Witwit, M R M

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Indian J. Phys. 70B (3), 223-230 (1996)I J P  B- an international journalEnergy eigenvalues and splitting between even and odd energy levels of a double-well potentialM R M WitwitDepartment of Applied Mathematics, University of Hull, Hull, HU6 7RX, UKReceived J December 1995, accepted 11 April 1996Abstract : Accurate eigenvalues and splitting AE between the even and odd energy levels are calculated foi a double-well potential A) = -  Z*xl  + Ajc4, in the one-dimensionalsystem, using finite difference technique for various values of parameters Z2 *, AKeywords : Schrodinger equation, double-well potential, numerical results and discussion. PACS No. : 03 65.1. IntroductionThe double-well potential for one-dimensional quantum system has been the subject of intensive study in the last two decades. The literature on this topic, voluminous as it is, has recently been thoroughly referenced by several workers 11-10].There is a large number of physical and chemical systems whose properties can be studied (modelled) assuming that the potential function responsible for such properties is well represented by means of a function with two valleys and a barrier between them.Some of the problems to which the double-well potential model has been applied are ihe interpretation of the infrared spectra of the NH3 molecule, infrared and Raman spectra of hydrogen-bonded systems, inversion characteristics of isomers, structural phase transitions, formation of noble-gas monolayers on a graphite substrate, macroscopic quantum coherence in super conducting Joseohson devices, and so on 111,12]. In the theory of these problems, the most important characteristics are related to the separation between the two lowest-lying energy levels as it defines the tunnelling rate through the double-well barrier.© 19961ACS224 MRMWitwitThe fundamental system of interest to us here is the double-well potential described by Schrodinger equation- d2%X) + V(x.;Z*. A)1P(Jr) = EIPW, (1)where V j^ciZ^.A) = - Z 2.r2 +A x4. (2)The potential V[x\ Z2, A) consist of two potential wells separated by a barrier. If the barrier were impenetrable to a particle, there would be energy levels corresponding to the motion of the particle in one or other well, the same for both well. The fact that a passage through the barrier is possible, results in splitting of each of these levels into two neighbouring ones, corresponding to states in which the particle moves simultaneously in both wellsThe eigenvalue spectrum of the Schrodinger eq. (1) with the potential v (jc; Z 2, A) has the feature that the lower eigenvalues are closely bunched in one group if the vjalucs of the Z2 are sufficiently large. As Z2 increases, the magnitude of the splitting between the even and odd energy levels decreases. \In the present work, we have carried out a detailed investigation of the energy eigenvalues of double-well potential over a wide range of Z2 and A values. We have achieved our objective using the finite difference. The specific goal we set before us is, to study the splitting between the even and odd energy levels over a wide range of Z 2, A and nk. It is worth noting here that the finite difference technique is used in this work in the form of combination with a matrix diagonalization for numerical computations and transformed the Schrodinger equation into an algebraic eigenvalue problem^2. Formulation of the finite difference methodThe theory of the finite difference approach to find eigenvalues for the Schrodingci equation with potential given by cq. (2) usually starts from the central difference operator, which can be expressed as8 = ehPf2 __ e-hDfi = 2smh(/iD/2) (3)from eq. (3), it is easy to getwhere(4)( 5)Energy eigenvalues and splitting between even and odd energy etc 225Therefore, eq. (4) takes the following formh2D 2 = 8 2 - £ •  i !  58 12 + 90 “ 560 +g'Q3150407512 49351424772608 + 30828134425516 122551®150994944 + 86973087744 ’ (6)where h is the considered step length and 8 applies on function f'OO as follows. Our results show involving the fourth term in eq. (6). The accuracy improves by truncating (6) after the fourth term, giving an overall accuracy of nineteen significant digits.8 n x )  =  ¥^x + I -  ^ j r -  i h j ,  (7)8 2 tE(x)  = *F(x +  h) -  2  *F(x) +  V ( x - h ) , (8)8 4V (x )  = V (x  + 2h) -  4[ Y lx  + h) + ^ fx -A )]+ 6'F(x) + 'F (x -2 h ),  (9)8 2,F (x) = Y (x  + 3h) -  6[V (x + 2h) + 'T {x-2h )]+ 15fY ix  + h) + 'F (x -h )]  -  2 0 Y (x)  + ^ { x - 3 h ) ,  (10)5 8|P (*) = *r{x + 4h) _  8[»P(x + 3/i) + *P(jc -  3A)]+ 28[ >F(x + 2h) + 'F (x-2h )] -  S G ^ x  + h) + -  A)]+ 709'(x) + 'F (x -4 h ) .  (11)Although the solutions of (1) are explicitly defined in ( - « ,  it should be notedthat these solutions are either of even or odd parity i.e. *F(x ) = ± iF ( - x ) t so that the determination of *F{x) can be restricted to the region (0, +°°). Furthermore, one can suppose that the wavefunctions are restricted to obey the Dirichlet boundary condition W x) = 0 at some x value R. An acceptable /?-value will be guessed numerically.If wc replace D2 from eq. (6) and using eqs. (7-11), the Schrodinger equation Hans forms into an algebraic eigenvalue problem of the form[C m.n ~ G \ ] n n h )  =  0 , ( 1 2 )where G is proportional to E and its value G  = -5040 Eh2, 1 is the unity matrix. The form of matrix associated with eq. (12) can be expressed as226 M R M  Witwit'Co.o q .i Q,2 Q ,4 ^0.5 Q .6 Q ,7 - C o * ' ^ ( 0 )  "C..o C,., 1^.2 ^1.3 C,.4 C l.5 g ,.6 ^1.7 - C ln n h )^2.0 C2., ^2,2 ^2,1 C2.4 ^2.6 ^2.7 - C l * ¥ ( 2 h )^3,2 ^3.3 CM ^3.5 ^3,6 ^1.7 Y ^ h )CYo ^4,1 ^4,2 C4.4 C4,S C4.6 Q .7 ■ A - H'(Ah)^5.0 C5.1 C5.2 Cyy ^5,4 Cyt ^5.6 C$,1 - c s *Q ,0 Cft.l Q,2 Q .4 ^6 .5 Q .6 C6,7 ■ Q, •F (6A )V  ■ ■ - c m* , . W n h ) ,The values ol some coefficients are given below■^o.u ~ Yi) -  G , where yn = - 5040 ^  -  h^V(nh) ]■ \C (u = 101268, C’0,2 = 2016, C0.., = 256,\C„.4 = - I 8 ,f  0 .S = 0, Q).ft = 0 , C, 0 = 8064, G|.i = V\ + 7i ~ G.//, -  +1008, C, , = 8064+128, C, 4 = 1008+9, C|.4 = 128,C’i.s = -9, C 1/1 = 0 , C2.o = -I008, Gii = C1.3’c\ z  = & + 7, -0 , = + 9 , C2j3 = 8064, C24 = -1008,C25 = 128, C2.e, = -9, C,.„=I28. = -1008+9,II oc G.U = 7, -  G, C,,4 = 8064, C7 5 = -1008,Cv ,= 128, G’ -4,7 = -9, G 1,8 = 0, C4.0 = -9,C4, = 128. C4ii = -1008, C4|, = 8064, G4,4= u - g .C’4,5 = 8064, C4,6 = -1008, C47 = 128, G4.8 = -9,-P- O II O t \ „  = o. C,., = -9, C5 2 = 128,Cs > = -1008, C, 4 = 8064, c w = r4- c . C,it) = 8064,Csi7 = -1008, C58 = 128, Gs,9 = -9, Gs.io= OiCSM=:0.It should be noted that the double sign (+), which appear in some coefficients above, which mean, the states with even parity take (+) and stales with odd parity take (-). Also it should be mentioned that for even parity states the coefficients Co,*, (n> 1), become zero.To solve the algebraic eigenvalue problem for eq. (12), we first transform the matrix Cm,n 3 symmetric one. Due to the already special form of Cm tp this can be done by means of one similarity transformation. We then reduce this matrix to a tridiagonal form by preliminary transformations. To calculate eigenvalues of this symmetric tridiagonal matrix, we use the F02BEF Subroutine of the NAG-Library [13]. To select out the correct energy we require stability of our results with respect to the small variation of R at a given value of step length h,3. Results and discussionThe finite difference technique has been applied in this paper for a double-well potentials in one dimensional system. Eigenvalues for different values of Z \, X and state numbers nx are listed in the Tables 1 and 2.Energy eigenvalues and splitting between even and odd energy etc 227Table l. Ground- and first-excited stale energies for a double-well potential V(jt, Z j, X = I), for different values of Zj E*r (other) represents other values.A nx En( (present) E* 1 (other)0 011 060362090484182 3.7996730298013941.0606209048* .3.7997302980*0.5 010.8700175183716123.3337793298870060 870017518372* 3 33377932989*1 010 657653005180715 2.8345362021193040 6576.53005191* 2.83453620212*2 010.137785848188222 1 7130278977676750.1377858481882225s 1 713027897767676s4 01-1 710350450132638 -1.247922492066213-1 710350450132640s -1.247922492066215s5 0 -3.410142761239829 -3  410142761239830s1 -3.250675362289235 -3  250675362289236s25 01-149.219456142190888-149.219456142190888-149 2194561421913* -149.2194561421910*50 01-615.020090902757816-615.020090902757816-615.0200909027576* -615 0200909027576*100 01-2845.867880342075294-2845.867880342075294-2845 867880.342076* -2845.867880342076*200 01-9980 005002815982695 -9980.005002815982695-9980 00500281598269567 -9980 00500281598269567As suggested by the referee, comparison with the results of other methods [4-7] has been made in Tables 1 and 2. It is clear from Tables 1 and 2 that there is agreement between our results and the results produced by other methods.228 M  R M  WitwitIn Table 1, we compare ihe results obtained by the present method with those obtained by other methods [4-7]. The values of the energy are calculated over a wide range of 0 ^ Z \ £ 200 at A= 1, for the ground and first-excited state. As Z \  increases, the magnitude of the splitting between the energy levels for ground and first-excited state|£0 -  £, | decreases.In Table 2, we compare the results obtained by our method with those obtained by using power series method for two values of Z\ = 10, 100 and A = 0.5, 7.5. Generally speaking, the agreement between our values and those of power series is very good. SomeTabic 2. Numerical values of some energy eigenvalues for the double-well potentialvu , zj. A)nx Present method Power senes method0 -45 5791974486965547 -45.57919744869655473A1 -45.5791974486818590 -45.57919744868185901A2 -36 95.38380723252171 -36.953 8380723252171 Oo'-36 9538380677618931 -36 9538380677618930544 -28.6830875222325604 -28.6830875222325604275 -28 6830868969642709 -28.68308689696427091912 -1.4089110509977329 -1 40891105099773294213 -0  5217551377821886 -0  52175513778218866416 8 9976916989575165 8.9976916989575J647217 12 5121782111944881 12.512178211194488149IK 16 2532388098463624 16.253238809^4636236619 20 1747735655636969 20 1747735655636969840 -319 2671170279540906 -319 267117027954090601 -319 2671170279540906 -319 267117027954090602 -291.4454394228711989 -291.445439422871198963 -291.4454394228711989 -291.445439422871198964 -264.1075745401302216 -264.107574540130221665 -264.1075745401302216 -264.107574540130221666 -237 2797814701336811 -237.279781470133681117 -237 2797814701336811 -237 279781470133681118 -210.9923806353957530 -210 992380635395753039 -210.9923806353957506 -2 TO 9923806353957506910 -185 2808776964343650 -185 28087769643436504II -185.2808776964341016 -185.2808776964341016424 -27 0044852236812682 -27 0044852236812682125 -26.9211476446426514 -26.9211476446426514526 -10 0772932133725591 -10.0772932133725591627 -8.9598963608357662 -8.9598963608357662710 05100 7.5Energy eigenvalues and splitting between even and odd energy etc 229Table 2. (Corn'd,)Z \ A n , Present method Power scries method28 2.7350408154161046 2.7350408154161046329 8.1915969606041077 8.1915969606041077130 17.4123343420860030 17 4123343420860030131 26.2920453927462065 26.2920453927462065032 36 0236079545425696 36.0236079545425696133 46.1417178714088725 46.14171787140887259fine points, however may be noted, as Z?x increases, the energy levels for states of even and odd parity become effectively degenerate i.e. Eeven = Eodd. The splitting increases rapidly as one goes to higher levels.To show the dependence of the results on the choice of the R value, we present in Tabic 3 again for Z\ -  10 and A = 0.5, the results obtained with step length h = 1/20 butTable 3. The dependence of the results on the choice of the R valueA nx R h-45 357 148 594 361 401 06 4-45 579 197 233 453 902 44 510 05 1 -45 579 197 448 681 859 01 6 1/20-45.579 197 448 681 859 01 7-45 579 197 448 681 859 01 8-45.579 197 448 681 859 01 10with six different R values, i.e. R = 4, 5, 6, 7, 8 and 10. Our results in Table 3 do not change anymore whenever R > 5. Of course one should control for each parameters (Z x, A) choice the influence of the chosen R value on the results.AcknowledgmentThe author is grateful to the referee \or his valuable comments and suggestions in the light of which this paper has been revised.This research was supported in part by a research grant No. GR/K17750 from the Engineering and Physical Science Research Council.References[1] M R M  Wilwit J. Phys. A25 503 (1992)[21 C R Handy Phys. Rev A46 1663 (1992)[3] W E Caswell Ann. Phys. 123 153 (1979)[4] R Balsa, M Plo, J G Esteve and A F Pacheco Phys. Rev. D28 1945 (1983)230 M R M Witwit[5] F M Fernandez. A M Meson and E A Castro J Phys A18 1389 (1985)[6] F Arias de Saavedra and E Buendia Phys Rev. A42 5073 (1990)[7] M RM  Witwit and J P Killingbeck Can J Phys 71 475 (1993)[8] K Baneriee and S P Bhatnager Phys Rev. D18 4767 (1978)[9] R JW  Hodgson and Y P Varshm J Phys. A22 61 (1989)f 10] M Bansal, S Snvastava, Mamta and Vishwamittar Chent. Phys. l«ett 195 505 (1992) f 11] R L Somorjui and D F Homig J. Chem Phys 36 1980 (1962)[12] J Bnckinann and H Zimrnermann J Chem Phys 50 1608 (1969)[13] NAG Fortran User Manual. Mark 15 (Oxford Numerical Algorithms Group) (1991)

Energy Eigenvalues and Splitting Between Even and Odd Energy Levels of a Double-Well Potential

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Energy Eigenvalues and Splitting Between Even and Odd Energy Levels of a Double-Well Potential

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