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Quantum circular billiards: Further analytical results
Physica E 53 (2013) 59–62
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Quantum circular billiards: Further analytical results E. de Prunelé n Institut UTINAM, U.M.R. C.N.R.S. 6213, Université de Franche-Comté, 16 route de Gray, La Bouloie, 25030 Besançon, Cedex, France
A U T H O R - H I G H L I G H T S
A two-term recursion for mean values of powers of radial coordinate is given. A three-term recursion for mean values of powers of momentum is given. Analytical expressions for these mean values are given. Wave functions and orthogonality relations in momentum space are given. Comparison of these quantum results with classical ones is achieved.
art ic l e i nf o
a b s t r a c t
Article history: Received 26 February 2013 Received in revised form 19 April 2013 Accepted 22 April 2013 Available online 30 April 2013
Analytical results for circular billiards are obtained for different quantities. In particular a two-term recursion relation for mean values of powers of radial coordinate is given and a three-term recursion relation for mean values of powers of total momentum is given. Analytical expressions for these mean values are also determined, and briefly compared with the results of classical mechanics. & 2013 Elsevier B.V. All rights reserved.
Keywords: Circular billiard Analytical results Mean values Recursion relations
1. Introduction A particle in two-dimensional space, subjected to the only constraint that his wave function in configuration space be zero on and outside a circle of radius A forms a system called a circular billiard. This very simple low dimensional system has been considered in several works, see e.g. [1–5]. For 0 ≤ r ≤ A, the configuration space wave function in polar coordinates r; φ is proportional to a Bessel function Jm regular at origin (as defined in [6,7]), and a exponential function expðimφÞ 〈rψ n;m 〉 ¼ ψ n;m ðrÞ ¼ N n;m J m ðpn;m rÞ pffiffiffiffiffiffi : ð1Þ 2π m is a positive or negative angular momentum quantum number which can take any integers values (including zero). With zn;m ¼ pn;m A, the momentum pn;m is determined by the boundary condition J m ðzn;m Þ ¼ 0; n
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1386-9477/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2013.04.017
ð2Þ
leading to quantification of possible values of pn;m . Thus zn;m is the nth positive zero of Jm. Nn;m is a normalization constant. The radial wave functions satisfy the orthogonality relation Z A Nn′;m N n;m r dr J m ðpn′;m rÞJ m ðpn;m rÞ ¼ δn′n : ð3Þ 0
All the above results are well known. The main purpose of this brief report is to give analytical expressions for mean values of power of r and momentum p. For hydrogen atom, a much more complex system, the mean values of integer powers or r for example have been the subject of intensive works since the early days of quantum mechanics (see e.g. [8–13]). Of course, the hydrogen atom is physically more important than the circular billiard, but the latter is a model of interest for low dimensional systems. The normalization factors N n;m can be computed by numerical integration [4], and the momentum space functions by numerical Fourier transformation [1]. The only reference we have found where analytical results for normalization factors and momentum space wave functions can be extracted is a relatively recent paper [14] concerned with image processing and pattern recognition. This illustrates how academic problems in quantum mechanics
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may be connected with applied physics. A two-term recursion relation for mean values of powers of radial coordinate r, and analytical expressions for these mean values are given in Section 2. The normalized wave functions in momentum space together with their orthogonality relations are given analytically in Section 3. In that section, a very simple approximation for probability density with respect to the momentum p is given for high energy states. A three-term recursion relation for mean values of powers of total momentum p , and analytical expressions for these mean values are given in Section 4. Note that in Eq. (1), Nn;m J m ðpn;m rÞ could also, within a global phase factor, be written N n;jmj J jmj ðpn;m rÞ according to Eq. (8.404.1-2) of [7]. Therefore, from now on, including the appendix, the index of a Bessel function (e.g. μ in J μ is always supposed to be ≥0). The results for the cases m o0 then follow easily. Let us first determine the normalization factors N n;m . One obtains from Eq. (5.54.2) of [7] Z 1 ð4Þ x ½ J μ ðαxÞ2 dx ¼ x2 f½J μ ðαxÞ2 −J μ−1 ðαxÞJ μþ1 ðαxÞg; 2 taking into account the regularity of Jm at the origin and the boundary condition (2) Z
A 0
rJ m ðpn;m rÞJ m ðpn;m rÞ dr ¼
A2 f−J m−1 ðpn;m AÞJ mþ1 ðpn;m AÞg: 2
ð5Þ
One further obtains from Eq. (9.1.27) of [6] ν ν J′ν ðzÞ ¼ J ν−1 ðzÞ− J ν ðzÞ ¼ −J νþ1 ðzÞ þ J ν ðzÞ; z z
ð6Þ
and Eq. (2) J′m ðzn;m Þ ¼ J m−1 ðzn;m Þ ¼ −J mþ1 ðzn;m Þ:
ð7Þ
The normalization factor thus is, within an arbitrary complex multiplicative factor of module unity pffiffiffi N n;m ¼ 2=½AJ mþ1 ðzn;m Þ: ð8Þ
2. Mean values of powers of radial coordinate r With the change of variable x ¼ pn;m r, the mean value of r α is Z zn;m 2Aα ½J m ðxÞ2 x1þα dx: ð9Þ 〈r α 〉n;m ¼ 2 αþ2 ðJ mþ1 ðzn;m ÞÞ zn;m 0 From the behavior of J m ðxÞ at origin (Eq. (9.1.7) of [6]) , it follows that the integral (9) converges if and only if α 4 −2m−2, a condition which is implicit in this section. Eq. (11.2.17) of [16] " 2 # Z Z μþ1 μþ2 2 2 ðμ þ 2Þ z J ν ðzÞ dz−ðμ þ 1Þ ν − zμ J 2ν ðzÞ dz 2 ( ) 2 1 1 1 ¼ zμþ1 zJ′ν ðzÞ− ðμ þ 1ÞJ ν ðzÞ þ z2 −ν2 þ ðμ þ 1Þ2 J 2ν ðzÞ : 2 2 4 ð10Þ together with the condition (2) allow to obtain a two-term recursion relation " # 〈r αþ1 〉n;m α þ 1 2 〈r α−1 〉n;m 2 −ðα þ 1Þ m − ¼ 1; ð11Þ ðα þ 2Þ 2 Aαþ1 Aα−1 z2n;m which directly gives the results for some integer α values. (Note that for hydrogen atom, the so-called Kramers Pasternak relation [10,11] is a three-term recursion relation). For α ¼ −2, Eq. (11) gives A3 〈r −3 〉n;m ¼ 4z2n;m =½ð2m−1Þð2m þ 1Þ. Repeated applications of Eq. (11) then gives the mean values of negative odd integer powers of r, 〈r −ð2kþ1Þ 〉n;m , in terms of rational functions of z2n;m with integer
coefficients. For example 〈r −5 〉n;m −5
A
¼
4z2n;m ð4m2 −1 þ 8z2n;m Þ : 3ð2m−3Þð2m−1Þð2m þ 1Þð2m þ 3Þ
ð12Þ
For α ¼ 1, Eq. (11) gives A−2 〈r 2 〉n;m ¼ ð−2 þ 2m2 þ z2n;m Þ=ð3z2n;m Þ. Repeated applications of Eq. (11) then gives the mean values of positive even integer power of r, 〈r 2k 〉n;m , in terms of rational functions of z2n;m with integer coefficients. For example 〈r 4 〉n;m 4
A
¼
½8ð4−5m2 þ m4 Þ þ 4ð−4 þ m2 Þz2n;m þ 3z4n;m : 15z4n;m
ð13Þ
For the mean values of negative even integer powers of r, 〈r −2k 〉n;m , or positive odd integer powers of r, 〈r 2kþ1 〉n;m , we do not find expressions significantly simpler than the general expression (16) that follows shortly. This does not mean that Eq. (16) cannot be expressed in terms of Bessel functions. For example Eq. (11.2.16) of [16] gives " # m−1
A2 〈r −2 〉n;m ¼ 1−ð J 0 ðzn;m ÞÞ2 −2 ∑ ðJ i ðzn;m ÞÞ2 =½mðJ mþ1 ðzn;m ÞÞ2 :
ð14Þ
i¼1
(From now on, in order to avoid too much formula, equations of standard references such as [6,7,16] will be referred by their number in the references, without writing the equations explicitly). In particular A2 〈r −2 〉n;1 ¼ −1 þ 1=ðJ 0 ðzn;1 ÞÞ2 . We now turn to a general expression involving hypergeometric function p F q p F q ða1 ; …; ap ; b1 ; …bq ; zÞ≡
ða1 Þk …ðap Þk zk ; k ¼ 0 ðb1 Þk …ðbq Þk k! ∞
∑
ð15Þ
valid also for non-integral values of α. From the indefinite integral (11.2.28) of [16], on obtains α
2m 2 F 3
〈r 〉n;m 2ðzn;m =2Þ ¼ Aα ðJ mþ1 ðzn;m ÞÞ2
1 α α m þ ; m þ þ 1; m þ 1; 2m þ 1; m þ þ 2; −z2n;m 2 2 2 : ðα þ 2m þ 2ÞΓðm þ 1Þ2
ð16Þ The expression (16) gives a general closed form expression for the mean values of r α . It is however not directly transparent from Eq. (16) that, for example, 〈r 0 〉n;m ¼ 1. To verify this equality, notes first that for α ¼ 0, 2 F 3 reduces to 1 F 2 . Then, it can be verified from Eq. (1.4.1.18) of [16] that 2ðx=2Þ2m 1 F 2 m þ 12 ; m þ 2; 2m þ 1; −x2 2mJ mþ1 ðxÞJ m ðxÞ ¼ J m ðxÞ2 − x ð2m þ 2ÞΓðm þ 1Þ2 þ J mþ1 ðxÞ2 ; and therefore 〈r 0 〉n;m ¼ 1. 3. Momentum space wave function for the circular billiard The normalized momentum space wave function χ n;m ðpÞ is now obtained as the Fourier transform of the normalized configuration space wave function Z 1 2 〈pψ n;m 〉 ¼ χ n;m ðpÞ ¼ ð17Þ d r expð−ip rÞ〈rψ n;m 〉: 2=2 ð2πÞ Since p r ¼ pr cos ðφ−φp Þ with φ, φp respectively the angle coordinate of r and p, the angular part of the two-dimensional integral requires the evaluation of the integral G Z 2π 1 G¼ dφ expð−ipr cos ðφ−φp ÞÞ expðimφÞ: ð18Þ 2π 0 Eqs. (8.511.4, 8.404.1-2, 8.476.1) of [7] give: G ¼ expðimφp Þð−iÞm J m ðprÞ. Thus Z A expðimφp Þ pffiffiffiffiffiffi ð−iÞm drrJ m ðprÞJ m ðpn;m rÞ; ð19Þ χ n;m ðpÞ ¼ N n;m 2π 0
E. de Prunelé / Physica E 53 (2013) 59–62
which can be evaluated according top Eq. ffiffiffiffiffiffi(5.54.1) of [7] and Eqs. (2), (7), (8): χ n;m ðpÞ ¼ P n;m ðpÞðexpðimφp Þ= 2π Þ pffiffiffi J ðpAÞ P n;m ðpÞ ¼ − 2ð−iÞm pn;m 2m 2 p −pn;m
ð20Þ
For p ¼ pn;m , both numerator and denominator are zero, but the value P n;m ðpn;m Þ is well defined pffiffiffi using l'Hôpital's rule and Eq. (7): P n;m ðpn;m Þ ¼ ð−iÞm AJ mþ1 ðzn;m Þ= 2 ¼ ð−iÞm =Nn;m . By the Parseval Plancherel theorem, the orthonormalization relations for radial wave functions Rn;m ðrÞ give the radial momentum space orthonormaliR∞ zation relations 0 dppP n′;m ðpÞP n;m ðpÞ ¼ δnn′ . In terms of integrals of Bessel functions, these orthonormalization relations read Z
∞ 0
2
dxxJ m ðxÞ=½ðx2 −z2n;m Þðx2 −z2n′;m Þ ¼ δnn′ =ð2zn;m zn′;m Þ:
ð21Þ
If the orthogonality relation (3) for Bessel function is well known [16], the orthogonality relation (21) does not seem to have been noticed in the literature. It is of interest to remark that the probability density divided by A, i.e. pjP n;m ðpÞj2 =A can be expressed as a function of the variable y¼Ap only A−1 pjP n;m ðpÞj2 ¼ 2z2n;m y½J m ðyÞ=ðz2n;m −y2 Þ2 ≡f n;m ðyÞ: As a result, since dp¼ dy/A, the graph of the probability density shrinks by a factor A in abscissa p when A increases, and expands in ordinate by a factor A, so that the area remains unity. From the viewpoint of classical mechanics, the probability density is a Dirac delta distribution for all values of A. Indeed, the momentum is constant for a classical trajectory. A simple approximated expression g n;m ðyÞ for f n;m ðyÞ is obtained from asymptotic expansion [7] of the zeros zn;m when n⪢m : zn;m ≃ðn þ m=2− 14Þπ, and from the asymptotic behavior (Eq. (9.2.1) of [6]) of Bessel functions for large arguments 4π f n;m ðyÞ ≃
πm m 1 2 π þ n− −y þ cos2 2 4 2 4 ≡ g n;m ðyÞ: !2 2 m 1 þ n− −y2 π2 2 4
ð22Þ
For fixed value of the ratio m=n⪡1, the accuracy of this approximation decreases as m increases as can be seen in Fig. 1. For fixed large m values, the accuracy however increases as n increases.
61
4. Mean values of powers of momentum p With the change of variable x ¼pA, the mean value of pα is !2 Z ∞ J m ðxÞ 1þα dxx : ð23Þ 〈pα 〉n;m ¼ 2z2n;m A−α x2 −z2n;m 0 From the behavior of J m ðxÞ at the origin (Eq. (9.1.7) of [6]), it follows that the integral (23) diverges if α ≤ −2m−2 and from the behavior of J m ðxÞ at infinity (Eq. (9.2.1) of [6]), it follows that the integral (23) diverges if α≥3. Therefore the inequality −2m−2 o α o 3 is implicit in this section. The integrand in Eq. (23) is oscillatory and the numerical integration over the infinite interval ½0; ∞½ is difficult, so that analytical results are of particular interest. Now we turn to the determination of a three-term recursion relation for 〈pα 〉n;m . Eq. (20) gives pjP n;m ðpÞj2 pα ðp2 −p2n;m Þ2 ¼ 2p2n;m p1þα J 2m ðpAÞ, and by integration Aαþ4 〈pαþ4 〉n;m −2z2n;m Aαþ2 〈pαþ2 〉n;m
−1−α α Γ mþ1þ Γ 2 2 þ z4n;m Aα 〈pα 〉n;m ¼ z2n;m pffiffiffi
α α : πΓ m − Γ − 2 2
ð24Þ
The integral of p1þα J 2m ðpAÞ has been expressed in terms of gamma functions in Eq. (24) with the help of Eq. (6.574.2) of [7] and Eq. (6.1.18) of [6]. Let us first determine 〈p2 〉n;m . Since p2 acts as minus the Laplacian in configuration space and since J m ðpn;m rÞ expðimφÞ= p ffiffiffiffiffiffi 2π is a solution of the Schrödinger equation, −Δψ n;m ðr; φÞ ¼ p2n;m ψ n;m ðr; φÞ þ B where the term B involves a Dirac distribution centered on A since the derivative of ψ n;m ðr; φÞ with respect to r is discontinuous at r ¼A. The contribution of the B term to 〈ψ n;m ðr; φÞj−Δψ n;m ðr; φÞ〉 is zero since ψ n;m ðA; φÞ ¼ 0. Thus A2 〈p2 〉n;m ¼ z2n;m . Since 〈p0 〉n;m ¼ 1, the other A−2k 〈p−2k 〉n;m with k integer 0 4−2k 4 −2m−2 can be deduced from the three-term recursion (24). A−2k 〈p−2k 〉n;m is thus obtained as rational function of zn;m . For example A−2 〈p−2 〉n;m ¼ ð1 þ 1=mÞz−2 n;m ! 2 z −4 2 n;m A 〈p−4 〉n;m ¼ 1 þ þ ð25Þ z−4 : m 2ðm þ 1Þmðm−1Þ n;m The general formula for α even is ( −1 α − þkþ1 Aα 〈pα 〉n;m ¼ zαn;m 1 þ ∑ 2 k ¼ α=2
) ðk þ mÞ! −2k−2 zn;m ; 2−k−1 ð−k−1Þ! ð−k þ m−1Þ! ð−2k−3Þ!!
ð26Þ
where the summation over k contributes only if −1≥α=2. This expression indeed satisfies the recursion relation (24) and gives the correct values for 〈p0 〉n;m and 〈p2 〉n;m . A general expression valid also for non integer values of α is 3−α Γ pffiffiffi 2
α Aα 〈pα 〉n;m ¼ z2n;m π sin π − −m 2 3 α α α α ~ ; 2− ; −m− þ 2; m− þ 2; −z2n;m : ð27Þ 2 F 3 2; − 2 2 2 2 2 The regularized hypergeometric function 2 F~ 3 is defined by ~
p F q ða1 ; …; ap ; b1 ; …; bq ; zÞ ¼
Fig. 1. Graphs of f n;m ðyÞ (full curves) and of gn;m ðyÞ (dashed curves) for n ¼ 2m þ 1 with m ¼ 0; 1; …; 9. Each of these 20 curves has a pronounced local maximum near the Bessel zero zn;m . The positions of these 10 zeros are indicated by vertical lines. Starting from left, the first zero is z1;0 , the following succeed in order increasing m values. The dashed curves for g n;m ðyÞ are almost indistinguishable from the full curves for f n;m ðyÞ for the first values of m.
p F q ða1 ; …; ap ; b1 ; …; bq ; zÞ
Γðb1 Þ…Γðbq Þ
:
ð28Þ
The demonstration of Eq. (27) is rather involved and the method of proof is outlined in Appendix A . Within inequality −2m−2 o α o 3, 〈pα 〉n;m is finite, and the sine in the denominator of Eq. (27) is zero for integer even α. The function 2 F~ 3 then must be zero for integer even α, leading to an
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E. de Prunelé / Physica E 53 (2013) 59–62
undetermined form ∞ 0 in Eq. (27). Consider for example the case α ¼ 0. In that case 〈pα 〉n;m ¼ 1 and sin ðπððα=2Þ þ m−1ÞÞ is zero. It follows that 2 F~ 3 ð2; 32 ; 2; −m þ 2; m þ 2; −z2n;m Þ ¼ 1 F~ 2 ð32 ; −m þ 2; m þ 2; −z2n;m Þ must be zero and we may anticipate that it must be proportional to J m ðzn;m Þ which is zero by definition. Indeed Eq. (1.4.1.18) of [16] and Eq. (8.404.1-2) of [7] give 1 F~ 2 ð12 ; −m þ 1; m þ 1; −x2 Þ ¼ ð−1Þm J 2m ðxÞ. Then differentiation gives the result mþ1 2J m ðxÞJ′m ðxÞ ~ 3; −m þ 2; m þ 2; −x2 ¼ ð−1Þ : 1F 2 2 x In the cases where even integer values of α ¼ 2k, k∈Z in Eq. (27) lead to products of the form ∞ 0, Eq. (27) can be determined by l'Hôpital rule 2Γ 3−2k 2 A2k 〈p2k 〉n;m ¼ z2n;m pffiffiffi π ð−1Þkþm−1 3 α α α α d 2 F~ 3 2; − ; 2− ; −m− þ 2; m− þ 2; −z2n;m 2 2 2 2 2 : dα α ¼ 2k
ð29Þ
5. Concluding remarks Connections between classical and quantum circular billiards have been discussed elsewhere [2] and will not be repeated here. We only focus on mean values 〈pα 〉, 〈r α 〉. The momentum p is constant classically. Therefore 〈pα 〉c ¼ 〈p〉αc where the subscript c is for classical. We have seen how the quantal probability distribution approach a Dirac distribution when A-∞. Now the classical density probability Dc ðrÞ for r is proportional to dt=dr, with t the time, and can easily be computed [2]: Dc ðrÞ ¼ r= qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 r −m =p Þ with m=p the minimal value of r. The ð A −m =p RA normalization is m=p Dc ðrÞdr ¼ 1. If p2 is replaced by its quantum value z2n;m =A2 , comparisons with quantum results can be achieved. RA One computes 〈r α 〉c ¼ m=p r α Dc ðrÞdr and obtains for example A3 〈r −3 〉c ¼ z2n;m =m2 ,
A5 〈r −5 〉c ¼ z2n;m ðm2 þ 2z2n;m Þ=ð3m4 Þ,
A−2 〈r 2 〉c ¼
ð2m2 þ z2n;m Þ=ð3z2n;m Þ, A−4 〈r 4 〉c ¼ ð8m4 þ 4m2 z2n;m þ 3z4n;m Þ=ð15z4n;m Þ. These expressions correspond to the dominant contributions of the series expansions in power of 1=m of the quantum results. Some extensions of the present work concern spherical billiards, annular billiards, and two concentric spherical billiards. In the case of spherical billiards, many results are obtained by the replacement m-ℓ þ 1=2 in the present results, with ℓ the orbital quantum number.
Appendix A. Method of proof Eq. (27) The evaluation of the integral (23) follows a general method suggested in Section 13.61 of [15] and is now briefly outlined. Let R∞ 1þα X≡ 0 dxx ½J μ ðxÞ=ðx2 −z2 Þ2 . Here z is supposed to have a non-zero imaginary part, in order to avoid singularity of the integrand. At the end of the calculation, this imaginary part goes to zero and z goes to zn;m , and μ will be m. The starting point is the integral representation of the product of two Bessel functions along a Barne type contour [15]
2μþ2s Z ∞i Γð−sÞΓð2μ þ 2s þ 1Þ x 1 2 J μ ðxÞJ μ ðxÞ ¼ ds: ð30Þ 2πi −∞i Γðμ þ s þ 1Þ2 Γð2μ þ s þ 1Þ The poles of Γð−sÞ are on the right of the contour, while those of Γð2μ þ 2s þ 1Þ are on the left. After using the duplication formula
of gamma function (Eq. (6.1.18) of [6]) the integral is evaluated by change of the order of integrations. The integration over x gives 1 Z ∞i ðα þ 2μ þ 2sÞ Γð−sÞΓ μ þ s þ 1 2 h
i X¼ 8π 1=2 i −∞i Γðμ þ s þ 1ÞΓð2μ þ s þ 1Þ sin π α þ μ þ s 2 1 −ðα=2Þ−μ−sþ1 − 2 ds: ð31Þ z By swinging round the contour in (31) so as to enclose the poles on the right, one can apply the residues theorem. Since the gamma function has no zero, the singularities of the integrand of (31) inside the contour are the poles of Γð−sÞ and the zeros of the sine in the denominator : s¼ n and s þ α=2 þ μ ¼ n with n≥0 an integer. We suppose that α is different from an even integer so that the two series of poles are distinct for integer μ. The residue of Γð−sÞ at s¼n is ð−1Þnþ1 =n!. Let X ¼ X 1 þ X 2 with X1 the contribution of the set of poles s ¼n and X2 the contribution of the set of poles s þ α=2 þ μ ¼ n. The infinite sum over residues at s¼n gives after some calculations for X1 1 α −α=2−μþ1 Γ μ þ pffiffiffi Γ 2þμþ1 1 π 2 − 2 X1 ¼ 2 z sin π 2α þ μ 1 α α ~ ; þ μ þ 1; μ þ 1; 2μ þ 1; þ μ; −z2 ; 2F 3 μ þ 2 2 2 with the regularized hypergeometric function defined by Eq. (28). Now one can verify with the help of Eq. (1.4.1.18) of [16], that both members of 1 α α 2 ~ ; þ μ þ 1; μ þ 1; 2μ þ 1; þ μ; −z F μ þ 2 3 2 2 2 pffiffiffi −2μ π z J μ ðzÞððα þ 2μÞJ μ ðzÞ−2zJ μþ1 ðzÞÞ ¼ 2Γ μ þ 12 Γ 2α þ μ þ 1 have the same power series expansion in the variable z. For the case of interest, μ ¼ m and z ¼ zn;m and therefore the factor J μ ðzÞ gives zero, and thus X 1 ¼ 0. It remains to calculate the contribution X2 to X of the series of poles −s−ðα=2Þ−μ þ 1 ¼ −n. Using the same procedures as described for X1, one obtains pffiffiffi 3−α πΓ 2 3 α α α α α 2 F~ 3 2; − ; 2− ; − −μ þ 2; μ− þ 2; −z2 ; X2 ¼ 2 2 2 2 2 2 sin π − 2 −μ Eq. (27) is obtained. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13] [14] [15] [16]
R.W. Robinett, American Journal of Physics 64 (1996) 440. R.W. Robinett, American Journal of Physics 67 (1999) 67. R.W. Robinett, S. Heppelmann, Physical Review A 65 (2002) 062103. M.A. Doncheski, S. Heppelmann, R.W. Robinett, D.C. Tussey, American Journal of Physics 71 (2003) 541. M. Brack, R.K. Bhaduri, Semiclassical Physics, Frontiers in Physics, 2003. M. Abramowitz, A. Stegun, Handbook of Mathematical Functions, Dover Publications, INC., New York, 1970. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals Series and Products, Sixth ed., Academic Press, 2000. I. Waller, Zeitschrift für Physik 38 (1926) 644. J.H. Van Vleck, Proceedings of the Royal Society of London, Series A 143 (1934) 679. S. Pasternak, Proceedings of the National Academy of Sciences 23 (1937) 91. H.A. Kramers, Quantum Mechanics, North-Holland, Amsterdam, 1957. [translated by D. ter Haar from Quantentheorie des electron und der Strahlung (Akademische Verlagesellschaft, Leipzig, 1938)]. K. Bockasten, Physical Review A 9 (1974) 1087. G.W.F. Drake, R.A. Swainson, Physical Review A 42 (1990) 1123. Q. Wang, O. Ronneberger, H. Burhardt, IEEE Transactions on Pattern Analysis and Machine Intelligence 31 (2009) 1715. G.N. Watson, A Treatise on the Theory of Bessels Functions, second ed., Cambridge University Press, 1944. Y.L. Luke, Integrals of Bessel Functions, McGraw-Hill Book Company, INC, New York, 1962.
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Physica E journal homepage: www.elsevier.com/locate/physe
Quantum circular billiards: Further analytical results E. de Prunelé n Institut UTINAM, U.M.R. C.N.R.S. 6213, Université de Franche-Comté, 16 route de Gray, La Bouloie, 25030 Besançon, Cedex, France
A U T H O R - H I G H L I G H T S
A two-term recursion for mean values of powers of radial coordinate is given. A three-term recursion for mean values of powers of momentum is given. Analytical expressions for these mean values are given. Wave functions and orthogonality relations in momentum space are given. Comparison of these quantum results with classical ones is achieved.
art ic l e i nf o
a b s t r a c t
Article history: Received 26 February 2013 Received in revised form 19 April 2013 Accepted 22 April 2013 Available online 30 April 2013
Analytical results for circular billiards are obtained for different quantities. In particular a two-term recursion relation for mean values of powers of radial coordinate is given and a three-term recursion relation for mean values of powers of total momentum is given. Analytical expressions for these mean values are also determined, and briefly compared with the results of classical mechanics. & 2013 Elsevier B.V. All rights reserved.
Keywords: Circular billiard Analytical results Mean values Recursion relations
1. Introduction A particle in two-dimensional space, subjected to the only constraint that his wave function in configuration space be zero on and outside a circle of radius A forms a system called a circular billiard. This very simple low dimensional system has been considered in several works, see e.g. [1–5]. For 0 ≤ r ≤ A, the configuration space wave function in polar coordinates r; φ is proportional to a Bessel function Jm regular at origin (as defined in [6,7]), and a exponential function expðimφÞ 〈rψ n;m 〉 ¼ ψ n;m ðrÞ ¼ N n;m J m ðpn;m rÞ pffiffiffiffiffiffi : ð1Þ 2π m is a positive or negative angular momentum quantum number which can take any integers values (including zero). With zn;m ¼ pn;m A, the momentum pn;m is determined by the boundary condition J m ðzn;m Þ ¼ 0; n
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1386-9477/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2013.04.017
ð2Þ
leading to quantification of possible values of pn;m . Thus zn;m is the nth positive zero of Jm. Nn;m is a normalization constant. The radial wave functions satisfy the orthogonality relation Z A Nn′;m N n;m r dr J m ðpn′;m rÞJ m ðpn;m rÞ ¼ δn′n : ð3Þ 0
All the above results are well known. The main purpose of this brief report is to give analytical expressions for mean values of power of r and momentum p. For hydrogen atom, a much more complex system, the mean values of integer powers or r for example have been the subject of intensive works since the early days of quantum mechanics (see e.g. [8–13]). Of course, the hydrogen atom is physically more important than the circular billiard, but the latter is a model of interest for low dimensional systems. The normalization factors N n;m can be computed by numerical integration [4], and the momentum space functions by numerical Fourier transformation [1]. The only reference we have found where analytical results for normalization factors and momentum space wave functions can be extracted is a relatively recent paper [14] concerned with image processing and pattern recognition. This illustrates how academic problems in quantum mechanics
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E. de Prunelé / Physica E 53 (2013) 59–62
may be connected with applied physics. A two-term recursion relation for mean values of powers of radial coordinate r, and analytical expressions for these mean values are given in Section 2. The normalized wave functions in momentum space together with their orthogonality relations are given analytically in Section 3. In that section, a very simple approximation for probability density with respect to the momentum p is given for high energy states. A three-term recursion relation for mean values of powers of total momentum p , and analytical expressions for these mean values are given in Section 4. Note that in Eq. (1), Nn;m J m ðpn;m rÞ could also, within a global phase factor, be written N n;jmj J jmj ðpn;m rÞ according to Eq. (8.404.1-2) of [7]. Therefore, from now on, including the appendix, the index of a Bessel function (e.g. μ in J μ is always supposed to be ≥0). The results for the cases m o0 then follow easily. Let us first determine the normalization factors N n;m . One obtains from Eq. (5.54.2) of [7] Z 1 ð4Þ x ½ J μ ðαxÞ2 dx ¼ x2 f½J μ ðαxÞ2 −J μ−1 ðαxÞJ μþ1 ðαxÞg; 2 taking into account the regularity of Jm at the origin and the boundary condition (2) Z
A 0
rJ m ðpn;m rÞJ m ðpn;m rÞ dr ¼
A2 f−J m−1 ðpn;m AÞJ mþ1 ðpn;m AÞg: 2
ð5Þ
One further obtains from Eq. (9.1.27) of [6] ν ν J′ν ðzÞ ¼ J ν−1 ðzÞ− J ν ðzÞ ¼ −J νþ1 ðzÞ þ J ν ðzÞ; z z
ð6Þ
and Eq. (2) J′m ðzn;m Þ ¼ J m−1 ðzn;m Þ ¼ −J mþ1 ðzn;m Þ:
ð7Þ
The normalization factor thus is, within an arbitrary complex multiplicative factor of module unity pffiffiffi N n;m ¼ 2=½AJ mþ1 ðzn;m Þ: ð8Þ
2. Mean values of powers of radial coordinate r With the change of variable x ¼ pn;m r, the mean value of r α is Z zn;m 2Aα ½J m ðxÞ2 x1þα dx: ð9Þ 〈r α 〉n;m ¼ 2 αþ2 ðJ mþ1 ðzn;m ÞÞ zn;m 0 From the behavior of J m ðxÞ at origin (Eq. (9.1.7) of [6]) , it follows that the integral (9) converges if and only if α 4 −2m−2, a condition which is implicit in this section. Eq. (11.2.17) of [16] " 2 # Z Z μþ1 μþ2 2 2 ðμ þ 2Þ z J ν ðzÞ dz−ðμ þ 1Þ ν − zμ J 2ν ðzÞ dz 2 ( ) 2 1 1 1 ¼ zμþ1 zJ′ν ðzÞ− ðμ þ 1ÞJ ν ðzÞ þ z2 −ν2 þ ðμ þ 1Þ2 J 2ν ðzÞ : 2 2 4 ð10Þ together with the condition (2) allow to obtain a two-term recursion relation " # 〈r αþ1 〉n;m α þ 1 2 〈r α−1 〉n;m 2 −ðα þ 1Þ m − ¼ 1; ð11Þ ðα þ 2Þ 2 Aαþ1 Aα−1 z2n;m which directly gives the results for some integer α values. (Note that for hydrogen atom, the so-called Kramers Pasternak relation [10,11] is a three-term recursion relation). For α ¼ −2, Eq. (11) gives A3 〈r −3 〉n;m ¼ 4z2n;m =½ð2m−1Þð2m þ 1Þ. Repeated applications of Eq. (11) then gives the mean values of negative odd integer powers of r, 〈r −ð2kþ1Þ 〉n;m , in terms of rational functions of z2n;m with integer
coefficients. For example 〈r −5 〉n;m −5
A
¼
4z2n;m ð4m2 −1 þ 8z2n;m Þ : 3ð2m−3Þð2m−1Þð2m þ 1Þð2m þ 3Þ
ð12Þ
For α ¼ 1, Eq. (11) gives A−2 〈r 2 〉n;m ¼ ð−2 þ 2m2 þ z2n;m Þ=ð3z2n;m Þ. Repeated applications of Eq. (11) then gives the mean values of positive even integer power of r, 〈r 2k 〉n;m , in terms of rational functions of z2n;m with integer coefficients. For example 〈r 4 〉n;m 4
A
¼
½8ð4−5m2 þ m4 Þ þ 4ð−4 þ m2 Þz2n;m þ 3z4n;m : 15z4n;m
ð13Þ
For the mean values of negative even integer powers of r, 〈r −2k 〉n;m , or positive odd integer powers of r, 〈r 2kþ1 〉n;m , we do not find expressions significantly simpler than the general expression (16) that follows shortly. This does not mean that Eq. (16) cannot be expressed in terms of Bessel functions. For example Eq. (11.2.16) of [16] gives " # m−1
A2 〈r −2 〉n;m ¼ 1−ð J 0 ðzn;m ÞÞ2 −2 ∑ ðJ i ðzn;m ÞÞ2 =½mðJ mþ1 ðzn;m ÞÞ2 :
ð14Þ
i¼1
(From now on, in order to avoid too much formula, equations of standard references such as [6,7,16] will be referred by their number in the references, without writing the equations explicitly). In particular A2 〈r −2 〉n;1 ¼ −1 þ 1=ðJ 0 ðzn;1 ÞÞ2 . We now turn to a general expression involving hypergeometric function p F q p F q ða1 ; …; ap ; b1 ; …bq ; zÞ≡
ða1 Þk …ðap Þk zk ; k ¼ 0 ðb1 Þk …ðbq Þk k! ∞
∑
ð15Þ
valid also for non-integral values of α. From the indefinite integral (11.2.28) of [16], on obtains α
2m 2 F 3
〈r 〉n;m 2ðzn;m =2Þ ¼ Aα ðJ mþ1 ðzn;m ÞÞ2
1 α α m þ ; m þ þ 1; m þ 1; 2m þ 1; m þ þ 2; −z2n;m 2 2 2 : ðα þ 2m þ 2ÞΓðm þ 1Þ2
ð16Þ The expression (16) gives a general closed form expression for the mean values of r α . It is however not directly transparent from Eq. (16) that, for example, 〈r 0 〉n;m ¼ 1. To verify this equality, notes first that for α ¼ 0, 2 F 3 reduces to 1 F 2 . Then, it can be verified from Eq. (1.4.1.18) of [16] that 2ðx=2Þ2m 1 F 2 m þ 12 ; m þ 2; 2m þ 1; −x2 2mJ mþ1 ðxÞJ m ðxÞ ¼ J m ðxÞ2 − x ð2m þ 2ÞΓðm þ 1Þ2 þ J mþ1 ðxÞ2 ; and therefore 〈r 0 〉n;m ¼ 1. 3. Momentum space wave function for the circular billiard The normalized momentum space wave function χ n;m ðpÞ is now obtained as the Fourier transform of the normalized configuration space wave function Z 1 2 〈pψ n;m 〉 ¼ χ n;m ðpÞ ¼ ð17Þ d r expð−ip rÞ〈rψ n;m 〉: 2=2 ð2πÞ Since p r ¼ pr cos ðφ−φp Þ with φ, φp respectively the angle coordinate of r and p, the angular part of the two-dimensional integral requires the evaluation of the integral G Z 2π 1 G¼ dφ expð−ipr cos ðφ−φp ÞÞ expðimφÞ: ð18Þ 2π 0 Eqs. (8.511.4, 8.404.1-2, 8.476.1) of [7] give: G ¼ expðimφp Þð−iÞm J m ðprÞ. Thus Z A expðimφp Þ pffiffiffiffiffiffi ð−iÞm drrJ m ðprÞJ m ðpn;m rÞ; ð19Þ χ n;m ðpÞ ¼ N n;m 2π 0
E. de Prunelé / Physica E 53 (2013) 59–62
which can be evaluated according top Eq. ffiffiffiffiffiffi(5.54.1) of [7] and Eqs. (2), (7), (8): χ n;m ðpÞ ¼ P n;m ðpÞðexpðimφp Þ= 2π Þ pffiffiffi J ðpAÞ P n;m ðpÞ ¼ − 2ð−iÞm pn;m 2m 2 p −pn;m
ð20Þ
For p ¼ pn;m , both numerator and denominator are zero, but the value P n;m ðpn;m Þ is well defined pffiffiffi using l'Hôpital's rule and Eq. (7): P n;m ðpn;m Þ ¼ ð−iÞm AJ mþ1 ðzn;m Þ= 2 ¼ ð−iÞm =Nn;m . By the Parseval Plancherel theorem, the orthonormalization relations for radial wave functions Rn;m ðrÞ give the radial momentum space orthonormaliR∞ zation relations 0 dppP n′;m ðpÞP n;m ðpÞ ¼ δnn′ . In terms of integrals of Bessel functions, these orthonormalization relations read Z
∞ 0
2
dxxJ m ðxÞ=½ðx2 −z2n;m Þðx2 −z2n′;m Þ ¼ δnn′ =ð2zn;m zn′;m Þ:
ð21Þ
If the orthogonality relation (3) for Bessel function is well known [16], the orthogonality relation (21) does not seem to have been noticed in the literature. It is of interest to remark that the probability density divided by A, i.e. pjP n;m ðpÞj2 =A can be expressed as a function of the variable y¼Ap only A−1 pjP n;m ðpÞj2 ¼ 2z2n;m y½J m ðyÞ=ðz2n;m −y2 Þ2 ≡f n;m ðyÞ: As a result, since dp¼ dy/A, the graph of the probability density shrinks by a factor A in abscissa p when A increases, and expands in ordinate by a factor A, so that the area remains unity. From the viewpoint of classical mechanics, the probability density is a Dirac delta distribution for all values of A. Indeed, the momentum is constant for a classical trajectory. A simple approximated expression g n;m ðyÞ for f n;m ðyÞ is obtained from asymptotic expansion [7] of the zeros zn;m when n⪢m : zn;m ≃ðn þ m=2− 14Þπ, and from the asymptotic behavior (Eq. (9.2.1) of [6]) of Bessel functions for large arguments 4π f n;m ðyÞ ≃
πm m 1 2 π þ n− −y þ cos2 2 4 2 4 ≡ g n;m ðyÞ: !2 2 m 1 þ n− −y2 π2 2 4
ð22Þ
For fixed value of the ratio m=n⪡1, the accuracy of this approximation decreases as m increases as can be seen in Fig. 1. For fixed large m values, the accuracy however increases as n increases.
61
4. Mean values of powers of momentum p With the change of variable x ¼pA, the mean value of pα is !2 Z ∞ J m ðxÞ 1þα dxx : ð23Þ 〈pα 〉n;m ¼ 2z2n;m A−α x2 −z2n;m 0 From the behavior of J m ðxÞ at the origin (Eq. (9.1.7) of [6]), it follows that the integral (23) diverges if α ≤ −2m−2 and from the behavior of J m ðxÞ at infinity (Eq. (9.2.1) of [6]), it follows that the integral (23) diverges if α≥3. Therefore the inequality −2m−2 o α o 3 is implicit in this section. The integrand in Eq. (23) is oscillatory and the numerical integration over the infinite interval ½0; ∞½ is difficult, so that analytical results are of particular interest. Now we turn to the determination of a three-term recursion relation for 〈pα 〉n;m . Eq. (20) gives pjP n;m ðpÞj2 pα ðp2 −p2n;m Þ2 ¼ 2p2n;m p1þα J 2m ðpAÞ, and by integration Aαþ4 〈pαþ4 〉n;m −2z2n;m Aαþ2 〈pαþ2 〉n;m
−1−α α Γ mþ1þ Γ 2 2 þ z4n;m Aα 〈pα 〉n;m ¼ z2n;m pffiffiffi
α α : πΓ m − Γ − 2 2
ð24Þ
The integral of p1þα J 2m ðpAÞ has been expressed in terms of gamma functions in Eq. (24) with the help of Eq. (6.574.2) of [7] and Eq. (6.1.18) of [6]. Let us first determine 〈p2 〉n;m . Since p2 acts as minus the Laplacian in configuration space and since J m ðpn;m rÞ expðimφÞ= p ffiffiffiffiffiffi 2π is a solution of the Schrödinger equation, −Δψ n;m ðr; φÞ ¼ p2n;m ψ n;m ðr; φÞ þ B where the term B involves a Dirac distribution centered on A since the derivative of ψ n;m ðr; φÞ with respect to r is discontinuous at r ¼A. The contribution of the B term to 〈ψ n;m ðr; φÞj−Δψ n;m ðr; φÞ〉 is zero since ψ n;m ðA; φÞ ¼ 0. Thus A2 〈p2 〉n;m ¼ z2n;m . Since 〈p0 〉n;m ¼ 1, the other A−2k 〈p−2k 〉n;m with k integer 0 4−2k 4 −2m−2 can be deduced from the three-term recursion (24). A−2k 〈p−2k 〉n;m is thus obtained as rational function of zn;m . For example A−2 〈p−2 〉n;m ¼ ð1 þ 1=mÞz−2 n;m ! 2 z −4 2 n;m A 〈p−4 〉n;m ¼ 1 þ þ ð25Þ z−4 : m 2ðm þ 1Þmðm−1Þ n;m The general formula for α even is ( −1 α − þkþ1 Aα 〈pα 〉n;m ¼ zαn;m 1 þ ∑ 2 k ¼ α=2
) ðk þ mÞ! −2k−2 zn;m ; 2−k−1 ð−k−1Þ! ð−k þ m−1Þ! ð−2k−3Þ!!
ð26Þ
where the summation over k contributes only if −1≥α=2. This expression indeed satisfies the recursion relation (24) and gives the correct values for 〈p0 〉n;m and 〈p2 〉n;m . A general expression valid also for non integer values of α is 3−α Γ pffiffiffi 2
α Aα 〈pα 〉n;m ¼ z2n;m π sin π − −m 2 3 α α α α ~ ; 2− ; −m− þ 2; m− þ 2; −z2n;m : ð27Þ 2 F 3 2; − 2 2 2 2 2 The regularized hypergeometric function 2 F~ 3 is defined by ~
p F q ða1 ; …; ap ; b1 ; …; bq ; zÞ ¼
Fig. 1. Graphs of f n;m ðyÞ (full curves) and of gn;m ðyÞ (dashed curves) for n ¼ 2m þ 1 with m ¼ 0; 1; …; 9. Each of these 20 curves has a pronounced local maximum near the Bessel zero zn;m . The positions of these 10 zeros are indicated by vertical lines. Starting from left, the first zero is z1;0 , the following succeed in order increasing m values. The dashed curves for g n;m ðyÞ are almost indistinguishable from the full curves for f n;m ðyÞ for the first values of m.
p F q ða1 ; …; ap ; b1 ; …; bq ; zÞ
Γðb1 Þ…Γðbq Þ
:
ð28Þ
The demonstration of Eq. (27) is rather involved and the method of proof is outlined in Appendix A . Within inequality −2m−2 o α o 3, 〈pα 〉n;m is finite, and the sine in the denominator of Eq. (27) is zero for integer even α. The function 2 F~ 3 then must be zero for integer even α, leading to an
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E. de Prunelé / Physica E 53 (2013) 59–62
undetermined form ∞ 0 in Eq. (27). Consider for example the case α ¼ 0. In that case 〈pα 〉n;m ¼ 1 and sin ðπððα=2Þ þ m−1ÞÞ is zero. It follows that 2 F~ 3 ð2; 32 ; 2; −m þ 2; m þ 2; −z2n;m Þ ¼ 1 F~ 2 ð32 ; −m þ 2; m þ 2; −z2n;m Þ must be zero and we may anticipate that it must be proportional to J m ðzn;m Þ which is zero by definition. Indeed Eq. (1.4.1.18) of [16] and Eq. (8.404.1-2) of [7] give 1 F~ 2 ð12 ; −m þ 1; m þ 1; −x2 Þ ¼ ð−1Þm J 2m ðxÞ. Then differentiation gives the result mþ1 2J m ðxÞJ′m ðxÞ ~ 3; −m þ 2; m þ 2; −x2 ¼ ð−1Þ : 1F 2 2 x In the cases where even integer values of α ¼ 2k, k∈Z in Eq. (27) lead to products of the form ∞ 0, Eq. (27) can be determined by l'Hôpital rule 2Γ 3−2k 2 A2k 〈p2k 〉n;m ¼ z2n;m pffiffiffi π ð−1Þkþm−1 3 α α α α d 2 F~ 3 2; − ; 2− ; −m− þ 2; m− þ 2; −z2n;m 2 2 2 2 2 : dα α ¼ 2k
ð29Þ
5. Concluding remarks Connections between classical and quantum circular billiards have been discussed elsewhere [2] and will not be repeated here. We only focus on mean values 〈pα 〉, 〈r α 〉. The momentum p is constant classically. Therefore 〈pα 〉c ¼ 〈p〉αc where the subscript c is for classical. We have seen how the quantal probability distribution approach a Dirac distribution when A-∞. Now the classical density probability Dc ðrÞ for r is proportional to dt=dr, with t the time, and can easily be computed [2]: Dc ðrÞ ¼ r= qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 r −m =p Þ with m=p the minimal value of r. The ð A −m =p RA normalization is m=p Dc ðrÞdr ¼ 1. If p2 is replaced by its quantum value z2n;m =A2 , comparisons with quantum results can be achieved. RA One computes 〈r α 〉c ¼ m=p r α Dc ðrÞdr and obtains for example A3 〈r −3 〉c ¼ z2n;m =m2 ,
A5 〈r −5 〉c ¼ z2n;m ðm2 þ 2z2n;m Þ=ð3m4 Þ,
A−2 〈r 2 〉c ¼
ð2m2 þ z2n;m Þ=ð3z2n;m Þ, A−4 〈r 4 〉c ¼ ð8m4 þ 4m2 z2n;m þ 3z4n;m Þ=ð15z4n;m Þ. These expressions correspond to the dominant contributions of the series expansions in power of 1=m of the quantum results. Some extensions of the present work concern spherical billiards, annular billiards, and two concentric spherical billiards. In the case of spherical billiards, many results are obtained by the replacement m-ℓ þ 1=2 in the present results, with ℓ the orbital quantum number.
Appendix A. Method of proof Eq. (27) The evaluation of the integral (23) follows a general method suggested in Section 13.61 of [15] and is now briefly outlined. Let R∞ 1þα X≡ 0 dxx ½J μ ðxÞ=ðx2 −z2 Þ2 . Here z is supposed to have a non-zero imaginary part, in order to avoid singularity of the integrand. At the end of the calculation, this imaginary part goes to zero and z goes to zn;m , and μ will be m. The starting point is the integral representation of the product of two Bessel functions along a Barne type contour [15]
2μþ2s Z ∞i Γð−sÞΓð2μ þ 2s þ 1Þ x 1 2 J μ ðxÞJ μ ðxÞ ¼ ds: ð30Þ 2πi −∞i Γðμ þ s þ 1Þ2 Γð2μ þ s þ 1Þ The poles of Γð−sÞ are on the right of the contour, while those of Γð2μ þ 2s þ 1Þ are on the left. After using the duplication formula
of gamma function (Eq. (6.1.18) of [6]) the integral is evaluated by change of the order of integrations. The integration over x gives 1 Z ∞i ðα þ 2μ þ 2sÞ Γð−sÞΓ μ þ s þ 1 2 h
i X¼ 8π 1=2 i −∞i Γðμ þ s þ 1ÞΓð2μ þ s þ 1Þ sin π α þ μ þ s 2 1 −ðα=2Þ−μ−sþ1 − 2 ds: ð31Þ z By swinging round the contour in (31) so as to enclose the poles on the right, one can apply the residues theorem. Since the gamma function has no zero, the singularities of the integrand of (31) inside the contour are the poles of Γð−sÞ and the zeros of the sine in the denominator : s¼ n and s þ α=2 þ μ ¼ n with n≥0 an integer. We suppose that α is different from an even integer so that the two series of poles are distinct for integer μ. The residue of Γð−sÞ at s¼n is ð−1Þnþ1 =n!. Let X ¼ X 1 þ X 2 with X1 the contribution of the set of poles s ¼n and X2 the contribution of the set of poles s þ α=2 þ μ ¼ n. The infinite sum over residues at s¼n gives after some calculations for X1 1 α −α=2−μþ1 Γ μ þ pffiffiffi Γ 2þμþ1 1 π 2 − 2 X1 ¼ 2 z sin π 2α þ μ 1 α α ~ ; þ μ þ 1; μ þ 1; 2μ þ 1; þ μ; −z2 ; 2F 3 μ þ 2 2 2 with the regularized hypergeometric function defined by Eq. (28). Now one can verify with the help of Eq. (1.4.1.18) of [16], that both members of 1 α α 2 ~ ; þ μ þ 1; μ þ 1; 2μ þ 1; þ μ; −z F μ þ 2 3 2 2 2 pffiffiffi −2μ π z J μ ðzÞððα þ 2μÞJ μ ðzÞ−2zJ μþ1 ðzÞÞ ¼ 2Γ μ þ 12 Γ 2α þ μ þ 1 have the same power series expansion in the variable z. For the case of interest, μ ¼ m and z ¼ zn;m and therefore the factor J μ ðzÞ gives zero, and thus X 1 ¼ 0. It remains to calculate the contribution X2 to X of the series of poles −s−ðα=2Þ−μ þ 1 ¼ −n. Using the same procedures as described for X1, one obtains pffiffiffi 3−α πΓ 2 3 α α α α α 2 F~ 3 2; − ; 2− ; − −μ þ 2; μ− þ 2; −z2 ; X2 ¼ 2 2 2 2 2 2 sin π − 2 −μ Eq. (27) is obtained. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13] [14] [15] [16]
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