Witwit, M R M

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Indian J. Phys. 68B (6), 515 -  524 (1994)U P  B- an international journalHypervirial perturbation calculations for a spiked oscillatorM R M WitwitDepartment o f  Applied Mathematics, University o f Hull, HU6 7RX, England, United KingdomReceived 22 July 1994, accepted 7 November 1994A b stra c t : The renormalized hypervirial perturbation method is used to compute the energy levels for a spiked harmonic oscillator V(jc) = where a  is a real positive parameterResults are produced for a wide range o f  parameters a  , (10 < A ^ 10^) and o f  state numbers. Numerical results are compared for the special cases a  -  | ,  5/2K e y w o r d s Hypervirial perturbation theory, spiked oscillator, energy level calculationP A C S  N o . : 03.65.G e1. IntroductionThe study of the spiked harmonic oscillator (SHO) has evoked much interest because of its varied applications in chemical physics, nuclear physics and particles physics [1-6]. The literature available on the spiked harmonic oscillator is rather limited, when compared with that for the anharmonic oscillator of the type/ /  = - + N = 2 ,3 (1)The spiked harmonic oscillator system is defined by the quantum Hamiltonian/ /  = -dx^2 . -1 -Of+ jr + ^ (2)defined in the one-dimensional half space fO, «>J, the eigenfunctions obeying Dirichlet boundary conditions [7, 8]. The Hamiltonian is characterized by means of two parameters,© 1994 lACS516 M R M  W itw itA which plays the role of a coupling constant, and a  > 0 which controls the type of singularity of the potential at the origin [9].The Hamiltonian given by eq. (2) has been the subject of intensive study in the last two decades. In the literature, a variety of techniques have been used to investigate the eigenvalues and other properties of the SHO. Nearly all previous calculations have been limited to a few excited states; for instance Detwiler and Klauder [1] made variational studies of the ground state of the spiked harmonic oscillator and found upper bounds for the eigenvalue corrections which are proportional to A In A when a  = 3, and proportional to A vvhen a>  3. Afterwards, Harrell [2] developed a special perturbation theory called singular perturbation theory, and obtained the first terms of the small A expansion, which turned out to be a nonpower series expansion. Aguilera-Navarro er al [7] have applied a variational procedure and a large coupling perturbative calculation for the ground state energy. A perturbative study was carried out by the workers [8] in the two extremes of weak and strong coupling. Employing the Rayleigh-Ritz large-order perturbative expansions, Aguilera- Navarro etal [9] have been able to analyze the problem around the three regions A —>A 0, A —> -  oo, for the ground state of the nonsingular spiked harmonic oscillator, for the case a  =1, More recently Miller [10] has applied a nonperturbative algorithm (iterative technique) for obtaining the eigenvalue for (SHO). Est^vez-Bretdn and Est^vez-Bretdn [11] have derived an exact analytical ground state solution valid for the special case a  = 2 .In different context, the Hamiltonian given by eq. (2) has been investigated by Killingbeck [12] who used numerical algorithms to integrate the Schrodinger equation, based on finite differences to compute the ground state energy for small values of A, for two values of a  = 4, 6 . Also Korsch and Laurent [13] used the Milne method to find the energy for the case a  = 4, 6 .In the present work, we wish to point out that the hypervirial perturbation theory is used to study and calculate energy eigenvalues of the spiked oscillator for small and large values of A, for a wide range of values of a.Our calculation deal with several values of a , since the method used makes this extension easy to perform. Our main object is to demonstrate that the hypervirial approach is able to work and produce results even for large values of A, a  and the state number n . For purposes of clarity, this paper is divided into three sections. Section 2 gives the theory, Section 3 describes the formalism of the renormalized series method, and Section 4 gives the numerical results and their discussion.2. TheoryTo calculate energy eigenvalues for the (SHO), we considered the Schrodinger equationj2dx+ + Xx~“ I v (^jt) = Eyf(x). (3)Hypervirial perturbation calculations fo r a spiked oscillator 517The shape of the interaction potential for large values of A is very different from the shape in the case of small A values. Instead of having just a spike very close to the origin, the potential for large A resembles a wide valley extending from x = 0 to jc = oo with a minimum in themiddle.For the special case a  = 2, the Schrodinger eqation (3) may be solved exactly [14], = + 3 + 2/where A = /(/ + 1)./1 + 4A -1(4)(5)If we insert / = in eq. (4), we get the following form for the energy.= 2  + 4n + 2VAn11 + 8A 128A  ^ " 1024A  ^ 32768A“ 262144A*214194304A” -] A > —. 4(6)In the absence of other reported results, we have devised some internal checks on the reliability of our results; for example the closed forms above for the energy provide checks on our calculations for the case a  = 2.Reviews of various numerical perturbative and non-perturbative calculations for the (SHO) can be found in references [7-9] which suggested use of variational and perturbative methods for the (SHO) by expanding the potential about its minimum to calculate the energy eigenvalues. Here we treat more general (SHO) problems, by considering higher powerindices (a  = 3/2, 5/2 , 7 /2 ........), higher values of the state number n and higher A values(5 < A £ 10*). We expand V(x) around its minimum. Let x„ and V (x„) be the values of x and Hx), respectively, at the minimum.Let y = X -  x„; the expansion of V(x) around y = 0 can be written as a Taylor series.2 2 yV(y) = V(x„) + (a  + 2)y  ^ + (-1)' —(a ) ,P '' —/=3where ( a)i is the Pochhammcr symbol and1P = — ■(7)(8)The coefficients given by eq. (7) alternate in sign; the coefficients take (+ s.gn) for even /  values, and ( -  sign) for odd I values. We have expanded the potenUal as given by fcq. (7) to the limit in which any term beyond that limit makes no difference to our results. For ourcalculations, this limit was reached for (/ = 20).68B-(7)518 M R M W it w i tThe central idea of this work is to expand the potential V(x) in a Taylor series about its minimum value, and solve the resulting perturbed oscillator problem by hypervirial perturbation theory. The potential is expanded around the right-hand minimum, while the new coordinate y - x - x „  covers the full real axis. When A is large, the potential minimum occurs at a large value of x, so that the wave function centered at x„ does not penetrate far into the left-hand potential peaks; obviously this is not the case for small X. The highest power of y appearing in the potential is 20. The expanded potential V(y) of eq. (7) is of mixed parity type. It is clear from eq. (7) that we can regard the first two terms as the unperturbed terms and the other terms as the perturbation.3. Formulation of the problemThe hamiltonian for the potential described by eq. (2) isH  = + V(y).dy(9)Here we use units m = 1. The potential V(y) given by (7) can be rewritten in terms of the perturbation parameto’ A as20V(y)= 2 (10)7=020with the coefficients given by eq. (7). The sum > (-1) —(a ), —  is replaced by/=o ®AV(/). Starting from the hamiltonian given by eq. (9) and using the basic relations for commutatorsf d  1 dV"J "= j { N - l ) y ^ - ^  + Ny^-^[ / j  d  1 w dV N N 7 d MJ tr  "J = " V  ^ ^  2 N y ^ - \ V - H ) ,( 11)( 12)(13)the diagonal hypervirial requirement that < eq. (7) lead to the result.2E{N + \ ^ y " )  -  ^  XV{I) (2N + 2 + I) )> shall vanish for the eigenstates of207=0(14)The Hellmann-Feynman theorem provides a further relationships between the energy E  and the expectation-values <y^>. Let us assume that the energy E and expectation values <y'^> can be expanded in power series of the perturbation parameter A asE='LE{J)<y^> = Z R{N,H ypervirial perturbation calculations fo r  a spiked osciUator 5 , 9(15)(16)In order to improve the convergence proporties of the perturbation series, we use a rearrangement of terms in the potential given by (7). To illustrate this technique, it is necessary to write the potential t4)pearing in eq. (7) in renormalized form20Vly) = V{x„) + \ p - XK]y ^+  A ^ V ( l ) y ‘ ,/=3where // is given the numerical value /X = (a  + 2) + ^  .(17)(18)The use of the renormalization parameter K is helpful in improving convergence. If we use the perturbation expansions (15) and (16) in the hypervirial relation given by eq. (14), we obtain the recurrence relationM A/{2N + 2) E{J)R{N, M-J)  = -  — \n ^ R ( N -2 ,  M) -  {2N + 4)O20[pR(N-\-2,M) -  KR{N + 2, M - 1 ) ] + ' ^  K{7) (2)V + 2 + /)/=3(19)(20)R ( N ^ l , M - \ ) .Applying the Hellmann-Feynman theorem in the formdE l d H \  I dV^dX ~ \ d X /  ~ \ d X / 'we obtain a recurrence relation for the energy coefficients of the form20(M+l)E(M+l)  = 2 ]  V(l)E(l,M) -  KR(2,M)./=3The unperturbed energy corresponding to the SHO can be expressed asE(0) =  V ( j c J  + (2rt +1) V/i in - 0 ,  1 ,2 ...........).The zero-order contribution to £  is given by the fust term in eq. (22) and next con^bution comes from the harmonic oscillator term, the second term in (22). The SHO senes have(21)(22)520 M R M W itw i teven and odd powers of the perturbation parameter A. The actual series for the energy has the formE(K) = E(0) + ^ ( 1 )  + X^E(2) + X^E(3) + A^£(4) (23)From the recurrence relations (19) and (21) together with the unperturbed energy (22) and the initial coefficient value /?(0 , 0) = 1, we can calculate sequentially the perturbation series for the <y^> and the energy.4* Results and DiscussionThe renormalized series method has been used for calculating the eigenvalues of the (SHO) given by eq. (2). Eigenvalues for different values of A, a  and state number n are listed in Tables (1-^).T a b le  1. Ground state energy eigenvalues o f  the spiked harmonic oscillator for a  =  1, 5/2.For comparison* the second lines in the first and second colum ns display the energies corresponding to R efs. 9 and 8, resp ectively . The em pty spaces mean that the corresponding eigenvalues are not be reported.^a*t M ^ttMS/2 M0.523458102550751001502003004 0 050075010003.34.14.fl57?Q64.945.866.637.3777J84Q317419.377  10.577  10 7^748343 17.9283 27.40514  35.360350  42.4629180955.0965435232566.3742252641288.43224389401104.33633594703120.79206109643157.74209512062190.72330743978109  11 1114101515 14 12131917181718161415141281010885311118113.64.34Ji73Ii4.95.535.926.2956.2964727.2227.7357,73311110.488613.472415.698917.541989lL3dl£8220.57583323.08630627.217959430.6360929833.60571088539.817286930244.95548479961081215 14 1816 18171613161717333846685227262221.5192017.5 15108.275.4  43.58 810128In Table I numerical results are compared for two special cases a  = 1, 5/2 for the ground state energy eigenvalues for several values of X, (0.5  ^ A < 10^ ). It is clear from the Table 1 that there is some agreement between our results and the previous published results [7,9].Table 2. The energy for the ground and excited states of the spiked oscillator for four values of a, over a wide range values of A. The empty spaces mean that the corresponding eigenvalues cannot be calculated by our technique.HyperviricU perturbation calculations fo r  a spiked oscillator 52151010010005000a = l/2 a= 3 /2 a=5/2 a=7/27.9312.1967.25660691415.889785658011502.987202332316.969.3729.3971104.4102238S4259.07844700596.297.7417.542244.955484799689.691749234546.17.212.6726.10884944.9890390239icr* 2165.68031939241 384.07821353971 121.282856042479 57.2083469940210* 16495 4300708630 1426.57990603771 333.685605689407 129.07123821873710* 104070.732071571 5312.60615971726 924.715125454449 295.0922625120185 11.210 15.52100 70.4478161000 419.0565976755000 1506.1507294816810* 2618.8433145342210* 16498.5924622073910* 104073.894367248610.713.1333.156557 108.15657318 262.8219865100 387.82113922046 1430.32190436784 5316.3479086191910.511.921.76449.186374993.928518942125.5211584788337.92667180376928.95719823935810.511.417.3130.732249.6402852561.867940951133.74797411045299.77669073592451010010005000Id*10*10*5101001000500010*10*10*5101001000500010*10*10*19.273.666548422.227907665141509.315503362012622.0070262484316501.7549672131104077.05668094276.92425.40381512.481522625.17145407216504.91758586865104080.2190126520117.236.936119.0743425266.567378827391.565319856021434.064242673885320.0897489665340.74115.663270.31461395.310744361437.806920143025323.8316807004880.19 428.584 1515.6488 2628.336598 16508.08031816236 104083.3813623775744.52 119.425 274.0637 399.05741 1441.54993596524 5327.5737037623116.226.0653.4067561498.159734395129.75529689958342.166186613224933.19870616403830.1457.621102.38567133.985422346/404170972921937.43965199436934.29 61.83 106.6066 138.21168 350.64064531398 941.6800384756615.622.535.3046354.25719666.49974191138.411643985016304.45525057181526.439.8458.843771.10623143.06279166309.12805170989131.344.3463.403775.6896147.70192735313.7952007643522 M R M W itw itTable 2. (Coni'd).n X a = l /2 a=3/2 a=5/2 a ^ m510100 83.5 48.4 38.7 35,61000 431.768636674 123.185569 66.0191 49.163 5000 1518.81729130003 277.8144331 110.822637 67.9440210^ 2631.50245619404 402.80528129976 142.4341902319 80.25202210* 16511.2431640827 1445.29328933321 354.875629655027 152.329530356610^ 104086.5437301188 5331.31581809333 945.919868332854 318.456801386269Table 3. The energy for the ground and excited states of the spiked oscillator for four values of a, over a wide range values of X. The empty spaces mean that die corresponding eigenvalues cannot be calculated by our technique.a A Eo El E2 £ 310 7.249 11.63 16.4 22.33 100 14.6414 19.045 23.382 28.69500 25.7990039 30.22795 34.6252 39.0241000 33.31676152 37.7554906 42.166094 46.554210 6.64 11.52 16.2 22.44 100 11.276 16.11 21.58 24.9500 17.4764 22.2455 26.98 31.971000 21.369519 26.15539 30.886 35.65110 6.29 11.4 15.9 21.65 100 9.469 15.09 2 1.1 28.4500 13.422 18.54 23.92 29.821000 15.7647 20.834 26.017 31.52310 5.96 100 8.46 14.4 2 1 .1 28.9500 11.151 16.94 23.31 27.31000 12.724 18.19 24.21 30.81In Tables 2 and 3, we have calculated the energy eigenvalues for the ground stale and many excited states for several values of a  ^ a  ^  6) over a wide range values of X (5 ^ X £ 10^ ). The most studied (SHO) is a  = 5/2 for ground state only [7,10]. Our calculations deal with several values of a  and many excited states, since the method used makes this extension easy to perform.Currently there are no other eigenvalue-results available by any other method for the cases such as A = 500, 1000 and a  = 7/2; consequently, it is not possible to infer the accuracy of the full set of our present results by direct comparison. However, for some special cases, it is possible to check our calculations; for example at a  s  2, the results w oe  checked to high accuracy against those obtained by exact solution (6). Table 4 presents a selected set of energies for several sets of the perturbation parameter X (10 ^ X ^  10^ ) for four energy levels E q, E \, E^, E y  The agreement between the numerical and exact results is seen tobe good. Especially for larger values of X. the difference between the numerical and exact results decreases i.e.H ypervirial perturbation calculations fo r  a spiked oscillator 5 2 3'^numerical ^enactl ~  ^as is clear from the listed results in Table 4.(24)Tabic 4. The comparison of the renormalized series numerical calculation with exact calculation from eq. (6). The values of the renormalization parameter K  are given. M  is the order of the perturbation expansion with which we obtaip convergence.n A ^numerical M K \e  -  e  It num  exa |10 8.49 10 18 8.4031242374 0.8687576260CM)150 16.1780 30 24 16.1774468788 0.55312120000-030 100 22.0249 45 18 22.0249843945 0.84394500000-04250 33.63858441 64 16 33.6385840391 0.37089999690-06500 46.732538504 60 15 46.7325384927 0.11299995610-071000 65.2534584038 58 12 65.2534584035 0.29999114300-0910 12.8 8 20 12.4031242374 0.48008797140+0050 20.197 26 15 20.1774468788 0.19683971210-011 100 26.0261 20 10 26.0249843945 0.11175757280-02250 36.6386001 23 7 37.6385840391 0.I725318712I>O4500 50.732538925 37 6 50.7325384927 0.43252258010-061000 69.2534584155 34 5 69.2534584035 0.12152881370-0710 17.2 9 40 16.4031242374 0.79687576260+0050 24.18 16 12 24.1774468788 0.57612144440-012 100 30.0254 19 20 30.02498439450.11415605500-01250 41.63888 54 11 41.6385840391 0.29596090000-03500 54.732546486 54 12 54.7325384927 0.79932999950-051000 73.253458612 54 10 73.2534584035 0.20849999770-0610 21.2 10 20 20.4031242374 0.85444753370+0050 28.19 16 15 24.1774468788 0.20645562480-01100 34.08 15 20 34.0249843945 0.55015605500-01250 45.6411 33 18 45.6385840391 0.25159609000-023 500 58.732629 55 14 58.7325384927 0.90507300000-041000 77.2534608 50 10 77.2534584035 0.23964999940-05Our results for the spiked harmonic oscillator have the following consequences :First the renormalized series method works very well even for higher values of A, a  and state number n.Second, the renbrmalized series converges well (with a proper value of K)  for high values of X, but not for low values of X. Our work indicates the great importance of a good choice of the renormalization parameter K in order to get the best convergence. To select the correct convergence energy, we require stability against the various values of renormalization524 M R M W i t w i tparameter K.  The general consideration that governs our choice is that as X increases the values o fd ecrea se s.Third, our numerical investigations of the spiked oscillator show that applicability of the renormalized series method is limited to large values of X; this behavior is clear from our results in Tables 1-4. When applicable, the renormalized series approach has been shown to be veiy effective and more simple than standard matrix diagonalization, which woiild require the computation of standard matrix elements of the various powers of x^.  The hypervirial method gives the < y ^ >  and E  values for a selected state directiy.AcknowledgmentsThe author is grateful to the referee for his valuable comments and suggestions in the light of which this paper has been revised.References[1] L C Detwiler and J R Klauder Phys. Rev. Dll 1436 (I97S)[2] E M Hancll II Ann. Phys. 105 379 C1977)[3] F Rellich Perturbation Theory (tf Eigenvalue Problems (New Yoik: Gordon and Breach) (1969)[4] J R Klauder Science 199 725 (1978)[5] M de Uano Rev. Mex. Fis. 27 243 (1981)[6] R M Speclor J. Math. Phys. 8 2357 (1967)[7] VCAguilcra-Navarro,GAEsidvezandRGuardilaJ.Math.Phys. 3199(1990)[8] V C Aguilere-Navarroand R Guardila/ Math. Phys. 32 2135 (1991)[9] V C Aguilera-Navarro, F M F em ^ z, R Guardiola and J Ros /  Phys. A25 6379 (1992)[10] H G Miller J. Math. Phys. 35 2229 (1994)[11] Erika Sabrina Estdvez-Bietdn and Gentil A Estdvez-Bretdn /  Math. Phys. 34 437 (1993)[12] J P KiUingbeck J. Phys. BIS 829 (1982)[13] HJKorschandHLaurent/FA>j.814 4213(1981)[14] M R M Witwit /  Phys. A24 4535 (1991)

Hypervirial Perturbation Calculations for a Spiked Oscillator

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Hypervirial Perturbation Calculations for a Spiked Oscillator

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