The exact solutions of the Schr&#246;dinger equation for damped harmonic oscillator with pulsating mass and modified Caldirola-Kanai Hamiltonian are evaluated. We also investigated the case of under-damped for the twomodels constructed and the results obtained in both cases do not violate Heisenberg uncertainty principle

Akpabio, LE.

Antia, AD

Ibanga, EJ

Ikot, AN

Ituen, EE

English

AJOL - African Journals Online

Akaninyene D. Antia, Department of Physics, University of Uyo, Uyo, Nigeria Eno E. Ituen, Department of Physics, University of Uyo, Uyo, Nigeria Eno E. Ituen, Department of Physics, University of Uyo, Uyo, Nigeria Akpan N. Ikot, Department of Physics, University of Uyo, Uyo, Nigeria  Louis E. Akpabio, Department of Physics, University of Uyo, Uyo, Nigeria Eno J. Ibanga, Department of Physics, University of Uyo, Uyo, Nigeria  GLOBAL JOURNAL OF PURE AND APPLIED SCIENCES VOL 16, NO. 4 2010: 461- 467 COPYRIGHT© BACHUDO SCIENCE CO. LTD PRINTED IN NIGERIA. ISSN 1118-0579 www.globaljournalseries.com; Email: info@globaljournalseries.com QUANTUM THEORY OF DAMPED HARMONIC OSCILLATOR AKANINYENE D. ANTIA, ENO E. ITUEN, AKPAN N. IKOT, LOUIS E. AKPABIO AND ENO J.                IBANGA (Received 7 June 2009; Revision Accepted 2, February 2010)   ABSTRACT  The exact solutions of the Schrödinger equation for damped harmonic oscillator with pulsating mass and modified Caldirola-Kanai Hamiltonian are evaluated. We also investigated the case of under-damped for the two models constructed and the results obtained in both cases do not violate Heisenberg uncertainty principle.   KEY WORDS: Damped oscillator, Harmonic oscillator with pulsating mass, modified Caldirola-Kanai Oscillator,           uncertainty relation, wave function.   I INTRODUCTION   The aim of this paper is to evaluate the damped harmonic oscillator. The damping is here considered in the frame of Caldirola-Kanai model (Caldirola, 1941 and Kanai, 1948) and the recently developed model (Ikot et al, 2008). However, the problem of quantum oscillator with time-varying frequency had been solved (Um et al, 1987). The Hamiltonian of this model is usually quadratic in co-ordinates and momentum operators (Ikot et al, 2008).   The quantum calculation is applied because it will give the information about the particle at  intermediate levels (discrete energy and wave function can be determined at a particular point in time) unlike the mechanic or statistical mechanic that will give a continuum result (say from -∞ to ∞ without telling us what happen to the particle in between the two boundaries). Our main goals will be to construct two Lagrangians for this simple damped system and use the constructed Lagrangians to evaluate the equation of motion for the damped Harmonic oscillators, and also evaluate the minimum uncertainty relation for under damped regime in each case of Lagrangian.  II. REVIEW OF CALDIROLA-KANAI OSCILLATOR  The Caldirola-Kanai oscillator with a variable mass 9 has Hamiltonian of the form (Caldirola, 1941, Kanai, 1948 and Kim et al, 2003).   )1(221ˆ 222 qempmeHmtmtckγγω+=  or in the form (Caldirola, 1941, Kanai, 1948, and Um and Yeon, 2002).       )2(ˆ21ˆ)(21 22222 pemqtmeH ttckγγ ω −+=)                   461 462       AKANINYENE D. ANTIA, ENO E. ITUEN, AKPAN N. IKOT, LOUIS E. AKPABIO AND ENO J. IBANGA  where m is the mass of the oscillator, γ  is the damping coefficient, q̂  and p̂  are the co-ordinate and momentum operators and w(t) is time-dependent frequency of the oscillator. The Lagrangian associated with equation (1) and equation (2) are given as:    ( ) )3()(2,, 22  ++= qtqmqmetqqLmtωγγ&&&     ( ) [ ] )4()(22,, 22 qtqqmetqqLtωγγ++= &&&  respectively. The equation of motion for the classical co-ordinate q and momentum p of Equation (3) and Equation (4) are of the forms    )5(,0)()( 2 =++ qtqmtq ωγ&&&     )6(,0)(2)( 2 =++ qtqtq ωγ &&&   The time-dependent Schrödinger Wave Equation (SWE) describing this system is given as:         ( ) ( ) )7(.,, tqEtqH nnn ψψ =   The wave function and energy eigen value of equation (1) can be obtained as:    ( )= qmeHnmty mtnnn21212121!2),(hh ωπωψγ    )8(2121 22− +−×+qementiExp mtmγγ ωωh     )9(21mtenEnγω − += h  respectively.  And for equation (2), the corresponding wave function and energy eigen value is given as:    ( ) +−= − tnnteniExpnmtqγωπωψ 221!2),(2121h           )10(21 2122×− qmeHqem tnthhωω γγ and        )11(21 2 tn enEγω  += h  respectively.   III. Harmonic Oscillator With Pulsating Mass An Dmodified Caldirola-Kanai Oscillator    In this section we are going to construct two Lagrangians and use in evaluating the equation of motion for the damped Harmonic Oscillators and the uncertainty relation for the two models will be obtained.            QUANTUM THEORY OF DAMPED HARMONIC OSCILLATOR       463       A. HARMONIC OSCILLATOR WITH PULSATING MASS   We write the time-dependent Hamiltonian with time varying frequency as:  )12()(2212ˆ2ˆ 2220022 qttCoshmmptCoshH ωγγ+= −  using equations (7) and (12) we have the wave function of this oscillator as    =2cosh!222102121210tqmHnmnnnγωπωψhh         )13(2cosh212coshexp 402 − +− −tqmtntiγωωγh The lagrangian of equation (12) becomes:         [ ] )14()(221 22220 qtqtCoshmL ωγ−= &  and its equation of motion for the classical co-ordinate q and momentum p takes the form      )15(0)()(2tanh)( 2 =++ qttqttq ωγγ &&&  The solution of Equation (12) is     [ ] )16(,)( 2tanh2 titittBeAeetqΩ−Ω−+=γγ where 2tanh4)()( 222 tttγγω −=Ω   We summarized the general solution of Equation (15) for the over damped (OD), critically damped (CD) and under-damped (UD) as:        ( )[ ] )17(,)( 22 tanh tBSinhtACoshetqttΩ+Ω= −γγ           ( )[ ] )18(,)( 22 tanh BtAetqtt+= −γγ and        ( )[ ] )19(,)( 22 tanh tBSinhtACoshetqttΩ+Ω= −γγ respectively.   B. Modified Caldirola-Kanai Oscillator   We write the modified Caldirola-Kanai mode as:    ( )( ))20(ˆ)(21ˆ2ˆ 22020qtmepmeHttSinttSinωγγγγ++−+=  and the Lagrangian of this modified oscillator is:    ( ) ( ) ( ) )21()(2,, 2220 qtqmetqqL ttSin ωγγ −= + &&        464       AKANINYENE D. ANTIA, ENO E. ITUEN, AKPAN N. IKOT, LOUIS E. AKPABIO AND ENO J. IBANGA  and its equation of motion takes the form:          The solution of equation (22) is           ( )[ ] )23(,)( 2 tititttCosBeAeetq Ω−Ω+−+=γγγ where  22)()( 2  +−=ΩγγγωtCostt    and approximating 1≈tCosγ  for small damping yields:      )24()()( 22 γω −=Ω tt  Substituting equation (22) into equation (21) results:        )25(.)(2222112+=−−− +−ωγωωγωγγγtititttCosBeAeetq  substituting equation (20) into equation (7) we have the wave function as  = + ttnnn qemHnmγγωπωψ sin2102121210!2hh ( ) ( ))26(2121expsin2sin20 +−−× +−+ teniqem tttt ωω γγγγh We can now summarized the general solution of Equation (22) for the over damped (OD), critically damped (CD) and under-damped (UD) as ( )[ ] )27()( 12 tBSinhtACoshetq tCostΩ+Ω=+− γγ ( )[ ] )28()( 12 BtAetq tCost+=+− γγ and    ( )[ ] )29()( 12 tBSintACosetq tCostΩ+Ω=+− γγ respectively.  IV. Investigation Of The Under-Damped (Ud) Oscillator For The Two Models Constructed  A. The Under Damped Oscillation Of Harmonic Oscillator With Pulsating Mass.   We consider the quantum damped oscillator with time-dependent varying frequency given by equation (16). Subjecting Equation (16) to continuity condition (Man’ko, unpublished), q(0)=1 and ,)0( Ω= iq&  we obtain the arbitrary constant A and B as:             )30(1==iBA  and the classical trajectory becomes:  [ ] )31(,)( 2tanh2 tiSintCosetqttΩ+Ω=− γγ      ( ) ) 22 ( 0 ) ( ) ( ) ( 2 = + + + q t ω t q t Cos t q & & & γ γ γ QUANTUM THEORY OF DAMPED HARMONIC OSCILLATOR       465      where    2tanh4)( 2222 ttγγω −=Ω   An invariant operator for the general time-dependent oscillator whose eigen function is an exact quantum state up to a time-dependent phase factor had been introduced by Lewis and Risendeld (Lewis and Riesendeld, 1969).  Following Lewis and Risendeld, we can now introduce a pair of operators first order in both position and momentum operators (Kim et al , 2003 and Kim and Page, 2001).   [ ] )32(ˆ)(ˆ)()(ˆ qtptita ∗∗ −= εε &    [ ]qtptita ˆ)(ˆ)()(ˆ εε &−=+  And they are require to satisfy the quantum Liouville-Von Neumann equation defined as:    [ ] 0)(ˆ),(ˆ)(ˆ =+∂∂tHtatatih    [ ] )33(0)(ˆ),(ˆ)(ˆ =+∂∂ ++ tHtatatih  where )(tε in equation (32) must satisfies the classical damped equation of equation (15).   The operator in equation (32) and its Hermitian conjugate satisfy at any time t the boson commutation relation, and )(tε  must also satisfy the Wronskian condition (Kim, 2004)   [ ] )34(ˆ)()()(2ln2iqtttetCoshdd=− ∗∗ εεεεγββ&  The number operator defined by (Ikot et al, 2008)      )35()(ˆ)(ˆ)(ˆ tatatN +=  also satisfies equation (28), such that each number state      )36(,,)(ˆ 〉=〉 tnntntN  is also an exact quantum state of the time dependent Schrödinger equation    )37(),()(ˆ),( txtHtxti ψψ =∂∂h  The quantum dispersion coordination is obtained as (Ikot et al, unpublished).      )38(2)()(2ˆ 2tanh002ttemttmqγγεε−∗Ω=Ω=〉〈hh And the uncertainty in momentum is given as   2tanh02022)()()(2ˆttemtttmmpγγεε Ω=′Ω=〉〈 ∗h&&h x )39(222tanh2112222−−Ω+tSechtt γγγγ where )(tm′ is the reduced mass of the oscillator, which is defined as (Ikot et al, unpublished).   )40()( 2ln02 tCoshddemtmγββ=′         466       AKANINYENE D. ANTIA, ENO E. ITUEN, AKPAN N. IKOT, LOUIS E. AKPABIO AND ENO J. IBANGA  and β  is a variable parameter that takes values 1,2,3,E;n. By setting 1=β equation (39) is obtained.  The generalized uncertainty relation has the value:        ( ) )41(222tanh21422222 +Ω+=∆∆tSechttpqγγγγh Equation (41) is a generalized uncertainty relation and it satisfies the Heisenberg uncertainty relation. The product of   equations (38) and (39) gives a generalized Heisenberg relation which reduces to the exact when the damping coefficient γ  is set to zero.   B. Under Damped Oscillation of Modified Caldirol-Kanai Oscillator   Considering the quantum damped oscillator equation of equation (30) and imposing the boundary conditions we obtain the arbitrary constants A and B as:    Ω−=21γiA  and )42(2Ω=γiB  Substituting equation (42) into equation (29) we obtain the classical trajectory of the form:       ( ))43(221)(12ΩΩ+ΩΩ−=+−tSinitCosietqtCost γγγγ The quantity )(tε must satisfy the Wronskian condition        ( )[ ] )44(;)()()()( itttte ttSindd=− ∗∗+εεεεβγβγβ&&  By using the same procedure as in part A, we have the uncertainty in the co-ordinate as:    ( ) )45(212ˆ 222102ΩΩ+ΩΩ+Ω=〉〈 +− tSintSinemq tCostγγγγh and the dispersion of momentum by setting the variable parameter β  to unity takes the form:  ( )Ω+ΩΩ+=〉〈 +2222102 )(12ˆttSinemp tCostσγγγh −Ω−ΩΩ−ΩΩ1212222γγγtSintCos  )46(,2222)(2 ΩΩ+ Ω−ΩΩΩ+ tSintCostSintγγσγ where ( )[ ]ttSintCost γγγγσ ++−= 12)(  and the reduced mass in this case is defined as:  ( )( ))47()( 100ttttSinddememtm βγγβγβγβ ++==′  The uncertainty relation has the value    ( ) [ ] )48(,)(1422tgpq +=∆∆h where  tSintSintSintg ΩΩ+ΩΩ+ΩΩ+= 222222121)(γγγ   QUANTUM THEORY OF DAMPED HARMONIC OSCILLATOR       467      −Ω−ΩΩ−ΩΩΩ+ 1212221)(222γγγσtSintCost )49(2222)(ΩΩ+ Ω−ΩΩΩ+ tSintCostSint γγγσ  Equation (48) obeys basic quantum principle and when the damping term is set to zero it will return to the exact solution of time independent harmonic oscillator which is already known.  V. CONCLUSION   We have evaluated within the frame of Caldirola-Kanai model the damped harmonic oscillator in the  under-damped   regime.   Here,   we   obtained   the uncertainty relation for the under damped oscillator for the two models constructed and the results obtained satisfy the basic quantum principle (Heisenberg uncertainty principle). In this work, we restricted ourselves to the investigation of under damped oscillation. However, there is need for further investigation of the oscillation of over damped and critically damped oscillators.    REFERENCES Caldirola, P., 1941. Forze Nonconservative Nella  Meccanica Quantistica; Nouovo Cimento, vol.  18, 393.  Ikot, A. N., Akpabio, L. E., Akpan, I. O., Umo, M. I., and Ituen, E. E. (unpublished). Quantum Damped  Mechanical Oscillator.   Ikot, A. N., Ituen, E. E., Essien, I. E. and Akpabio, L. E., 2008. Path Integral Evaluation of Time-dependent  Oscillator in an External Field, Turk J. Phys. 32,  305-313.   Kanai, E., 1948. On the Quantization of the Dissipative  Systems, Prog, Theoretical Physics, Vol. 3, No.  440.                           Kim, S. P., 2004. Time-Dependant Displaced and  Squeezed Number States, Journal of the  Korean Physical  Society, Vol. 44 pp 446-450.  Kim, S. P. and Page, D. N., 2001. Classical and  Quantum Action – Phase Variables for Time- Dependent Oscillators, Phys. Rev. A. 64,  012104.  Kim, S. P., Santana, A. E., and Khana F. C.,  2003.  Decoherence of Quantum Damped Oscillator, J.  Korean Phys. Society, 43, 4.   Lewis Jr, H. R. and Riesendeld, W. B., 1969. An Exact  Quantum Theory of the Time-Dependent  Harmonic Oscillator and of a Charged Particle in  a Time Dependent Electromagnetic Field, J.  Math. Phys. (N.Y) 10,1458.  Man’ko, O. V. (unpublished). Damped Oscillator with  Delta Kicked Frequency.   Um, C. I. and Yeon, K. H., 2002. Quantum Theory of the  Harmonic Oscillator in Nonconservative  Systems, Journal of the Korean Physical  Society, Vol. 41, pp 594-616.     Um, C. I., Yeon, K. H., and Kahng W. H., 1987. The  Quantum Damped Driven Harmonic Oscillator,  J. Phys. A. Math gen. 20,611.     

Quantum theory of damped harmonic oscillator

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Quantum theory of damped harmonic oscillator

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