (ˆ p i · · · ˆ p i L ) Lℓ are themain results of this paper. IV. APPLICATIONS
As discussed in [9] and II, a useful class of three-dimensional Fourier transforms (that of functions like p n ˆ p i · · · ˆ p i L )can be conveniently done after separation of the various angular momenta in the tensor product. With a Fouriertransform pair Φ( ~p ) and Ψ( ~r ) defined throughΨ( ~r ) = Z d p (2 π ) e i~p · ~r Φ( ~p ) , (47a)Φ( ~p ) = Z d r e − i~p · ~r Ψ( ~r ) , (47b) An expression consistent with (42) is given in [19, 20] but more general because not restricted to three dimensions of space. Theconsistency of (42) with Theorem 1 of [19, 20] is established through use of the identity ∆ n p i · · · p i L = 2 n n ! p L − n X L,L − ni ··· i L where ∆is the Laplacian. it is generally true that the angular momenta of Φ( ~p ) and Ψ( ~r ) are the same. It follows that the transform pairs canbe represented by Φ( ~p ) = φ ( p ) Y mℓ (ˆ p ) ⇐⇒ Ψ( ~r ) = ψ ( r ) Y mℓ (ˆ r ) (48)or Φ( ~p ) = φ ( p ) (ˆ p i · · · ˆ p i L ) Lℓ ⇐⇒ Ψ( ~r ) = ψ ( r ) (ˆ x i · · · ˆ x i L ) Lℓ (49)where the radial functions φ ( p ) and ψ ( r ) are related by ψ ( r ) = i ℓ π Z ∞ dp p j ℓ ( pr ) φ ( p ) , (50a) φ ( p ) = 4 π ( − i ) ℓ Z ∞ dr r j ℓ ( pr ) ψ ( r ) . (50b)In the case that φ ( p ) = p n the transform is ψ ( r ) where ψ ( r ) = i ℓ π χ nℓ r n +3 − ( ℓ + 3) < n < ℓ , i ℓ (2 ℓ +1)!! r ℓ δ ( ~r ) n = ℓ . (51)Since the transform pair (50a), (50b) is essentially the Hankel transform, [21, 22] many additional φ , ψ pairs arealso available, for example those that relate momentum space and coordinate space versions of the Coulomb wavefunctions. [23] Results for the examples given in I are immediate consequences of angular decomposition and thetransforms contained in (51).An interesting use of Fourier transforms of the type considered here is to find unusual differential identities. Considerthe Fourier transform of f ( p ) ( p i · · · p i L ) Lℓ . The momentum vectors can be converted into derivatives when acting onthe exponential in the Fourier transform, leading to Z d p (2 π ) e i~p · ~r f ( p ) ( p i · · · p i L ) Lℓ = ( − i ) L ( ∂ i · · · ∂ i L ) Lℓ Z d p (2 π ) e i~p · ~r f ( p ) . (52)On the other hand, the transform can be evaluated explicitly using (51). Comparison leads to a differential identity.For example, comparison of the two approaches for the transform of p ( p i · · · p i k ) kk leads to the identity( ∂ i · · · ∂ i k ) kk r = ( − k (2 k − r k +1 (ˆ x i · · · ˆ x i k ) kk . (53)(The same identity expressed in terms of spherical harmonics has been given by Rowe. [24] General derivatives ofinverse powers of r have been worked out by Estrada and Kanwal from a distribution point of view. [25]) For k = 2,and with use of the Poisson equation ∂ r = − πδ ( ~r ), one obtains the familiar identity [26] ∂ i ∂ j r = − π δ ij δ ( ~r ) + 3 r (cid:18) ˆ x i ˆ x j − δ ij (cid:19) , (54)as also follows from (3b) of Sec. I. A second interesting identity of this class follows from consideration of the transformof ( p i · · · p i k ) kk : ( ∂ i · · · ∂ i k ) kk δ ( ~r ) = ( − k (2 k + 1)!! r k (ˆ x i · · · ˆ x i k ) kk δ ( ~r ) . (55)Other differential identities can be obtained as easily. Acknowledgments
This material is based upon work supported by the National Science Foundation through Grant No. PHY-1404268.
References [1] G. Breit, Phys. Rev. , 553 (1929).[2] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, Berlin, 1957),Sec. 39.[3] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii,
Quantum Electrodynamics (Pergamon Press, Oxford, 1982), Sec. 83.[4] J. A. R. Coope, R. F. Snider, and F. R. McCourt, J. Chem. Phys. , 2269 (1965).[5] J. A. R. Coope and R. F. Snider, J. Math. Phys. , 1003 (1970).[6] A. J. Stone, Mol. Phys. , 1461 (1975).[7] A. J. Stone, J. Phys. A: Math. Gen. , 485 (1976).[8] J.-M. Normand and J. Raynal, J. Phys. A: Math. Gen. , 1437 (1982).[9] G. S. Adkins, arXiv:1302.1830 [math-ph].[10] G. N. Watson, A Treatise on the Theory of Bessel Functions , 2nd ed. (Cambridge University Press, Cambridge, 1922),p. 368.[11] P. M. Morse and H. Feshbach,
Methods of Theoretical Physics (McGraw Hill, New York, 1953), p. 1466.[12] G. B. Arfken and H. J. Weber,
Mathematical Methods for Physicists , 5th ed. (Academic Press, San Diego, 2001), p. 770.[13] Arfken and Weber, op. cit. , Sec. 11.7.[14] Arfken and Weber, op. cit. , Sec. 12.8.[15] I. S. Gradshteyn and I. M. Ryzhik,
Table of Integrals, Series, and Products (Academic Press, New York, 1980), Integral6.561(14), p. 684.[16] V. Hnizdo, Eur. J. Phys. , 287 (2011).[17] S. G. Samko, On the Fourier transforms of the functions Y m ( x/ | x | ) / | x | n + α , Izvestiya VUZ. Matematika, (7), 73-78(1978).[18] J. M. Bowen, Am. J. Phys. , 511 (1994), Appendix A.[19] A. Bezubik, A. Dabrowska, and A. Strasburger, J. Nonlinear Mathematical Physics, Supplement, 167 (2004).[20] A. Bezubik and A. Strasburger, Symmetry, Integrability and Geometry: Methods and Applications, , 033 (2006).[21] F. Oberhettinger, Tables of Bessel Transforms (Springer-Verlag, Berlin, 1972).[22] B. Davies,
Integral Transforms and their Applications (Springer-Verlag, New York, 1978), Sec. 15.[23] B. Podolsky and L. Pauling, Phys. Rev. , 109 (1929).[24] E. G. P. Rowe, J. Math. Phys. , 1962 (1978).[25] R. Estrada and R. P. Kanwal, Proc. R. Soc. Lond. A , 281 (1985).[26] C. P. Frahm, Am. J. Phys.51