Pattnayak, G C

Rath, B

Torasia, S

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Indian]. Phyt, MB (3), 232-236 (1390)Anharmonic oscillator and Bogoliubov transformationG C PattnayakEkamra College, Bhubaneswar-751 002, Orissa, IndiaandS TorasiaDepartment of Science and Technology, Govt, of Orissa,Bhubaneswar-751 001, Orissa, IndiaandB RathDepartment of Physics, S. B. Women's College,Cuttack-753 001, Orissa. IndiaReceived 1 December 1989, accepted 28 March 1990The anharmonic oscillator occupies a cornerstone in many problems in Physics. Besides its various applications, It is an interesting system in which to the studies of new effects and phenomena in quantum theory, such as, divergence of perturbation series, convergence of pade approximants, singularities in the coupling constant plane and level crossing, convergence of strong coupling expansions, perturbation theory in large order and instantons can be made. However, from the literature it is observed that none of the authors have tested Bogoliubov transformation to study anharmonic oscillator. Bogoliubov transfor­mations are applicable to cases which satisfy commutational as well as anti- commutational relationships. An example of the later is the case of supercon­ductivity where weak coupling regime is predominant. An example of the former is the case of superfluidity where also weak coupling regime predominates. It is therefore, evident that before testing the range of coupling constant for the applicability of Bogoliubov transformation, one must apply it to the anharmonic oscillator.In this communication, groundstate energy of the anharmonic oscillator is studied using Bogoliubov transformation. The Hamiltonian considered here isp 2  Y 2H = y + ^ + A X *Using the coordinate transformation X-hlP0 =( 1)(2)232Anharmonlc oscillator and Bogoiiubov transformationth e  H am ilto n ian  in  eq. (1 )  is w r itte n  asH = a + f l + l + ^ | o ‘ + ( o + ) *  +  4 ( f l + ) ' * o + 4 f l ^ f l * + 6 ( a + ) * a * 4  + 6 (o + )»  + 1 2 0 *0  +  3 ]233(3 )6o«(4 )The eq . (4 )  is w r it te n  in norm al o rder using th e  com m u tatio n  re la tion .[ o o + ]  =  1 (5 )Let us n o w  pass from  Bose operators 0 and o * to  n ew  Bose operators (Burnerand R yvkin  1980 ) <c and <c* by means o f B og o iiub o v  transform ationo =  U * - V < *o + = . U * * - V *  (6)U “ - V ® ^ 1  t<  <t*]-1 *  I 0 > - 0to  ensure d ia g o n a liza tio n  o f H am ilto n ian  in eq . (4 ) . Before s im plify ing  theH am ilto n ian , w e  su b s titu te  (R ath  1984 , Banerjee and  Bhattacharjee 1 9 8 4 ) the  fo llo w in g ,a * ^ -3 U V a ^  (7 )( < j * ) * = - 3 U V ( o ^ ) *a * o » - 3 V “o “( a + ) ' * f l - 3 V » ( o * ) »UV,(a* ) *o® -[o *  +  (o + ) “] - f  2 V “fl*o .U sing th e  transform ation s  in eq (7 ) , th e  H a m ilto n ia n  in eq . (4 )  is w r itte n  asw hereandH  =  2  +  4  A f lM  A ( o + ) M  Bo*o,2  4A - ( 1 2 V » - 6 U V + 6 ) 2B =  ( 3 V » + 3 )A 4  1.(8)(9)( 10)2 3 4  G C Pattnayak S Torasla and B Rath N o w  th e  eq . (8 ) in  term s o f <  and can bo w r itte n  asH  =  [A U  • +  A V  -  BUY]** +  [A U  ■ + AV* -  B U V ](< +  )*-t- [BV* -  2AUV]**+ + [BU ■ -  2AU V] ^ ^The g ro un d state  energy o f th e  o sc illa to r is£ = < 0 | H | 0 >= -2AUV+BV* + ^ + g  4  2(11)( 12)T he  constant U or V is  determ ined  by m aking  th e  co effic ien t o f or zero i.e .AV*+AU*-BUV=0 j(13)Further th e  eq . (13) is s im p lified  using ths re la tio n  U“ =  1 +  V* and is w r itte n  as^( V » ) “( 1 8 a » +  12A) +  ( V * ) « | i 4  2 1 A + ? ^ ^ * )+  V » | ^ + 9 A + l ) - ® ^ " = 0 (1 4 )I t  is seen th a t g roundstate  energy c a lc u la te d  from  th e  eq. (1 2 )  using th e  solution  o f V  from  eq. (1 4 )  y ie ld s  reasonable good agreem ent w ith  th e  previous com putations  and are g iven  in T a b le  1.Table 1. Groundstate energy of the anharmonic oscillator.A Present Previous(Caswell1979)Previous (Rath 1984, Banerjee and Bhatta- charjee 1984)0.01 0.50728 0.50725 0.507330.1 0.56037 0.55914 0.562701 0.8125001 0.8037706 0.8358410 1.88618 1.50497 1.60727From  th e  ta b u la te d  resu lt, one concludes th a t B og o liub o v  transform ation  is ap p licab le  to  problem s in vo lin g  w e a k  coup lin g  as w e ll as m oderate ly  strong  co up lin g  constants.H o w e v e r, p rev io u s ly  it  w a s  tested  on ly  aga inst problem s invo lv ing  w eak  co u p lin g  constants. A p a rt from  th is , th e  energy expression in  eq . (1 2 )  is neither varia tio n a l nor p e rtu rb a tive  in nature. Th is  is m a in ly  due to  th e  transform ation  fromAnhormonic oscillator and Bogollubov transformation 235non*interacting Bose operator w ith  a 10 )  = 0  to  interacting Bose operator a l O )  =  - V | l > b y  using the Bogoliuiwv transformation.R«fer«ncoiBanerjee K and Bhattacharjee J K 1984 Phys. Rev. 0291111 Caswell W E 1979 Ann. Phys. 123184 Rath B 1984 Rad. Eff.80183Burner Yu B and Ryvkin M Sh 1980 Therm odynam ics and Statisticof Physics and Kinetics (Moscow: Mir Publisher) p 361

Anharmonic Oscillator and Bogoliubov Transformation

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Anharmonic Oscillator and Bogoliubov Transformation

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