Appendix A Time Reversal Invariance Let us briefly discuss the time reversal property. In Chap. 4, we introduced the scattering and transfer matrices S and M. We have seen also that in a scattering process as the one shown in Fig. A.1, these matrices satisfy the following relations u ðx Þ u ðx Þ u ðx Þ u ðx Þ ol 1 ¼ S il 1 or 2 ¼ M il 1 ; ðA:1Þ uorðx2Þ uirðx2Þ uirðx2Þ uolðx1Þ the matrix S relates the incoming waves with the outgoing ones, while the matrix M connects the state vector at x1 with the state vector at x2. If we take the complex conjugate of the previous relations, we have à à à à uolðx1Þ Ã uilðx1Þ uorðx2Þ Ã uilðx1Þ Ã ¼ S à à ¼ M à : ðA:2Þ uorðx2Þ uirðx2Þ uirðx2Þ uolðx1Þ Let us now consider the time inversion operator T. The action of this operator changes the sign of time, thus the physical quantities depending linearly on time as the velocity and the wave number k ¼ p=h, change sign. Under the time inversion operation, particles that were moving to the right, move towards the left and vice versa. A wave function like eikx, under T, gets transformed into eÀikx. This is equivalent to complex conjugate. Therefore TuðxÞ¼uÃðxÞ: ðA:3Þ If the time inversion operator T acts on the system of Fig. A.1, we will have à à Tuilðx1Þ¼uilðx1Þ¼/olðx1Þ; Tuirðx2Þ¼uirðx2Þ¼/orðx2Þ; ðA:4Þ Ã Ã Tuolðx1Þ¼uolðx1Þ¼/ilðx1Þ; Tuorðx2Þ¼uorðx2Þ¼/irðx2Þ: ðA:5Þ As one can see in Fig. A.2, the wave functions that were incoming functions transform into outgoing functions and viceversa. Reversing time in the scattering system implies reversing time every where. This includes the interactions that define the scattering and transfer matrices. If the system remains invariant under time inversion, the matrices S and M should also remain invariant. Therefore, if a P. Pereyra, Fundamentals of Quantum Physics, Undergraduate Lecture Notes in Physics, 305 DOI: 10.1007/978-3-642-29378-8, Ó Springer-Verlag Berlin Heidelberg 2012 306 Appendix A: Time Reversal Invariance Fig. A.1 The incident wave V(x) ϕ ϕ or functions, and the reflected il and transmitted wave func- tions by a dispersion potential VðxÞ ϕ ϕ ol ir x1 x2 Fig. A.2 The incident wave functions, and the wave functions that are reflected and transmitted by the dis- persion potential VðxÞ system is time reversal invariant, we must have / ðx Þ / ðx Þ / ðx Þ / ðx Þ ol 1 ¼ S il 1 or 2 ¼ M il 1 : ðA:6Þ /orðx2Þ /irðx2Þ /irðx2Þ /olðx1Þ with S and M that we had in (A.1). These relations, written in terms of the wave functions u, take the form à à à à uilðx1Þ uolðx1Þ uirðx2Þ uolðx1Þ Ã ¼ S à à ¼ M à : ðA:7Þ uirðx2Þ uorðx2Þ uorðx2Þ uilðx1Þ If we multiply the first of these equations, from the left by SÀ1, and we observe that à à à à uirðx2Þ 01 uorðx2Þ uolðx1Þ 01 uilðx1Þ Ã ¼ à and à ¼ à ; uorðx2Þ 10 uirðx2Þ uilðx1Þ 10 uolðx1Þ ðA:8Þ it becomes clear, after comparing with (A.2), that 01 01 SÀ1 ¼ Sà and M ¼ MÃ: ðA:9Þ 10 10 For the transfer matrix, this means that dc aà bà ¼ : ðA:10Þ ba cà dà Therefore we conclude that, for systems that are time reversal invariant, the transfer matrices have the structure Appendix A: Time Reversal Invariance 307 ab M ¼ : ðA:11Þ bà aà Notice that this holds when the wave number k on the right side is the same as on the left side of the scattering region. If that is not the case, these structures may change but (A.9) still holds. Appendix B Laguerre’s Polynomials With respect to the Laguerre polynomials, there is in the literature such a great diversity of definitions and notations that we feel justified to include this appendix, and deduce some basic relations and properties. The polynomials that Laguerre 0 studied are the polynomials LmðxÞ¼Lm ðxÞ defined by 1 dm L ðxÞ¼ ex xmeÀx; ðB:1Þ m m! dxm or as the expansion coefficients of the generating function no xt X1 G ðt; xÞ¼exp ¼ð1 À tÞ L ðxÞtm; ðB:2Þ L t 1 m À m¼0 and they are solutions of the differential equation d2y 1 dy m þ À 1 þ y ¼ 0: ðB:3Þ dx2 x dx x The Laguerre polynomials of lower orders are L0ðxÞ¼1; 1 À Á L ðxÞ¼ 1 À x ; 1 1! 1 À Á L ðxÞ¼ 1 À 4x þ x2 ; 2 2! 1 À Á L ðxÞ¼ 6 À 18x þ 9x2 À x3 ; 3 3! 1 À Á L ðxÞ¼ 24 À 96x þ 72x2 À 16x3 þ x4 : 4 4! Related to these polynomials we have the generalized (or associated) Laguerre l polynomials LmÀlðxÞ defined by P. Pereyra, Fundamentals of Quantum Physics, Undergraduate Lecture Notes in Physics, 309 DOI: 10.1007/978-3-642-29378-8, Ó Springer-Verlag Berlin Heidelberg 2012 310 Appendix B: Laguerre’s Polynomials 1 dk dmþk dk LkðxÞ¼ðÀ1Þk ex xmþkeÀx ¼ðÀ1Þk L ðxÞ: ðB:4Þ m ðm þ kÞ! dxk dxmþk dxk mþk These polynomials are also the coefficients of a generating function, and one can easily verify that a particular solution of the differential equation d2y dy k þ 1 x mðm À kÞ x þðk þ 1 À 2mÞ þ n þ À þ y ¼ 0 ðB:5Þ dx2 dx 2 4 x is Àx=2 m k yn ¼ e x Ln: ðB:6Þ In this way, if m ¼ k=2, the function Àx=2 k=2 k yn ¼ e x Ln; ðB:7Þ is a solution of k þ 1 x k2 xy00 þ y0 þ n þ À À y ¼ 0; ðB:8Þ n n 2 4 4x n that can be written as d d k þ 1 x k2 x y þ n þ À À y ¼ 0: ðB:9Þ dx dx n 2 4 4x n k A very useful representation of LmðxÞ is the Rodrigues formula Xm 1 dm ðm þ kÞ! xr LkðxÞ¼ exxÀk xmþkeÀx ¼ ð1Þr : ðB:10Þ m m! dxm m r !r! r k ! r¼0 ð À Þ ð þ Þ Some generalized Laguerre polynomials are: 0 L0ðxÞ¼1; 0 1 L1ðxÞ¼1 À x; L1ðxÞ¼2 À x; 1À Á 1À Á 1À Á L0ðxÞ¼ 2À4xþ x2 ; L1ðxÞ¼ 6À 6xþ x2 ; L2ðxÞ¼ 12À 8xþ x2 : 2 2 2 2 2 2 Like other orthogonal polynomials, the Laguerre polynomials satisfy a three- term recurrence relation. The relation that the Laguerre polynomials satisfy is k k k ðn þ 1ÞLnþ1ðxÞþ ðx À k À 2n À 1ÞLnðxÞþ ðn þ kÞLnÀ1ðxÞ¼ 0 n ¼ 1; 2; ... ðB:11Þ This recurrence relation can also be written as Appendix B: Laguerre’s Polynomials 311 k k k nLnðxÞþ ðx À k À 2n þ 1ÞLnÀ1ðxÞþ ðn þ k À 1ÞLnÀ2ðxÞ¼ 0 n ¼ 1; 2; ... ðB:12Þ With the help of these definitions and relations we will obtain two very useful results for the evaluation of normalization integrals and the calculation of expectation values for the Hydrogen atom. Orthogonality of Laguerre’s Polynomials If we multiply equation (B.9)byym, subtract the same equation (with the indices n and m interchanged), and integrate from 0 to 1, we have Z Z 1 1 dy 0 dy 0 ym xyn À yn xym dx þðn À mÞ ymyndx ¼ 0; 0 dx dx Z 0 1 ðB:13Þ 0 0 1 ðymxyn À ynxymÞ 0 þðn À mÞ ymyndx ¼ 0: 0 Àx=2 k=2 k k 1 x Àk dn nþk Àx As yn ¼ e x Ln, and as LnðxÞ¼n ! e x dxn x e is a polynomial of degree n, the first term of the previous equation is zero at x ¼ 0ifk [ À 1. Thus Z 1 Àx k k k ðn À mÞ e x LnLmdx ¼ 0: ðB:14Þ 0 From here we conclude that Z 1 Àx k k k e x LnLmdx ¼ 0 if n 6¼ m: 0 Normalization of Laguerre’s Polynomials To obtain the normalization constant of the Laguerre polynomials we need the recurrence relations (B.11) and (B.12). If we consider the relation (B.11) k k multiplied by LnÀ1ðxÞ, subtract (B.12) multiplied by LnðxÞ, multiply the result by eÀxxk and integrate from 0 an 1 (using the orthogonality condition), we will obtain Z Z 1 Âà 1 Âà Àx k k 2 n þ k Àx k k 2 e x LnðxÞ dx ¼ e x LnÀ1ðxÞ dx: ðB:16Þ 0 n 0 One can repeat this relation n À 1 times to obtain 312 Appendix B: Laguerre’s Polynomials Z Z 1 Âà 1 Âà Àx k k 2 n þ k n À 1 þ k 2 þ k Àx k k 2 e x LnðxÞ dx ¼ ÁÁÁ e x L1ðxÞ dx: ðB:17Þ 0 n n À 1 2 0 k Since L1ðxÞ¼1 þ k À x it is possible to show that Z 1 Âà Àx k k 2 e x L1ðxÞ dx ¼ðk þ 1Þ! ðB:18Þ 0 Therefore Z 1 Âà Àx k k 2 ðn þ kÞ! e x LnðxÞ dx ¼ for n ¼ 2; 3; ... ðB:19Þ 0 n! It is possible to explicitly show that this formula is also valid for n ¼ 0 and n ¼ 1. Thus Z 1 Âà Àx k k 2 ðn þ kÞ! e x LnðxÞ dx ¼ for n ¼ 0; 1; 2; 3; ... ðB:20Þ 0 n! This integral and the orthogonality condition are summarized in the following equation Z 1 m! k Àx k k x e LmðxÞLm0 ðxÞdx ¼ dm;m0 : ðB:21Þ ðm þ kÞ! 0 Finally, if we consider the recurrence relation (B.11), we multiply it by k Àx k x e LnðxÞ and integrate from 0 to 1 we get: Z 1 kþ1 Àx k k ðn þ kÞ! x e LnðxÞLnðxÞdx ¼ðk þ 2n þ 1Þ : ðB:22Þ 0 n! References 1.