A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution is usully depicted by the rrow of unit length see Fig Figure : Grph of the Dirc distribution δx 2 The sifting property of the Dirc distribution my serve s nother possible definition: Let s suppose tht funciton fx is continnous over the intervl x, x 2 or tht it hs t most finite number of finite discontinuities over tht intervl Then fx δx x dx x x2 2 [ fx + fx + ], if x x, x 2, 2 fx+, if x x, 2 fx, if x x 2,, if x x, x 2 Of course, if the function is continuous, the first of the reltions 2 reduces to the form x2 x fx δx x dx fx, if x x, x 2, 2 which is the most frequently ppering form of the sifting property see Fig 2 2 Figure 2: Sifting property of the Dirc distribution 3 Very often the Dirc distribution is defined s the limit of the sequence of functions δ p x The function δ p x hve to stisfy two conditions: δx lim δ px
2 A DIRAC DISTRIBUTION δ p x lim δ p x dx nd lim 3 lim x δ p x In most cses the functions δ p x stisfy more severe conditions: δ p x dx nd lim δ px 3 A2 Exmples of functions δ p x Probbly the most obvious exmple of functions δ p x is δ p x p rect px see Fig 3 Evidently, these functions stisfy the conditions A3 Figure 3: Grph of the function δ p x p rect px b Another obvious exmple provide the functions δ p x p tri px 2 see Fig 4 Also these functions obviously stisfy the conditions A3 Figure 4: Grph of the function δ p x p tri px c An importnt exmple is the sequence of functions p δ p x π exp px2 3 see Fig 5 Let us show, tht lso these functions stisfy the conditions A3: p δ p x dx exp px 2 dx exp t 2 dt π π The integrl I exp t2 dt is evluted s follows: I 2 exp x 2 dx,
A DIRAC DISTRIBUTION 3 Figure 5: Grph of the function δ p x p/π exp px 2 so tht I 2 4 4 π π/2 exp x 2 dx exp y 2 dy 4 exp r 2 r dϕ dr 2π exp s ds π exp r 2 r dr exp [ x 2 + y 2 ] dx dy Hence I π The second condition A3 is stisfied s well: lim δ px p lim π exppx 2 2x 2 π lim exp px 2 p x x d Also the functions see Fig 6 stisfy the conditions A3: δ p x dx π δ p x π p dx + p 2 x 2 π lim δ px π lim p + p 2 x 2 p + p 2 x 2 4 dt + t 2 π rctg t π lim x 2px 2 t t x e In clcultions nd proofs of theorems bout the Fourier trnsform we often meet formlly different expressions of the function δ p x 2π p p exp±itx dt π p see Fig 7 The first of the conditions A3 is stisfied: cos tx dt π x p π px 5
4 A DIRAC DISTRIBUTION Figure 6: Grph of the function δ p x π p +p 2 x 2 Figure 7: Grph of the function δ p x πx δ p x dx π 2 π x sin y y dx π dy 2 π π 2 cf eg [2], 372, [3], 5225 The second condition A3 is however, not stisfied becuse the corresponding limit does not exist: If x, the function δ p x πx tkes the vlues from the intervl πx, πx, which does not depend on p The condition A3 is, of course, stisfied becuse lim x δ p x p π, nd hence δ p x lim lim x δ p x lim π lim πx p px x x A3 Properties of the Dirc distribution sin y y dy Let us denote by x n the roots of the eqution fx nd suppose tht f x n Then
A DIRAC DISTRIBUTION 5 δfx n δx x n f x n Proof: Let us choose the numbers n, b n, in neighbourhood of ech root x n in such wy tht n < x n < b n nd the function fx is monotonous in the intervl < n, b n > Then where gxδfx dx n I n, 2 I n b n n gxδfx dx By the substitution x x n f x n t we get I n f x n b n b n x nf x n n x nf x n n gxδx x n f x n dx 3 g t f x n + x n δt dt 4 If f x n <, the upper limit of integrtion is greter thn the lower one nd it is I n f x n n x nf x n b n x nf x n g t f x n + x n δt dt gx n f x n From 2 nd 3 nd from the sifting property of the Dirc distribution A2 it follows gxδfx dx n gx n f x n n b n f gxδx x n dx 5 x n n If f x n >, the reltion 5 follows imeditely from 2, 4 nd A2 Thus, the eqution is proved Importnt consequences of eqution re: δ x δx, 6 b It is ie where δx x δ δ sin π x π δx 2 2 δx 2 m x x, 7 δx m, 8 δx + δx + 9 2 d x, dx x δx dhx dx, Hx 2 + x x 2
6 A DIRAC DISTRIBUTION is the Heviside function Proof: 2 d x dx x 2 lim d 2 dx π rctg px π lim p + p 2 δx 3 x2 cf A24 c The following properties of the Dirc distribution re frequently used eg while evluting convolutions nd cross coreltions: fx δx f δx, 4 d c δx δx b dx δ b, c < min, b, d > mx, b 5 A4 The Dirc distribution obtined from complete system of orthonorml functions Interesting nd often useful expressions of the Dirc distribution cn be obtined from complete systems of orthogonl functions Let functions ψ n x, n being integers, form complete orthonorml system of functions on n intervl x, x + nd let x nd x be inner points of tht intervl Then ψnx ψ n x δx x, n where the summtion goes over ll n for which the orthonorml system {ψ n x} is complete Proof: To prove we shll demonstrte tht the left hnd side of eqution hs the sifting property of the Dirc distribution ie tht x+ x fx δx x dx fx, x+ x fx n ψ nx ψ n x dx fx 2 To prove 2 we expnd function fx into the system of orthonorml functions {ψ n x}, ie fx m c m ψ m x, 3 where c m x+ x fx ψ mx dx Now we insert the series 3 into the left hnd side of eqution 2, exchnge the order of integrtion nd ddition nd mke use of the condition of orthonormlity x+ x ψ nx ψ m x dx δ m,n : x+ x m c m ψ m x n x+ ψnx ψ n x dx c m ψ n x ψnx ψ m x dx m n x c m ψ n x δ m,n m n c m ψ m x fx m
A DIRAC DISTRIBUTION 7 Thus, we hve got the right hnd side of 2 nd the sttement is proved The functions ψ n x exp in2π x, n, ±, ±2, form the complete orthonorml system on ny intervl of the length nd hence lso on the intervl /2, /2 Therefore, ccording to it is n exp in2π x x δx x, x, x 2, 2 Every summnd of the infinite geometric series on the left hnd side of the foregoing reltion is periodic function with the period Consequently the sum of the series hs the sme period nd for ll x, x it holds n exp in2π x x m δx x m 4 Reltion 4 is importnt for the proof of the fct, tht the Fourier trnsform of the lttice function is proportionl to the lttice function chrcterizing the reciprocl lttice cf section 43 This is true for the lttices of ny dimensions N, N being integer N To be prepred for the proof in the spce of the dimension N 2 we denote the length of the intervl, so tht in eqution 4 my be both positive nd negtive The series t the left hnd side of 4 my be rewritten in vrious forms For exmple + 2 n cos n2π x x m δx x m 5 The series t the left hnd side of 4 is geometric series of the rtio exp i2π x x We my replce it by the limit n exp in2π x x lim By summing 2p + terms of the limit we get p n p exp in2π x x lim p n p exp in2π x x lim lim { { exp exp ip2π x x [ ]} exp i2p + 2π x x exp i2π x x ip2π x x [ exp i2p + π x x ] exp iπ x x ] [ ]} exp i2p + π x x exp [ i2p + π x x lim exp iπ x x ] sin [ 2p + π x x sin π x x exp iπ x x Hence lim sin [ ] 2p + π x x sin π x x m δ x x m 6
8 A DIRAC DISTRIBUTION A5 The Dirc distribution in E N In Crtesin coordintes From the fct tht D f xδ x x d N x f x, if x D D fx, x 2,, x N δx x δx 2 x 2 δx N x N dx dx 2 dx N fx, x 2,, x N, it follows tht in Crtesin coordintes Obviously N δ x x δx x δx 2 x 2 δx N x N δx k x k 2 k δ x δ x 3 N b Generl coordintes Let the Crtesin coordintes x, x 2,, x N be connected with generl coordintes y, y 2,, y N in E N by reltions x x y,, y N, x 2 x 2 y,, y N, x N x N y,, y N with the Jcobin Jy,, y N x x y N y,, x N y,, x N y N If x P, x P 2,, x P N then nd yp, y P 2,, y P N re coordintes of point P nd if JyP,, y P N, δx x P δx 2 x P 2 δx N x P N Jy,, y N δy y P δy 2 y P 2 δy N y P N 4 If, however, Jy P,, y P N nd the point P is specified by k coordintes yp, y P 2,, y P k tht mens tht N k coordintes y k+, y k+2,, y N re superfluous for the specifiction of the point P, we denote by J k y,, y k Jy,, y N dy k+ dy N the integrl over the N k superfluous coordintes nd it holds
A DIRAC DISTRIBUTION 9 δx x P δx 2 x P 2 δx N x P N J k y,, y k δy y P δy 2 y P 2 δy k y P k 5 c Exmple: Polr coordintes in E 2 x r cos ϕ, x 2 r sin ϕ, Jr, ϕ cos ϕ sin ϕ r sin ϕ r cos ϕ r i At points P r P, ϕ P, r P, it is δx x P δx 2 x P 2 δr rp δϕ ϕ P r ii At the point P r P the coordinte ϕ is superfluous nd so tht J r α+2π α r dϕ 2πr, δx δx 2 δr 2πr d Exmple: Sphericl coordintes in E 3 see Fig 8 x r sin ϑ cos ϕ, x 2 r sin ϑ sin ϕ, x 3 r cos ϑ, Figure 8: Sphericl coordintes Jr, ϑ, ϕ x r x 2 r x 3 r x ϑ x 2 ϑ x 3 ϑ x ϕ x 2 ϕ x 3 ϕ sin ϑ cos ϕ r cos ϑ cos ϕ r sin ϑ sin ϕ sin ϑ sin ϕ r cos ϑ sin ϕ r sin ϑ cos ϕ cos ϑ r sin ϑ r 2 sin ϑ i At the point P with coordintes r P, ϑ P, ϑ P π, ie with Jr P, ϑ P, ϕ P, it is
A DIRAC DISTRIBUTION δx x P δx 2 x P 2 δx 3 x P 3 δr rp δϑ ϑ P δϕ ϕ P r 2 sin ϑ ii At the point P with coordintes r P, ϑ P, or ϑ P π, the Jcobin Jr P, ϑ P, ϕ P nd Therefore J 2 r, ϑ 2π r 2 sin ϑ dϕ 2πr 2 sin ϑ δx δx 2 δx 3 x P 3 δr rp δϑ 2πr 2 sin ϑ iii At the point P with r P, it is Jr P, ϑ P, ϕ P nd Therefore J r π 2π r 2 sin ϑ dϕ dϑ 4πr 2 δx δx 2 δx 3 δr 4πr 2 e Exmple: Obligue coordintes importnt for the Fourier trnsform of lttices in E N, N 2 Let More explicitly this cn be rewritten s x i ik y k, det ik det A x y + + N y N, x N N y + + NN y N, or in the mtrix form x x N,, N N,, NN y y N, ie x A y As the Jcobin is x i y k ik, Jy,, y N,, N N,, NN det A nd
A DIRAC DISTRIBUTION ie δx x P δx N x P N det A δy y P δy N y P N, δ x x P det A δ y yp A6 Notes nd fetures δ p x p 2 + p 2 x 2 3/2 δ p x 22n 3 n!n 2! 2n 3! p π + p 2 x 2, n 2, 3, 2 n 2 3 δ p x p π px δy y y J m xyj m xy x dx 4 δ p x, y p π exp{ p[ exp x2 y 2 ]} 5 δ p x, y p2 π circ p x 2 + y 2 δ p x, y p 2 J p x 2 + y 2 4π p 7 x 2 + y 2 δxy 6 δx + δy x2 + y 2 8 References [] Dirc P A M: The Principles of Quntum Mechnics 4th edition At the Clrendon Press, Oxford 958, 5 [2] Grdshteyn I S, Ryzhik I M: Tble of Integrls, Series, nd Products Acdemic Press, New York nd London 994 [3] Abrmowitz M, Stegun I A: Hndbook of Mthemticl Functions Dover Publictions, Inc, New York 972