Banoyopadhyay, A

Sahoo, S

Sil, N C

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Indian J. Phys. 73B (2), 279-283 (1999)U Pan internat ional journalT he c a lc u la tio n  o f  th e  g r o u n d  s ta te  e n e r g y  o f  th e  p o s itro n iu m  n e g a tiv e  io n  P s -A Bandyopadhyay. S-Sahoo and N C SilDepartment o f  Theoretical Physics, Indian Association for  the Cult ivation o f  Science, Ca lcu tta-700  032, IndiaA b s t r a c t  W ith the in troduction of an effective hanultonian ,  a regular  perturbationmethod is developed systematically  for the calculation o f  the ground state energy o f  the positronium negative ion Results upto the second order is very encouragingK e y w o r d s  Positronium negative ion Ps , g iound state energy,  perturbation approach P A C S  No. 16 10 DrI he positronium negative ion (Ps'), the system composed of two electrons and a positron, has been reinvestigated theoretically. The calculation of the binding energy of the system has a long history that can be traced back to the early work of Wheeler [ I ]. Subsequently many authors [2, 3] with varying degrees of sophistication have reported reasonably good estimate ol the ground state energy. Despite these moderate to superior calculations, the interest in this system remains unabated perhaps because of challenging quest for even simpler and more transparent method. It is in this spirit that we present here a simple intuitive method which is a combination of an effective soluble hamiltonian and Ray Icigh-Schrodingcr perturbation theory (RSP F). The essence of the method, which leads to a good estimate of the GS energy, lies in a suitable partitioning of the exact hamiltonian into a soluble one with a complete basis set and a perturbation.The hamiltonian of the positronium negative ion is given by//  = - - 1 v 2, ___L y 2.2 wi ri 2m- 1 v i -2m., r '-JH K  “ R\(i)where r r '  and J? denote the position vectors of the electrons 1,2 and positron respectively. mvanc* mp arc the masses of the electron and the positron respectively.0  19991ACS280 A Bandyopadhyay, S Sahoo and N C SilIn reduced atomic unit (where energy is scaled b y — ^ l m t ) and length by a (mlH))."oThe hamillonian is reduced to/ /  = -I2 1+PV. .V.(2)m m t.where the reduced mass p = — -—— , p - m  /m ,  andr., r ,  arc the vectors from the positivem  + in '/' >'particles in units of (mt, / p)a0. We essentially rearrange the hamiltoman in the following way,- (1 - J )11 + pV V (3)where Z is the effective charge of the positive particle.For reasons to be apparent presently we introduce the following simple scalingp, = Z r, and p2 = Z r2 (4)with the results thatwhereandH = H0 + Ht ,H0 = - - ( V ;  +V; ) -0 0 \  Pi P i  11_ J _{Pi * PiHI = JL (z ~\)  | — + — z  P\ P i ) P 12+ —  ——  V .V1+pIn this form, H0 is exactly solvable with a basis(5)(6)(7)w  ( o  o ) =  )^/(P2> + ^ ( P , )]> i * j" K r  2) 0((p,)0,(p2). (S)0’s being the Coulomb functions. In the discrete and the continuum state these are respectively given by=  ~ Nni p  <2Y„P) yim ( p ) . n =  (n. /, m) (9)The calculation o f  the ground state energy etc 281N„/ = —r  (2 / n)‘n~Yn = ! /« •-|l/2(« —/ — 1)!(n + /)!} 'i :^ ](2ynp) is the Lagucrc polynomial [4] and K/w(p) is the spherical harmonic and <t>k ( p )  =  Nk exp ( i k . p )  ,f, ( / /k A J k p - i k . p ) . ( 10)w h ere  /V. = ------e x p  f —  I T  (1 -  /A)* (2n )> ~ V 2k 'and , /*[ (// A\ 1, i kp - i k .p )  is the confluent hypcrgeometric function [5|.We now estimate the ground state (GS) o f //  with //, as perturbation. The unperturbed energy E() is given byii A>; >; VII 1 (IDith <P, P : ) =  (PV(P]) <P,(P2 ) and (12)(f>v(p)= exp (-p ). (13)where the perturbation Hf has untill now an unknown parameterZ which has to be determined.Wc fix this parameter by the prescription that the first order perturbation correction to the GS energy.£*" = < y/KK(p ,.p : )iH(); y/KK(pr p2 )> to be /.cro.I 'll us, w e  g e t( 2 Z - 11 /8) = 0 , Z = J 1 /16 . (14)1 he next order correction £<2) is given by<2) <v/w (p,,p2)|W/)> , /(Pr P2)>!where as usual prime in the summation indicates that the interm ediate state ^ , /P jP 2) = WK^ P \ ' P 2 ) is excluded andH o Vi,(PpP2>==£i/^,/(PrP2)-In the evaluation o f # 21 we essentially split it into three terms282 A Bandy opadhyay, S Sahoo and N C Si Iwhere-(2)/ |< v v p , .p 2 . ( p , - p , ) >!<  V ' . J P i - P ,  >|W /„ ( |V '„ < ( P | . P 2 ) >E. - E A*.; ^ w,(P|-P2) K „ |v '<r(Pr P2): </* d k '.To simplify the calculation of these three terms, we have gainfully employed the contour integral representation of the Lagucrre polynomial [61 and the confluent hypergeometric function 17J. It turns out that the matrix elements in E{nz] are obtained in closed analytic form while those in E™ and E^!  are reduced to one dimensional integral which are evaluated numerically. The discrete summation has been performed only upto n = 5 and the second order perturbative correction turns out to beEa) = —0.0398 / r .Therefore, the ground state energy of the positronium negative ion upto second order RSFT is given byE = (E0 +En) + Ea ’)Z2 = - ( l  I /16)2 -0.0398 = -0.5123.The result is quite encouraging. Further contribution from the other bound states are likely to increase the result. The most accurate ground state energy of Ps“ in the literature is that of Frolov and Yeremin [8] who used a 700 terms exponential variational expansion wave function. They obtained £  = -0.52401404656 ± lx  10~12. Bhatia and Drachman [31 obtained £  = -0.5240101300 Ryd by the use of 220 term hylleraas -  type wave function with two non­linear parameters. Ho [9] reported E =-0.52401014041 Ryd with 946 terms hylleraas type wave functions. Though the qualities of the wave functions used by them arc very accurate, the calculations they performed are very complicated. We have carried out a simple calculation for the GS energy of the positronium negative ion. Our success due to the dominant contribution from 2nd order perturbation term is a very interesting feature of our method. Higher order corrections are likely to improve the result. With the expected improvement of the result in our method, further investigation of the corrections due to the other bound states is under progress.R e fe re n c e s111 J A Wheeler  Ann N. Y A ca d  Set 48 219 (1946)[2)  Y K Ho J  Ph ys  B16 1503 (1983)[31 A K Bhatia and R J Drachinan P h ys . Rev  A28 2523 p 28 (1983) f4 J L  I Scif f  Q uantum  M ech a n ics . 3rd edn. (New York : Mcgraw Hill) (1968)The ca lcu la tion  o f  the g ro u n d  s ta te  energy etc 28315 1 1- D Landau and E M Lifshit/ Quantum Mechanic Vol. 3 2nd edn (revised) (London , Fergamon) (1965)|(i | H Margenau and G M Murphy 77if Mathematus of Pltvms and Chemistry 2nd edn (New Delhi Eas( W est) (1956)(7 1 A Nordsieck Plm Rev 93 785 (1954)| s 1 A M Frolov and A Yu Yeremin J Plm B22 1263 (1989)|0 | Y K Ho Plm Leu A 144 237 (1990)

The Calculation of the Ground State Energy of the Positronium Negative Ion Ps

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/5531/1/The%20Calculation%20of%20the%20Ground%20State_By%20A%20Banoyopadhyay.pdf

The Calculation of the Ground State Energy of the Positronium Negative Ion Ps

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