Ozcelik, Soleyman

Simsek, Mehmet

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Indian J. Phys. 71B (1), 59-67 (1997)IJ P  B- an international journalO n th e  b o u n d  sta tes o f  th e  g e ne ra lize d  in v e rs e -p o w e r p o te n tia ls  in  (V -d im ens ionsSuleyman tiz^elik and Mehmet SimsekDepartment of Physics, Faculty of Arts and Science,Gazi University, 06500 Ankara, TurkeyReceived 26 April 1996, accepted 21 November 1996Abstract : We have presented bound state solutions to the Schrbdinger equation for general inverse-power potentials, V(r) s  ‘n ^-dimensions. By using a simplealgebraic method, it is shown that bound states of these potentials exist when the parameters A v satisfy certain constraintsKeywords : Bound states, inverse-power potentialsPACS No. : 03.65.Ge 11. IntroductionThe problem of quantum inverse-power potentials has been the subject of many discussion, Tor decades, both from an analytical and numerical points of view, because of its important applications in molecular physics [1], atomic physics [2], solid-state physics [3] and scattering theory [4]. Some special forms of the inverse-power potential describe iniermolecular short and long-range interactions [1], and also, the spin-orbit and spin-spin interactions [5]. The induction energy for the interaction between a molecule with a permanent dipole moment and a non-polar molecule can be described with inverse-power potentials. The dipole-dipole dispersion energy for two molecules in their ground states is therefore attractive and inversely proportional to the sixth power of the intermolecular separation. Also, the dispersion energy arising from dipole-quadrupole interactions, quadrupole-quadrupole interactions, etc., is in a more complete form if it includes the terms ot inversely proportional to higher power of the intemuclear separation [1]. On the other hand, the inverse-power potentials can be used for the short and long-range interactions between two colliding particles [4]. Also, the various forms of the inverse-power potential ♦or the short-range repulsive energy have been adopted in order to study the crystalline properties of alkali halides apd other ionic solids 13],© 1997 IACS61) Suleyman Oz^elik and Mehmet SimsekU should he noted that, nonrclativistic quantum mechanics with local interactions loses its general properties. Therefore, there arc a number of cases where use of singular potentials is of interest. Recently, solutions to the Schrodinger equation for particular cases of inverse-power potentials have been studied by several authors [5-8] in one- and three- dimensions. In this paper we have extended the method |7,9] to use the solutions of the radial Schrodinget equation for the general inverse-power singular potentials,This potential can be reduced to well known potential by choosing appropriate A v and L  For example, for L  = 4 it reduced to eighth-order inverse power potential (EPP). EPP models to the induction energy for the interaction between a molecule with a permanent dipole moment and a non-polar molecule, or the induction energy for identical neutral molecules, including terms of upto quadrupole-induced dipole interaction which is proportional to r H 11]. Also, the crystal properties for ionic crystals have been calculated using special types of EPP. Thus, the potential V ( r )  = A  + A  + + nas been usejto calculate crystal properties of alkali halides and other ionic solids, where A  and n  are potential parameters, A  i is the Madclung constant, A6 and A8 are van der Walls constants for dipole-dipole and dipole-quadrupole interaction, respectively |3]. On the other hand, the EPP can be used as the interaction potential in the scattering of charged particles by neutral targets oi m collisions between neutral particles (4|. The long-range pails of the interaction potential includes higher power of r, namely, + , [4]For L  = 2, the eq. (I) turns to fourth-order inverse power potential (FIP) [6-8|, V(r)  “ Tf + T t + T f  + T1- describes zeroth moment, dipole moment, dipole-dipole and dipole-quadrupole interactions of two molecules [2]. For A, = 0 , the FIP reduces to the potential V(r) = ^ - 4 ^ 4  4 ,  which contains magnetic spin-spin and spin-orbit interactions between two spin particles [5], For A 4 = 0 and Ay  -  0, the FIP reduces to the Kratzer f 1OJ, atomic Fucs 111 ] or a Mie-typc [12] potential of the form V( r)  = + -j-It can be used to investigate the rotation-vibration spectrum of a diatomic molecule [10,12|. Also, lor A a = 0, A j = 0 and A 2 = 0, the potential in eq. (1) turns to the Coulomb potential.2. Solutions in iV-dimensional spaceThe radial Schrodinger wave equation for a sphcrically-symmetric potential V ( r )  in N- dimensional space is2 L( 1)(2)On the bound states o f the generalized inverse-power potentials etc 61l/V-l)The transformation 0(r) = R(r) reduces eq. (2) to the form- § [ | r  -  l * r ' * ? - 3)] « r )  = (E-V(r)Wr,where M = N + 21. Choosing the following factorized form for $ r)<t>(r) = f ( r)  exp|g(r)]jml substituting it in cq. (3), one can obtain0"(r) g"(r) + g'(r)2 + f " ( r) + 2/ ' (  r)g'(r)fU )(3)(4)(5)Comparing eq. (3) and cq. (5) we obtain (A f-l)(M -3 )U(r) + - 4 r2 - e  = g"(r) + g'{r)2 +r f r ) + 2 / V t e V )f i r) (6 )where / '(r i = ^-ViD  and e -  Eq. (6) for_/fr) = constant is one of the forms of theKieeali equation. Our purpose is to find the fraction of right hand side of eq. (6) corresponding to the potential and energy. Exponential part of wave function, #(r), and oilier part /(/*), which is a polynomial in r must ensure the normalizability of 0 (r) [0 (r —» <<•} —» 0 and 0 (r —> 0) —> 0].Eq. (6) in /V-dimensional space for the reduced radial wave function 0 (r) has the Nilucturc of the one-dimensional Schrodinger equation for a spherically symmetric potential V(t ) To solve the transformed Schrodinger equation, eq. (6), for V( r)  in eq. (1) we choose /(/tin the nodal form,n/ „ ( 0  = n  ['■-“ C ] ' * = 2...... /o<0 = 1 (7)and g{r) in the polynomial form as,L i -/tg(r) = a0r + ] ^  ~ l ^ T  + blnr'where crj"' values correspond to the nodes of the wave function 0 (r) that are determined loots ol /(/ ). We can rewrite cq. (6) by substituting g(r) and f n(r) foi ground and excited Mates respectively, as given below.(9)62 Suleyman Ozi'elik and Mehmet Simsek,hc first term ol the right-hand side in eq. (9) is the equivalent energy (£ = — aj). others corresponding to the parametric inverse-power potential Ue„, and, 2a„b . b (b -l)  , V  ^ 2b~ J )= « 2o + - r  + —p -  + l i 7T[2a«+  r )1 2a f n 1 1 2+ S  ~  l^f r-a\n\ + ( r-a^Mr-a}"*)'^ , | Ik^ 2 \ i=i ' ~i /For the first node, parametric potential and energy are obtained from eq. (10) for n = 1 asLl a nt) h (h  -  11Ueff(r) ~ £\( 10)*!(;§  £ ) ♦ £ ( § * £where c, = -  Aq, al = b and= 0.(II)( 12)In general case, parametric polcnlial can be written asj --2  V }Z.-I ( L- 1L- 2 (  L-+ S  [S Hf -i ~ " "6 • ( £" p  )'(13)(14)On comparing cq. (13) and eq. (I) we find the relations between the corresponding potential parameters and wave function coefficients (ak, a (,,)) as2^h2 A« = X  “A -j  + ov.,(2 n -v  + l) + 2£ 0j X a i"y ' ‘j=0 j=0 L ,=1On the bound states o f the generalized inverse-power potentials etc 63v-2 ( /.im=0 V '^ 7\a W"‘a w - - i  _ v _ , 2 3... 1- (15a)m ak1 = 0 - v K - iL+ X V v - y -  v = L +  l;j = v - L(15b)hi \ - iw ,J -  v- Lr, v = L +  2, L +  3, .... 2L\ (15c)where ci\ = b  (2 b  = 21 + N  -  1). N ote that for v  = 2 , \  (M -  1) (M -  3) is second contributor oi the param eter jjjr A2. T here are the relations betw een the wave function param eters a k ml a]n) (for the first three sets) asi L (  ■ ^ L - 1 f  n )+  I^ T ( x f r,)l a \ n ) L k ~'k = 0 l  i =  l J * = 0 , ' < j  J(16a)n > L-2 » ^^ a (n,L‘ oJn|.•*J J *=0 y+L - 1  {  n \= o, (16b)L  !r  „■ L - 2 r  „  ^ty  o r j n ^ * l o r J #l^  *J t - o  l J k=  0 jt =0/ rtl = 0 . (16c)Hence, the unnorm alized wave functions are given by<t>n(r) = f l  ( r - a iln))r(I'+A'-|>'2 exp a 0r + £ (17)The index n denotes the num ber o f  nodes o f  the wave function in 0  < r  < «>. The eqs. (1 5 a -  0 contain the relations betw een the potential param eters A v and wave function param eters ak «md a (ln). From  eqs. (1 5 a -c ) and eqs. (1 6 a -c ) it is possible to obtain the coefficients a k as a function o f  A v. H ence, fo r the  po ten tia l in eq. (1), the exac t energy  and the corresponding w ave functions in N -dim ensional space can be determ ine from  eq. (14) and tlci (17), respectively.The general results obtained above can be sim plified for special cases o f  L :64 Suleyman Ozfelik and Mehmet SimsekFor L = 4, the potential in eq. ( I) reduces to EPP asV(r) r8 p  rb r s r4 r 3 ( 18)From eqs. (15a-c), relations between potential parameters and wave function coefficients can be obtained as (h = 2/i = 1)4, = 2 a0(b + n)\n 1A2 = b(b + 2 n - \)  + 2aQ(a2 + '£ i a {ln)) + n ( n - \ ) - j ( M - \ ) ( M - 3 ) \i=in nA, = 2ana  ^+ 2a2(b + n - \ )  + (2b + 2 n -2 )  £  a,(,,)+ 2a0 £  a p ; (19)J=l 1=1n /iA4 = 2fl0a4+fl3(2^ + 2 n -3 ) + a^+2flQ ^ a /(n) + (2b + 2 n - 2 ) ^ a ilny + 2a2 ^  a (n) + 2 ^  a.\n)a (}n)\i=l k ;A5 = 2a2a3+2a4(b -2 ); A6 = ri2 + 2a2a4; A7 = 2a3fl4; Ag = a 2.Also, there are four constraint relations on the potential parameters as follows :r - /, > pA2 = n{n + 2l + N - 2 )  + 2n + 2, + N _ l 4 ^ 6^ 8 ~ ^7 ±4/ti2 X  oJ-»V 1=1(20a)A3 = ±A. A, 4A„AR -  A72—  ± - 6- 8 ,- ? (21 + N + 2 n -3 )A%(21 + N + 2n -  \) 8A*2A.2/I + 2/+  /V- 1n n£  a ‘")3 + (21 + /V + 2n -  3) £  a,("1, (20h)±2;4| V ^  ± —^ 2— <21 + N + 2n 4) + ^ V ^ 2)2>14. 9/ 4. a/_  i 1   ^ I——t +1\ + in  4) + >^1 --I4 2n + 2/ + A/ - 1 ^  -  6 4 ^ “+ 24 V « - ^ 7nV4 / 4 82 .i = l- s '  *  V v « + » - » (20d)The energy for any n and l isOn the bound states of the generalized inverse-power potentials etc 65£  . __________ A?(2n + 2l + N - l ) 2 (21a)Also, one can write the energy expression in different forms in terms of other potential parameters from eqs. (19-20), for exampleA1-n (n  + 2 l+ N -2 )« ln)j(21b)In order to use last form of energy expressions, ajn) parameters must be found from eqs. ( |6a-c) with above restrictions. Note that, there is no simple form of a (n) as seen in eqs.(16). For the first excited state (n = l) this parameter can be obtained from2A,a \ir + (21 + N - l )  a}0’ ± A,4A6A8-A 724A*V<1)2(2/ + yv + i)±  a,0) ±  = ° ’and corresponding wave function is*.(r) = (r -a ! l))r,2'4A'-')/2 exp f-----^----- r± 4AA ; ^  I1 '  ’ ' V [(2l + N + \) i A i  rA,(22)± ' ±  + V^L 1±4-yJ\ r2 3 r3F;or L = 3, from cq. (1) we haven r) = 4  + ^  + i f  + ^  + 4  + ^L.(23)(24)In this case, relations between the parameters can be determined with eqs. (16, 19-20) for a4 = 0 (A s = A7 = 0). The wave functions are»  = r(2/+AM>/2 exp r A ,[(21 + N + l)* 2 ? Cwith the consistency relations^ 5 _ 1  + J_n{n + 21 + N — 2) +2n + 2/ + N - l(25), (26a)7lB(l)-lo66 SUleyman Oz^elik and Mehmet SimsekA .  =  ±  ( 2 / + W  +  2 I . - 3 )i "  2/ + N + 2n - 1 2 ^ \^  -  ± f t a . . 2 i * » - A )  *  ^  I  <+ (2ff+2/ + /V-3) £  a j"*2 ± £  a '" ’ + 2 £  a ,")aj'°  (26c)"TiT VA> , = i ,<iThe energies are of same form os in eq. (21) with restrictions in eqs. (26a, b, c). Also, parameters a\n) can be found from eqs. (16a-c) for several values of nWe note that, if the potentials in eq. (18) and eq. (24) do not consist of the Coulomb term (i.e. A, = 0 ui b = -  n), the energy (e = n(2, ) is determined in terms of other potentialparameters, from cq. (19). Also, the potential in eq. (24) for = A$ = 0 (orflj = 0) reduces to the FIP which can be obiained from eq. (1) for L = 2. In this case, there arc two restrictions between potential parameters : A2 = fn+ii+N-i + (X[fn)) + n (n + 21+ N -  2) and Ay = (2/ + N -  . These results are identical with the previous works [6-8] in three-dimensional (N = 3) spaces. This potential for several values of A{ turns to some special potentials, i.e., for A\ = 0, the HIP can be reduced to the potential as given for c = -  * -n in (7] in three dimensions. For Ay = A2 = 0, one can obtain the potential “  + thatr r4includes only Coulomhic interaction and interaction of an ion with a neutral atom or polarization term of atom or molecule, where En = a ^ b  = 1, Aj = 2oq ( n + 1) and 2oq (J a4 + Z"., a jn)) + n (n + 1) +£(2/ + N -  1) (2/ + N -  3) = 0. Moreover, one can obtain Kratzer and atomic Hues (or a Mie-type) potentials for L = 1 (i.e. = As = A4 = Ay = 0 ineq. (24) of ay =a? = 0) by the same way. For this potential, there is no constraint on the potential parameters.3. ConclusionsA large area of mathematical physics is concerned with exact solutions of singular systems Irom the viewpoint of quantum mechanics. We have given solutions to the Schrodinger equation (eqs. (2)-<6)) for general inverse-power potentials, eq. (1), in A/-dimensions. We take all the solutions in the form 0(r) =J(r) exp l#(r)], which suitably restricts fir) and g(r) to incorporate the boundary conditions. The wave functions and energies of different forms ol the mvci.se-powcr potentials can be determined by using this algorithm for several values ol Our solutions have involved some restrictions between the potential parameters. The number of restrictions on the 2L potential parameters are L except L = 1. We note that, the solutions of general inverse-power potentials can be reduced to some molecular, atomic and67spin systems. Thus, the difference between the present work and the previous works [6-9] is that the previous works are the special solutions of the general solution derived in this work, and its attempt to extend the class of quasiexactly solvable potentials from polynomial towards essentially singular potentials.AcknowledgmentWe thank to refree for useful suggestion and Professor S Erkog (from METU) for carefully reading. This work was supported by TUBITAK/project Grand no TB AG-1362.References( 1 1 G C Maitland, M Rigby, E B Smith and W A Wakcham Intermolecular Forces (Oxford 1 Oxford Univ Press) (1987)(2 J L D Landau and E M Liftz Quantum Mechanics Vol 3 3rd edn. (Oxford ■ Pcrgamon) (1977); F Reuse Met Phys. Aua 51 157(1978)[31 A Rosemary, Coldwell-Horsfall Phys Rev. 129 22 (1963); A K Sarkor and S Sengupla Phys. Srat Sol 36 359 (1969). J Shanker, A P Gupta and O P Sharma Phil. Maf{. B37 329 (1978); B N Jha and K P Thakur Plmicu B137 408 (1991), A M Sherry and M Kumar J. Chem. Solids 52 1145 (1991); B N Jha Physica HI92 253(1993)[41 R G Newton Scattering Theory of Waves and Particles (New York ■ Springer-Vcrlag) (1982); R S/mylkowski J Phys A28 7333 (1995)15] AO Barut J Math. Phys. 21 568 (1980); H H Aly and A O Barut Phys Lett. A145 299 (1990)|0 j A O Barut, M Berrondo and G Garcia-Calderon J. Math Phys. 21 1851 (1980)[7] S ()7 e^hk and M §im$ek Phys Ljett. A152 145 (1991)[K] E Pap Phys. Lett. A157 192 (1991)|9| M §un e^k and S to^elik Phys. Lett. AI8 6  35 (1994)|I01 A Kralzer Z Physik 3 289 (1920)III! G Simons J Chem Phys. 55 756 (1971)II2J G Mie Ann Phys. (Leipzig) 11 657 (1903)On the bound states o f the generalized inverse-power potentials etc

On the Bound States of the Generalized Inverse-Power Potentials in N-Dimensions

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On the Bound States of the Generalized Inverse-Power Potentials in N-Dimensions

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