First-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries

Transcription

1 c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische Physik 1, Universität Stuttgart, 755 Stuttgart, GERMANY Received 17 March 26) In the reference [B. Grémaud, Phys. Rev. E 65, )] first-order corrections to Gutzwier s trace formua for systems with a smooth potentia were presented. Here we present an extension of that theory to systems with discrete symmetries. We appy the method to the two-dimensiona hydrogen atom in a uniform magnetic fied and expoit the C 4v -symmetry in the cacuation of the correction terms. The numerica resuts for the semicassica vaues are compared with vaues extracted from exact quantum mechanica cacuations. The comparison shows a very good agreement. PACS numbers: 3.65.Sq, 5.45.Mt Keywords: semicassica theories, Gutzwier s trace formua, expansions, discrete symmetries, diamagnetic hydrogen atom 1. Introduction Gutzwier s trace formua [1] provides a semicassica approximation of the quantum eve density in terms of cassica periodic orbits. In a systematic expansion of the eve density in powers of it can be considered as the eading order. Higher orders of this asymptotic expansion have been deveoped in severa studies [2 4], but for a ong time were ony tested for biiard systems, i.e., systems with hard was instead of smooth potentias. By extending an expansion which was derived by Gaspard et a. [3, 4], Grémaud [5] obtained corrections to Gutzwier s trace formua for quantum systems with a smooth potentia. However, in [5] no symmetries of the Hamitonian were considered. Preiminary resuts for the diamagnetic hydrogen atom, which has discrete and continuous symmetries were pubished in [6] but the infuence of the symmetries on the numerica cacuations which are required to obtain the correction terms was not discussed. E-mai: Hoger.Cartarius@itp1.uni-stuttgart.de In this paper we extend the theory presented in [5] and investigate systems with discrete symmetries. The main focus wi be on the cassica Green s function, which is an essentia part of the correction terms and has the most compicated symmetry properties. If there are discrete symmetries, the eigenstates of the quantum system spit up into severa subspaces. In these subspaces, cassica orbits which are not periodic without a symmetry transformation contribute to the eve density. We wi show how discrete symmetries of the Hamitonian have to be taken into account for the cacuation of the first-order corrections, and appy the method to the twodimensiona diamagnetic hydrogen atom, which has a C 4v symmetry but no continuous symmetry. A continuous symmetry woud require a further study. The corrections wi be cacuated for seected periodic orbits taking into account that discrete symmetry. The semicassica vaues wi be compared with the anaysis of exact quantum cacuations. The agreement between the resuts of both methods turns out to be very good. The outine of the paper is as foows. In section 2 we wi briefy present the most important 24 Noninear Phenomena in Compex Systems, 9:3 26)

2 First-Order Corrections to Gutzwier s Trace Formua terms of the corrections for systems without discrete symmetries. In section 3, we wi introduce the hydrogen atom in a uniform magnetic fied with the aspects reevant for the cacuation of the correction terms. Then we wi give an introduction to the extension to discrete symmetries, cacuate resuts for the two-dimensiona hydrogen atom and compare them with exact quantum mechanica cacuations in section Semicassica approximation of the quantum eve density Grémaud presented a semicassica approximation of the quantum Green s function, which incuded the first-order correction. The resut reads [5]: G E) = 1 i [ i Sc) ] T p exp T ) iπ 2 µ det m T ) 1) { 1 + i C1 T ) + C T E 1 T ) ) + O 2)}, 1) where the eading order, beonging to the 1 in the cury brackets, is known from Gutzwier s trace formua [1]. In this expression, S c) T ) is the reduced action and µ represents the Masov index. The stabiity matrix is denoted by m and T stands for the time period with T p for the primitive periodic orbit. The first-order correction consists of two terms. The first has the expicit form C 1 T, t ) = 1 T dt V,i1 i 8 2 i 3 i 4 G i1 i 2 t, t; t )G i3 i 4 t, t; t ) + 1 T T dt dt V,i1 i 24 2 i 3 V,j1 j 2 j 3 3G i1 i 2 t, t; t )G i3 j 1 t, t ; t )G j2 j 3 t, t ; t ) + 2G i1 j 1 t, t ; t )G i2 j 2 t, t ; t )G i3 j 3 t, t ; t ) ) + V,jt ) T 2 q c) dt V t ) 2,i1 i 2 i 3 G ji1, t; t )G i2 i 3 t, t; t ), 2) where the V,i1...i n are derivatives of the potentia evauated at the point q c) t) on the cassica orbit. G ij are the components of the cassica Green s function, which is a soution of the inearized equation of motion 1 d2 dt 2 2 V q t)) ) c) Gt, t ) = 1δt t ) q q 3) and fufis the boundary conditions G, t ) = GT, t ), P t G, t ) = P t GT, t ) =, Q t G, t ) = Q t GT, t ), ij 4) with the projection operators ) qt ) qt ) P t ) ij = qt ) 2 = q it ) q j t ) qt ) 2, 5) Noninear Phenomena in Compex Systems Vo. 9, No. 3, 26

3 242 H. Cartarius et a. and Q t = 1 P t, where 1 is the f-dimensiona unity matrix. At time t = t, the additiona condition ) ) ) G t, t ) G + t, t ) G t, t = ) G + t, t +, 6) ) 1 with G t, t ) = Gt, t ) for t t, and G + t, t ) = Gt, t ) for t t T has to be fufied. The second contribution to the correction is given by: C T E 1 T ) = 1 2W 2) T ) ) C 1) T ) 2 + C 2) T ) 1 W 3) T )C 1) T ) 2 W 2) 1 T ) 2 8 W 4) T ) W 2) T ) W 3) T ) 2 W 2) T ), 7) 3 where W n) W n) and C n) are the derivatives T ) = n W c) T ) T n, C n) T ) = n C c) T ) T n of the action W c) T ) and of the ogarithm of the ampitude of the trace of the propagator ) T p C T ) = n, 8) E T det m T ) 1) respectivey. A quantities required for the corrections can be obtained by the soution of ordinary differentia equations as is discussed in [5] in detai. The number of the differentia equations can become very arge. For exampe, in a twodimensiona system the correction term C 1 eads to 97 ordinary differentia equations. 3. The hydrogen atom in a uniform magnetic fied The diamagnetic hydrogen atom was often used as an exampe of a quantum system whose cassica dynamics is chaotic see e.g. [7] or [8] for an overview). As a rea physica system it was the topic of studies in experimenta physics [9, 1]. It has even been used for the numerica test of the correction terms C 1 and C1 T E [5]. Because of its simpe scaing property, which is aso fufied for the corrections, it is possibe to compare the semicassica resuts for individua orbits with exact quantum mechanica cacuations The two-dimensiona hydrogen atom The hydrogen atom with three degrees of freedom in a uniform magnetic fied has a continuous symmetry, namey the rotationa invariance around the magnetic fied axis, which can be used to formuate the dynamics in a two-dimensiona coordinate system. The cacuation of the cassica vaues in the two-dimensiona coordinate system works very we for the eading order but it eads to new difficuties for the first-order correction [6, 11]. In the foowing sections we ony ook at the two-dimensiona hydrogen atom for the cacuation of the first-order corrections where no continuous symmetry does exist. For a uniform magnetic fied represented by γ, the cassica Hamitonian of the two- Noninear Phenomena in Compex Systems Vo. 9, No. 3, 26

4 First-Order Corrections to Gutzwier s Trace Formua dimensiona diamagnetic hydrogen atom in atomic units and semiparaboic coordinates [7, 12] is given by a) b) ν H = 1 2 p2 µ p2 ν ɛ µ 2 + ν 2) ν µ µ2 ν 2 µ 2 + ν 2) = 2, 9) where ɛ is the scaed energy ɛ = γ 2/3 E and µ, ν) are treated as Cartesian coordinates which can aso be negative contrary to the reguarization of the three-dimensiona hydrogen atom. The Schrödinger equation associated with the cassica Hamitonian 9) is given by: { 2 + ɛµ 2 + ν 2 ) 1 } 8 µ2 ν 2 µ 2 + ν 2 ) ψµ, ν) { = γ 2/3 1 )} 2 2 µ ν 2 ψµ, ν). 1) Note that γ 1/3 takes the pace of, which is equa to one in atomic units. It is often caed effective. The potentia of the two-dimensiona diamagnetic hydrogen atom in semiparaboic coordinates see equation 9)) V µ, ν) = ɛ µ 2 + ν 2) µ2 ν 2 µ 2 + ν 2) 11) has a C 4v -symmetry. This symmetry can be seen in figure 1 a), in which a few equipotentia contours of the potentia 11) are potted. One has to integrate differentia equations aong the periodic orbits of the cassica system in order to cacuate the required cassica vaues. It is known that, because of the symmetry, the cassica cacuations can be reduced to a fundamenta domain [13], which is shown in figure 1 b). It consists of one eighth of the fu coordinate pane. The restriction to the fundamenta domain and the introduction of symmetry operations ead to new periodic orbits, which are ony periodic if µ FIG. 1. a) Equipotentia contours of the potentia 11) for different scaed energies ɛ. b) The shadowed area in the coordinate system marks the fundamenta domain. one expoits the symmetry properties of the system. The periodic continuation can be achieved by refections at the borders of the fundamenta domain. An exampe is the orbit +, which is shown in figure 2 a). Starting for exampe on the µ-axis the orbit has to be refected at the ange bisector. After returning to the µ-axis a second refection this time at the µ-axis) is necessary such that the momenta at the initia and fina points agree. For practica purposes it is often easier not to restrict the cacuation to the fundamenta domain but to find a periodic continuation of the orbit by mapping the fina point of the orbit on its initia point via a symmetry transformation from C 4v, namey rotations by mutipes of 9 degrees, refections at the coordinate axes, and refections at the ange bisectors. For exampe, figure 2 b) shows the orbit from figure 2 a) in the case where it is not restricted to the fundamenta domain. The periodic continuation is done by a cockwise rotation by 9 degrees. This method requires ony one symmetry operation to render an orbit periodic, and is in genera easier to impement as the restriction to the fundamenta domain. Noninear Phenomena in Compex Systems Vo. 9, No. 3, 26

5 244 H. Cartarius et a. a) ν µ b) ν FIG. 2. The periodic orbit + in semiparaboic coordinates for a scaed energy of ɛ =.1. The dot marks the nuceus at the origin. Both orbits shown correspond to the singe repetition in the fundamenta domain Symmetry properties of the wave functions and cacuation of the quantum spectra Because of the C 4v -symmetry of the Hamitonian 1), the eigenfunctions of the system spit up into subspaces beonging to a representation of the symmetry group. The symmetry group C 4v has four one-dimensiona representations, namey A 1, A 2, B 1 and B 2, and a two-dimensiona representation, which is caed E. Appying a symmetry eement from C 4v to a wave function with symmetry E eads in genera to a inear combination of two energeticay degenerate) wave functions. At fixed scaed energy ɛ, equation 1) can be considered as a generaized eigenvaue probem in the variabe γ 2/3. The eigenvaues can be cacuated by diagonaizing a matrix representation of the Hamitonian in a compete basis set with the Lanczos agorithm [5, 14]. Using ony the wave functions of one of the subspaces, which means that the bock diagona form of the Hamitonian is expoited, reduces the dimension of the eigenvaue probem and eads to separate spectra for each of the subspaces, which is necessary for the anaysis of the infuence of the symmetry. The anaysis of the quantum spectra can be µ performed with the harmonic inversion method [8, 15]. It provides an efficient possibiity to extract the ampitude of Gutzwier s trace formua, the action S c) T ) and the sum C = C 1 + C T E 1 12) of the two first-order correction terms for individua orbits. Detais can be found in [5] or [11]. 4. Discrete symmetries and the correction terms 4.1. Symmetry transformations in the cacuation of the correction terms The cassica quantities which contribute to the semicassica eve density have to be cacuated for periodic orbits and, as we have seen in section 3, some orbits are ony periodic after the appication of a symmetry operation during or at the end of the integration aong the cassica orbit. It is we known which symmetry operations refections or rotations of vectors) have to be impemented for the cacuation of the phase space coordinates but, as was mentioned in section 2, one has to sove a arge number of additiona differentia equations for a new set of coordinates if one wants to cacuate the first-order correction. These quantities are the monodromy matrix Mt, T ), which can be obtained from the inearized Hamitonian equations of motion 2 H Ṁt, T ) = Σ X X Mt, T ), 13) the derivatives of the phase space coordinates of the orbit X 1) t, T ), X 2) t, T ), X 3) t, T ) with respect to the time period T, and the derivatives M 1) t, T ) and M 2) t, T ). One has to search for the correct transformations of the additiona coordinates. In particuar, the cassica Green s function, which in our case is a 2 2 matrix, has compi- Noninear Phenomena in Compex Systems Vo. 9, No. 3, 26

6 First-Order Corrections to Gutzwier s Trace Formua cated symmetry properties, as one can see in figure 3. The function potted in figure 3 beongs to the orbit which corresponds in the fundamenta domain to four repetitions of the orbit shown in figure 2. The symmetry of the orbit is not obvious in the eements of the Green s function. This behaviour is due to the boundary conditions see equations 4) and 6)) which are different for an orbit and its mutipe repetitions and affect the whoe Green s function. The symmetry transformations have to be appied during the integration of the inearized equations of motion 13). Each component of the monodromy matrix transforms the same way as the corresponding eement of the phase space vector [11]. If one appies the symmetry operations to the eements of the monodromy matrix at every time when one transforms the coordinates of the orbit during the integration, a boundary conditions of the cassica Green s function are fufied because the monodromy matrix is the ony part that foows from the soution of a differentia equation. A other vaues foow from formuas which incude aready the boundary conditions and are not affected by the symmetry transformations during the integration. In order to iustrate the infuence of the symmetry transformation, the eements of the cassica Green s function of the orbit in the fundamenta domain is shown in figure 4 a). The Green s function was rotated in such a way that the G 1j τ, τ ) components represent the direction aong the cassica orbit at the initia point, which is marked by a cross in the figure. One recognizes the discontinuities at the positions of the refection which appear due to the change in the meaning of the components accordingy to the change of the variabes by using the symmetry transformation. In figure 4 a), a boundary conditions which were mentioned in section 2 are fufied. At the fina point of the orbit, a components have the same vaues as at the initia point. The G ij τ,τ, ) G 11 G 12 G 21 G τ/t FIG. 3. The picture shows the cassica Green s function of the orbit ++++, which corresponds to four repetitions of the orbit + in the fundamenta domain, for τ =.5 T. In the eements of the Green s function, no obvious symmetry is visibe. The eements of the Green s function are chosen in such a way that the components G 1j correspond to the direction aong the orbit at the initia point, which is marked by a cross. condition G 1j, τ ) = G 1j T, τ ) =, which is expected from equation 4), is aso fufied. The discontinuities in the eements G 2j, which appear due to equation 6), are ceary visibe. Figure 4 b) shows the cassica Green s function for the same orbit, which is not restricted to the fundamenta domain but cosed by a rotation of its fina point. Therefore, the ony discontinuity appears at the end point of the orbit. To show this, the first and the ast point after the symmetry transformation) of each component are marked with the same symbo. Again, the boundary conditions are fufied. Under symmetry operations even a further variabes of the differentia equations which contribute to the first-order correction transform in the same way as the corresponding phase space coordinates of the orbit [11]. With this knowedge it is possibe to cacuate the correction terms for orbits which require a symmetry operation to become periodic. Noninear Phenomena in Compex Systems Vo. 9, No. 3, 26

7 246 H. Cartarius et a. a) G ij τ,τ, ) G 11 G 12 G 21 G τ/t a) A b) A S/2π S/2π b) G ij τ,τ, ) G 11 G 12 G 21 G τ/t FIG. 4. The cassica Green s function of the orbit + is shown in two cases for τ =.55 T. a) Orbit in the fundamenta domain. b) The same orbit in the fu pane of semiparaboic coordinates, which becomes periodic by the appication of a rotation of the fina point. The eements of the Green s function are chosen as described in the text of figure Exampes for some orbits In the different subspaces of the quantum wave functions, the individua ampitudes A of Gutzwier s trace formua can be found with a prefactor. This prefactor is given by the character of the eement from C 4v which provides the periodic continuation of the orbit in the representation of the corresponding subspace [13]. As an exampe one can see the Fourier transformation of the quantum spectrum which incudes a eigenvaues from the subspace A 1 in figure 5 a). In FIG. 5. a) The Fourier transform of the eigenvaues from subspace A 1 incudes a orbits which are periodic after the appication of a symmetry transformation. S: action, A: ampitude of Gutzwier s trace formua. b) If one ooks at the Fourier transformation of the spectrum which incudes the eigenvaues of a subspaces, one can ony find the orbits which are periodic in the fu pane of semiparaboic coordinates. this subspace a orbits contribute with a prefactor of 1 independenty of the symmetry eement required to find the periodic continuation. The orbit + and some of its mutipe repetitions are marked. If one uses the quantum spectrum which consists of the eigenvaues of a subspaces, ony the orbits which are periodic in the fu pane of the semiparaboic coordinates appear see figure 5 b)). For exampe, the orbit ++++ is periodic. One possibiity to obtain the first-order terms from the quantum spectrum is to subtract the eading order of the individua orbits from the whoe spectrum. If one uses A with its prefactor for this method, the harmonic inversion of the quantum spectrum eads to the correct firstorder terms C. Tabe 1 shows the resuts for the orbit + in a one-dimensiona subspaces of C 4v, which are compared with the cassicay cacuated vaue. The cassica resuts for the correction terms of a few orbits which require a symmetry opera- Noninear Phenomena in Compex Systems Vo. 9, No. 3, 26

8 First-Order Corrections to Gutzwier s Trace Formua Tabe 1. Moduus C qm) and phase argc qm) ) of the sum C of the correction terms see equation 12)) of the orbit + in different subspaces obtained by the anaysis of the quantum spectrum. They are in good agreement with the cassica vaue C c) =.9443 as is shown in the ast coumn. Subspace C qm) arg C qm) ) qm) C C c) 1 A π.13 A π.5 B π.13 B π.1 tion to become periodic, are given in tabe 2. Figure 6 shows the comparison of these resuts with vaues C qm) extracted from exact quantum cacuations. Ony the eigenvaues of the subspace beonging to the representation A 1 were used for the harmonic inversion of the quantum spectrum. As one can see, the agreement of the ampitudes from cassica and quantum cacuations is very good. In most cases the differences are ony of the order 1 3. If one compares the phases of the compex) quantum mechanicay cacuated correction terms, one sees that they reproduce the correct signs of the cassica vaues. The differences are typicay of the order Concusion and Outook In this paper we have studied the infuence of discrete symmetries on the correction terms to Gutzwier s trace formua. The main focus was on the symmetry properties of the cassica Green s function. A compete anaysis of a parts of the correction terms can be found in [11]. We appied symmetry transformations which made possibe the cacuation of the correction terms for a number of orbits which coud not be incuded without symmetry operations. Nevertheess, these orbits contribute to some of the sub- Tabe 2. The first-order corrections C see equation 12)) for some orbits which are ony periodic with a symmetry transformation obtained by cassica cacuations. Orbit C 1 C1 T E C c) C h c) qm) S /2π FIG. 6. The moduus C qm) of the first-order corrections obtained by harmonic inversion of the quantum spectrum fied circes) is compared with the cassica vaue C c) open squares). One can see that the agreement is very good. spaces of the quantum spectrum. The cassica resuts for the correction terms coud be compared with vaues which were extracted from exact quantum cacuations. An exceent agreement between the resuts of both methods was found. In spite of this success, it must be noted that before semicassica spectra incuding first-order corrections can be cacuated over the compete Noninear Phenomena in Compex Systems Vo. 9, No. 3, 26

9 248 H. Cartarius et a. spectrum a number of probems sti remain to be soved. On the one hand, as was aready mentioned in [5], the correction term C 1 cannot be cacuated in the form presented for orbits which have a turning point, but the incusion of these orbits is essentia for the correction of the eve density. The correction term C 1 diverges for orbits with turning points. On the other hand, besides the successfuy impemented discrete symmetries, physica systems often have a continuous symmetry. For exampe, it is necessary to take into account the rotationa invariance around the magnetic fied axis for the compete cacuation of a first-order corrections to the semicassica eve density of the three-dimensiona diamagnetic hydrogen atom. The probem can be considered as an additiona centrifuga term in the potentia. This term eads to diverging integras if the course of the orbit is not changed. A reguarization of these integras suggested in [6] eads to good resuts for a few individua orbits, however, a mathematica justification is acking. References [1] M. C. Gutzwier. Chaos in Cassica and Quantum Mechanics Springer-Verag, New York, 199). [2] G. Vattay and P. E. Rosenqvist. Phys. Rev. Lett. 76, ). [3] P. Gaspard and D. Aonso. Phys. Rev. A 47, R ). [4] P. Gaspard, D. Aonso, and I. Burghardt. Adv. Chem. Phys. 9, ). [5] B. Grémaud. Phys. Rev. E 65, ). [6] B. Grémaud. Phys. Rev. E 72, ). [7] H. Friedrich and D. Wintgen. Phys. Rep. 183, ). [8] J. Main. Phys. Rep. 316, ). [9] A. Hoe, G. Wiebusch, J. Main, B. Hager, H. Rottke, and K. H. Wege. Phys. Rev. Lett. 56, ). [1] J. Main, G. Wiebusch, A. Hoe, and K. H. Wege. Phys. Rev. Lett. 57, ). [11] H. Cartarius, J. Main, and G. Wunner. Annas of Physics, in press. Preprint: nin.cd/ ). [12] M. J. Engefied. Group Theory and the Couomb Probem Wiey-Interscience, New York, 1972). [13] P. Cvitanović and B. Eckhardt. Nonin. 6, ). [14] T. Ericsson and A. Ruhe. Mathematics of Computation 35, ). [15] J. Main, P. A. Dando, Dž. Bekić, and H. S. Tayor. J. Phys. A 33, ). Noninear Phenomena in Compex Systems Vo. 9, No. 3, 26