Ant¸nioc A. Strèklac

EpÐkouroc Kajhght c

Tm matoc Majhmatik¸n

PanepisthmÐou Patr¸n

Biografikì ShmeÐwmakai

Upìmnhma Episthmonik c Drasthriìthtac

PATRA 2008

1

1 Proswpik StoiqeÐa

Ep¸numo Strèklac'Onoma Ant¸niocHmeromhnÐa gènnhshc 15 OktwbrÐou 1951Tìpoc gènnhshc Aj naOikogeneiak katstash 'Eggamoc me dÔo paidi (agìria)

DieÔjunsh Panaqaðdoc Ajhnc 95, DasÔlion Ptra AqaÐacthlèfwno 2610 - 270442

Hlektronik DieÔjunsh http://math.upatras.gr/∼streklasemail - streklas@math.upatras.gr

2 TÐtloi Spoud¸n

1969 Apolut rio apì to A' Gumnsio Arrènwn AgÐwn AnargÔrwn Attik c.1975 PtuqÐo Tm matoc Majhmatik¸n thc Fusikomajhmatik c Sqol c tou

PanepisthmÐou Patr¸n me LÐan Kal¸c (bajmìc 8.2).1980 Didaktorikì DÐplwma apì thn Fusikomajhmatik Sqol tou

PanepisthmÐou Patr¸n (Kldoc Fusik c) me bajmì 'Arista.Epiblèpwn Kajhght c Astèrioc GiannoÔshc.

3 Epaggelmatik Drasthriìthta

2/1975 - 3/1980Ereunht c, me sÔmbash èrgou sthn 'Edra thc Jewrhtik cFusik c tou PanepisthmÐou Patr¸n (èmmisjoc me sÔmbashanajèshc ektelèshc èrgwn epÐ perÐpou 1.5 èth sunolik.)

4/1980 - 2/1982Stratiwtik JhteÐa.

2

6/1982 - 8/1982Epimelht c sthn Taktik 'Edra twn Genik¸n Majhmatik¸n(Kajhght c G. Dsioc) thc Fusikomajhmatik c Sqol ctou PanepisthmÐou Patr¸n.

9/1982 - 10/1985Lèktorac tou Tm matoc Majhmatik¸n tou PanepisthmÐouPatr¸n (Fek 227/22.10.82) sÔmfwna me tic metabatikècdiatxeic tou nìmou 1268/1982.

11/1985 - 4/1988Mìnimoc Lèktorac tou parapnw Tm matoc sÔmfwna me tictropopoi seic tou nìmou 1566/1985.

5/1988 - S meraMìnimoc EpÐkouroc Kajhght c tou parapnw Tm matoc kaiapì to 1994 pl rouc kai apokleistik c apasqìlhshc.

9/1992 - 7/1994Kajhght c sto Tm ma DioÐkhshc Epiqeir sewn thcSqol c DioÐkhshc kai OikonomÐac tou T.E.I thc Ptrac.

9/1995 - 2/1996Kajhght c sto Tm ma Touristik¸n Epiqeir sewn thcSqol c DioÐkhshc kai OikonomÐac tou T.E.I thc Ptrac.

4 'Allec Episthmonikèc Drasthriìthtec kai

DiakrÐseic

• UpotrofÐa apì to IKU sto B',G' kai D' ètoc Spoud¸n.• Mèloc thc Majhmatik c EtaireÐac

Sumetoq se Sunèdria1. Balkanikì Sunèdrio twn Efarmosmènwn Majhmatik¸n

sthn JessalonÐkh to 1976TÐtloi anakoin¸sewni) Basic Theorems in Operator Algebraii) On some properties of Operators

2. Diejnèc Sunèdrio Adronik c Fusik c (Ptra 1986)3. 4o Panell nio Sunèdrio Majhmatik c Anlushc (Ptra 1994)

3

5 Ekpaideutik Drasthriìthta

DidaskalÐa twn majhmtwn

1982 - 19831) Majhmatik Anlush I sto Fusikì Tm ma G' ètoc.2) Majhmatik Anlush II sto Fusikì Tm ma G' ètoc.

1983 - 19873) Majhmatik gia QhmikoÔc III sto Qhmikì Tm ma B' ètoc.

1983 - 19844) Majhmatik gia QhmikoÔc IV sto Qhmikì Tm ma B' ètoc.11) Sun jeic Diaforikèc Exis¸seic I sto Majhmatikì Tm ma G' ètoc.1983 - 19995) Kbantomhqanik I sto Majhmatikì tm ma G' ètoc.6) Kbantomhqanik II sto Majhmatikì tm ma G' ètoc.

1999 - 20087) Eisagwg sthn Kbantomhqanik sto Majhmatikì tm ma D' ètoc.

1994 - 20068) Pragmatik Anlush IV sto Majhmatikì tm ma B' ètoc.

1995 - 20069) Eisagwg sthn SÔgqronh Fusik sto Majhmatikì tm ma B' ètoc.

2006 - 200810) Eisagwg sthn Hlektrodunamik sto Majhmatikì tm ma D' ètoc.2008 -11) Sun jeic Diaforikèc Exis¸seic I sto Majhmatikì Tm ma G' ètoc.

1994 - 200812) Sqetikistik Kbantomhqanik kai Kbantik JewrÐa twn PedÐwn

sto Metaptuqiakì Prìgramma tou MajhmatikoÔ Tm matoc.

1992 - 199413) Genik Majhmatik sto Tm ma DioÐkhshc sto T.E.I. thc Ptrac.1995 - 1996 (Qeimerinì exmhno)14) Genik Majhmatik sto Tm ma TourismoÔ sto T.E.I. thc Ptrac.

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EpÐbleyh Ergasi¸n gia Metaptuqiakì DÐplwma EidÐkeushc

1) H Klassik jewrÐa tou hlektromagnhtismoÔ kai h epÐdrashaut c sthn Kbantikh JewrÐa. (2004) Andrèac Gkbranitz.

2) H mh antistreyimìthta tou qrìnou. (2006)Panagi¸thc Gewrgakìpouloc.

3) H summetrÐa bajmÐdac kai to pedÐo Gink - Millc.H ergasÐa eÐnai se exèlixh. Elewnìra EustajÐou.

Mèloc epitrop¸n gia Didaktorikèc Diatribèc

1) Asumptwtik sumperifor thc enèrgeiacbiskoelastik¸n kumtwn. Ptra 1990, Filarèth ZafeiropoÔlou.

2) Kbantik jewrÐa hmiagwg¸n se uyhlMagnhtik pedÐa. Sumbol sthn melèthtou kbantikoÔ fainomènou Hall. Ptra 1991. Ge¸rgioc Kl roc.

Didaktik BiblÐa

1) Eisagwg sthn Kbantomhqanik . Ekdìseic PanepisthmÐou Patr¸n.2) SÔgqronh Fusik . Ekdìseic PanepisthmÐou Patr¸n.

Panepisthmiakèc paradìseic

3) Dianusmatik Anlush kai Seirèc Fouriè. Ekdìseic Panep. Patr¸n.4) Dianusmatik Anlush me stoiqeÐa apì to Mathematica,

gia to mjhma Majhmatik II gia to Qhmikì tm ma.

'Ola ta parapnw biblÐa brÐskontai anhrthmèna sto diadÐktuo sthndieÔjunsh: http://math.upatras.gr/∼ streklas/didaskalia.htm

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6 Episthmonikì 'Ergo

Diatrib

H ditaxic twn telest¸n eic thn Kbantomhqanik .EgkrijeÐsa upì thc Fusikomajhmatik c Sqol c tou PanepisthmÐouPatr¸n. Epiblèpwn Kajhght c Astèrioc GiannoÔshc.Ptrai 1980

PerÐlhyh Diatrib c

Eic thn diatrib , melet¸men tac efarmogc thc jewrÐac diatxewc twntelest¸n eic thn kbantomhqanik n

Dia twn antistoiqi¸n twn diatetagmènwn telest¸n proc tac sun jeic miga-dikc sunart seic, ai kbantomhqanikaÐ sqèseic lambnoun thn sun jhn klas-sik n twn morf n. To sÔsthma perigrfetai katèujeÐan eic ton q¸ron twnfsewn apì mÐan sunrthsin katanom c kai ta parathr sima megèjh, parÐstan-tai amfimonoshmntwc apì sun jeic migadikc sunart seic. H summetrik Weyl antistoiqÐa èqei melethj perissìteron, diìti h antÐstoiqoc sunrthsictou telestoÔ thc m trac puknìthtoc sumpÐptei me thn gnwst n katanom ntou Wigner. Melet¸men thn perigraf n aut thc kbantomhqanik c dia thn mhsqetikistik n perÐptwsin kai thn epeiteÐnomen kai eic thn sqetikistik n toiaÔ-thn. OrÐzomen ton sqetikistikìn Wigner telest kai eurÐskomen tìson tacidisunart seic ìson kai tac idiotimc tou. ApodeiknÔomen ìti h idiosunrthsiceÐnai mia m tra me stoiqeÐa ton f metasqhmatismon tou ginomènou dÔo spinortou Dirac, en¸ ai idiotimaÐ eÐnai h diafor twn idiotim¸n duo exis¸sewn touWigner.

Af etèrou h ditaxic twn telest¸n dieukolÔnei kai aplopoieÐ thn epÐdrasintwn telest¸n epÐ twn diafìrwn sunart sewn. ApodeiknÔetai ìti dunmeja nagrywmen ton telest n qronik c exelÐxewc enìc sust matoc upì ekjetik nmorf n. AnaptÔssomen akoloÔjwc ton telest n autìn eic mÐan katllhlondiatetagmènhn morf n, toiaÔth ¸ste h epÐdrasic tou en logw telestoÔ epÐ thndèlta sunrthsin upologÐzetai me ènan aplìn kai eujÔ trìpon. H prokÔptousasunrthsic eÐnai o propagator tou sust matoc kai paÐzei ousi¸dh rìlon eicthn parstasin Feynman thc kbantomhqanik c h opoÐa anaptÔssetai epÐshc.Di qronik¸c exartwmènouc anexrthtouc telestc tou t, oi opoÐoi eÐnaipolu¸numa deutèrac txewc wc proc tac kanonikc suzugeÐc metablhtc, hditaxic epitugqnetai dia thc mejìdou thc parametrik c paragwgÐsewc.

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DhmosieÔseic se Diejn Episthmonik Periodik me Kritèc

1. Relativistic Wigner Operator and its Distribution.

A.Jannussis, A. Streclas, D.Sourlas and K. Vlachos.

Published in: Lettere al Nuovo Cimento Vol.18, 11 (1977), pp. 349 -351.

2. On the Equation of Motion.

A.Jannussis, A. Streclas, D.Sourlas and K. Vlachos.

Published in: Lettere al Nuovo Cimento Vol.19, 5 (1977), pp. 163 -164.

3. Some Properties of Commutators and the Equation of Motion.

A.Jannussis, A. Streclas, D.Sourlas and K. Vlachos.

Published in: Physica Scripta Vol.15, (1977), pp. 163 - 166.

4. Wigner Operator of Angular Momentum in Phase Space.

A.Jannussis, A. Streclas, D.Sourlas and K. Vlachos.

Published in: Lettere al Nuovo Cimento Vol.20, N. 7 (1977), pp. 238 -240.

5. On some properties of Weyl relations.

A.Jannussis, A. Streclas, D.Sourlas and K. Vlachos.

Published in: Lettere al Nuovo Cimento Vol.20, 11 (1977), pp. 460 -462.

6. Ordering of the Exponential of Quadratic forms in Boson Operatorsand Some Applications.

A.Jannussis, A. Streclas, N. Patargias, D.Sourlas and K. Vlachos.

Published in: Physica Scripta Vol.18, (1978), pp. 13 - 17.

7. Quantum Friction in a Uniform Magnetic Field.

A.Jannussis, G. Brodimas and A. Streclas.

Published in: Lettere al Nuovo Cimento Vol.25, N. 6 (1979), pp. 283 -286.

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8. Propagator with Friction in Quantum Mechanics.

A.Jannussis, G. Brodimas and A. Streclas.

Published in: Physics Letters Vol.74A, N 1,2 (1979), pp. 6 - 10.

9. The Lie Algebra of Wigner Operators of Angular Momentum.

D.Sourlas, A.Jannussis and A. Streclas.

Published in: Hadron Journal 3, (1980), pp. 1431 - 1445.

10. Fermi - Dirac Statistics for Free Electrons in Uniform Electic and Ma-gnetic Fields.

A.Jannussis, A. Streklas and K. Vlachos.

Published in: Physica 107A (1981), pp. 575 - 586.

11. Wigner Representation of Bloch Electrons in Uniform Fields.

A.Jannussis, A. Streklas and K. Vlachos.

Published in: Physica 107A (1981), pp. 587 - 597.

12. Some Remarks on The Caldirola - Montaldi Equation.

A.Jannussis, A. Streclas, A. Leodaris, N. Patargias, V. Papatheou,P. Fillippakis, T. Fillippakis. V. Zisis, and N. Tsagas.

Published in: Lettere al Nuovo Cimento Vol., N 18 (1982) pp. 571 -574.

13. Some Remarks on non - negative Quantum Mechanical DistributionFunction.

A.Jannussis, N. Patargias, A. Leodaris, P. Fillippakis and T. Fillippa-kis, A. Streklas, and V. Papatheou.

Published in: Lettere al Nuovo Cimento Vol. 34 N. 18 (1982) pp. 553- 558.

14. Foumdation of the Lie Admissible Fock Space of the Hadronic Mecha-nics.

A.Jannussis, G. Brodimas, D. Sourlas, A. Streclas, P. Siafaricas, L.Papaloucas and N. Tsangas.

Published in: Hadronic Lournal 5, (1982), pp. 1923 - 1947.

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15. Statistical properties of one dimensional non canonical oscillators.

M. Mijatovic, A.Jannussis and A. Streklas.

Published in: Hadronic Lournal V. 8 N. 6 , (1985), pp. 327 - 330.

16. Boltzmann Statistics of Quantum Friction.

M. Mijatovic, A.Jannussis and A. Streklas.

Published in: Physics Letters, V. 122, 1 (1987) pp 31 - 35.

17. Deformed Harmonic Oscillator for Non - Hermitian Operator and theBehavior of pt and CPT Symmetries

A. Jannussis, K. Vlachos, V. Papatheou, A. Streklas.

Published in: Int. J. of Mod. Phys. B, Vol. 20, 16. (2006), pp. 2313 -2322.

18. Harmonic Oscillator in non Commuting two Dimensional Space

Antony Streklas.

Publiched in: Int. J. of Mod. Phys. B, Vol 21 no. 32, (2007), pp 5363- 5380

19. Deformed damped harmonic oscillator

Antony Streklas.

Published in: Physica A 377 (2007) pp 84 - 94

20. Quantum damped harmonic oscillator on non - commuting plane

Antony Streklas.

Published in: Physica A 385 (2007) pp 124 - 136

'Allec dhmosieÔseic

1. Quantum deformed problem of electrons in the Dirac theory.

A. Jannussis, V. Papatheou, A. Streklas, K. Vlachos.

Published in: Handbooks. Treatises. Monographs, 49. Lagrange andHamilton Geometries and their Application.

In memory of professor Gregor Tsagas

Ed: Radu Miron. (2004), pp. 133 - 143.

9

Eteroanaforèc stic ergasÐec apì llec ergasÐec

ErgasÐa 1 1. Relativistic Quantum Statistics in the Wigner formalismand its application to the Toda oscillator. Hamo A., Vojta G.,Zylka C. Europhys Lett. 15 (8), (1991) 809 - 813.

ErgasÐa 2 1. Quantum and classical mechanics of Q - deformed sy-stems. Shabanov Sv. J. Phys. A - Math Gen 26 (11), (1993)2583 - 2606

2. The Poisson brackets for Q - deformed systems. Shabanov Sv. J.Phys. A - Math Gen 25 (22), (1992) 1245 - 1250

ErgasÐa 6 1. Thomas - Fermi theory in a weak slowly varyng vectorpotential. Haris Ra, Cina Ja. J. Chem. Phys. 79(3), (1983) 1381- 1383

ErgasÐa 7 1. Classically Equivalent Hamiltonians and Ambiguities of

Quantization - A Particle in a Magnetic - Field. Dodonov V.V,Manko V.I, Skarzhinsky V.D. Nuovo Cimento B Volume: 69 Issue:2 Pages: 185 - 205 (1982)

ErgasÐa 8 1. Nonperturbative Analysis of the Harmonic Oscillator wi-th a Time-Dependent Frequency. A. Fedoul, A.L .Marrakchi, S.Sayouri and A. Chatwiti. M. J. Condensed Matter Volume 8,Number 1 (2007) pp. 14 - 20

2. On the quantum motion of a generalized time-dependent forcedharmonic oscillator. A. Lopes de Limaa, , A. Rosasa, and I.A.Pedrosa, Annals of Physics (2007). Article in press.

3. Propagator for the time - dependent charged oscillator via linearand quadratic invariants. Abdalla, M.S., Choi, J. - R. Annals ofPhysics 322 (12), (2007) pp. 2795 - 2810

4. Path integral for a harmonic oscillator with time - dependent massand frequency. Pepore, S., Winotai, P., Osotchan, T., Robkob, U.2006 Science Asia 32 (2), pp. 173 - 179

5. Complete exact quantum states of the generalized time - depen-dent harmonic oscillator. Pedrosa, I.A. Modern Physics Letters B18 (24), (2004) pp. 1267 - 1274

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6. Squeezed states of the generalized minimum uncertainty state forthe Caldirola - Kanai Hamiltonian. Kim, S.P. J. of Physics A: 36(48), (2003) pp. 12089 - 12095

7. Quantum theory of the harmonic oscillator in nonconservative sy-stems. Um, C. - I., Yeon, K. - H. J. of the Korean Physical Society41 (5), (2002) pp. 594 - 616

8. Evolution of quantum systems with a scaling type time - depen-dent Hamiltonians. Samaj, L. International Journal of ModernPhysics B 16 (26), (2002) pp. 3909 - 3914

9. The quantum damped harmonic oscillator. Um, C. - I., Yeon, K.- H., George, T.F. Physics Report 362 (2 - 3), (2002) pp. 63 - 192

10. The Caldirola - Kanai model and its equivalent theories for a dam-ped oscillator. Huang Mc, Wu Mc. Chinese J. of Physics Volume:36 Issue: 4 (1998) Pages: 566 - 587.

11. Equivalence among the propagators of three - dimensional time -dependent quadratic systems and free particles. Filardo Bassalo,J.M., De Tarso Santos Alencar, P., Dorsa Cattani, M.S. NuovoCimento B 113 (6), (1998) pp. 691 - 698

12. Wave functions of a time - dependent harmonic oscillator withand without a singular perturbation. Pedrosa, I.A., Serra, G.P.,Guedes, I. Physical Review A 56 (5), (1997) pp. 4300 - 4303

13. A fresh look at the BCK frictional Lagrangian. Menon, V.J.,Chanana, N., Singh, Y. Progress of Theoretical Physics 98 (2),(1997) pp. 321 - 329

14. Equivalence among the propagators of three - dimensional time- dependent quadratic systems and free particles, by solving theSchrodinger equation. Bassalo, J.M.F. Nuovo Cimento B 111 (7),pp. 793 - 797 ( 1996)

15. Heisenberg - picture approach to the exact quantum motion of atime - dependent forced harmonic oscillator. Kim, H. - C., Lee,M. - H., Ji, J. - Y., Kim, J.K. Physical Review A - 53 (6), pp.3767 - 3772 (1996)

16. Equivalence among the propagators of time - dependent quadraticsystems and free Particles, by solving the schrodinger - Equation.

11

Bassalo JMF. Nuovo Cimento B. Volume: 110 Issue: 1 Pages:23-32 (1995)

17. An Anisotropic Charged Oscillator in the Presence of a ConstantMagnetic - Field. Abdalla Ms.Nuovo Cimento B Volume: 105Issue: 10 Pages: 1119 - 1129 (1990)

18. Feynman’s propagator for a charged particle with time - dependentmass in a crossed time - varying electromagnetic field. Alvaro deSouza Dutra and Bin Kang Cheng. Phys. Rev. A Volume 39Issue 11 5897 (1989)

19. Radiation Damping of a Quantum Harmonic - Oscillator. Castri-giano Dpl, Kokiantonis N. Journal of Physics A Volume: 20 Issue:13 Pages: 4237 - 4245 (1987)

20. The quantum damped driven harmonic oscillator. C I Uml, K HYeon and W H Kahng.J. Phys. A: Math. Gen. 20 (1987) 611-626.

21. Transformation of the Free Propagator to the Quadratic Propaga-tor. Junker G, Inomata A. Physics Letters A Volume: 110 Issue:4 Pages: 195 - 198 (1985)

22. Some Remarks on the Unitary Time Evolution Operator of theQuantum - Mechanical Forced Oscillator - The Weierstrass Pro-pagator. Grinberg H. Nuovo Cimento B Volume: 83 Issue: 1(1984) Pages: 17 - 22

23. Exact Evaluation of the Propagator for the Damped Harmonic -Oscillator. Cheng Bk. Journal of Physics A Volume: 17 Issue: 12Pages: 2475 - 2484 (1984)

24. On The Path Integral Quantization of the Damped Harmonic -Oscillator. Gerry Cc. Journal of Mathematical Physics Volume:25 Issue: 6 Pages: 1820 - 1822 (1984)

25. Propagators for Time - Dependent Linear and Quadratic Poten-tials and Inverse Weierstrass Transform. Bassetti B, Montaldi E,Raciti M. Physics Letters A Volume: 90 Issue: 7 Pages: 333 - 336(1982)

26. Classicai and Quantum - Mechanics of the Damped Harmonic -Oscillator. Dekker H. Physics Reports Letters Volume: 80 Issue:1 Pages: 1 - 112 (1981)

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ErgasÐa 11 1. Moment equation in the Wigner formulation for super -lattice band structures. Reich Rk, Ferry Dk. Phys. Lwtt. A 91(1)(1984) 31 - 32.

2. Quantum collision - theory with phase - space distribution. Car-ruthers P., Zachariasen F. Rev. Mod. Phys. 55(1), (1983) 245 -285

ErgasÐa 13 1. Symmetry transformation in extended phase space: Theharmonic oscillator in the Husimi representation. Bahrami, S.,Nasiri, S. 4, art. no. 014 (2008)

2. Symmetry, Integrability and Geometry: Methods and Applica-tions. Sadollah Nasiri. Vol. 2 (2006), Paper 062, 12 pages

3. Measurement - induced decoherence and Gaussian smoothing ofthe Wigner distribution function. Chun, Y. - J., Lee, H. - W.Annals of Physics 307 (2), (2003) pp. 438 - 451

4. Phase - Space Representations of Wave - Equations with Applica-tions to the Eikonal Approximation for Short - Wavelength Waves.McDonald Sw. Physics Reports. Letters. Volume: 158 Issue: 6(1988) Pages: 337 - 416

5. Phase - Space Eikonal Method for Treating Wave - Equations.McDonald Sw. Physical Review Letters. Volume: 54 Issue: 12(1985) Pages: 1211 - 1214

6. Obtaining non - Negative Quantum - Mechanical Distribution Fu-nction. Bertrand P, Doremus Jp, Izrar B, et al. Physics LettersA Volume: 94 Issue: 9 (1983) Pages: 415 - 417

ErgasÐa 14 1. Hopf - type deformed oscillators, their quantum doubleand a q - deformed Calogero - Vasiliev algebra . Paolucci A,Tsohantjis I. Physics Letters A Volume: 234 Issue: 1 (1997) 27 -34

2. On boson algebras as Hopf algebras. Tsohantjis I, Paolucci A,Jarvis PD. Journal of Physics A Volume: 30 Issue: 11 (1997)4075 - 4087.

3. Deformed boson algebras and the quantum double construction,McAnally DS, Tsohantjis I. Journal of Physics A Volume: 30 Is-sue: 2 (1997) 651 - 659.

13

4. An Alternative approach to deformed statistics. L. DeFalco, R.Mignani, R. Scipioni, Nuovo Cim.A108: (1995) 1029 - 1034.

5. An Introduction to Lie isotopic theories: Part 1: R. M. Santilli(IBR, Palm Harbor) . IBR - 1147, (1994). 38pp.

6. Generalized Deformed Oscillator Corresponding to the ModifiedPoschl - Teller Energy - Spectrum. Daskaloyannis C. Journal ofPhysics A Volume: 25 Issue: 8. (1992) Pages: 2261 - 2272

7. A Q - Deformed Completely Integrable Bose - Gas Model. Bogo-liubov Nm, Bulough Rk. Journal of Physics A Volume: 25 Issue:14 (1992) Pages: 4057 - 4071

8. Infinitesimally Deformed Field and String Theories. Ahmed E,Hegazi As. Journal of Mathematical Physics. Volume: 33 Issue:1 (1992) Pages: 379 - 381

9. Quantum Phase and a Q - Deformed Quantum Oscillator. EllinasD. Physical Review A Volume: 45 Issue: 5 (1992) Pages: 3358 -3361

10. Quantum phase and q oscillator. D. Ellinas . HU - TFT - 91 - 18,(1991). 9pp.

11. Quantized planes and multiparameter deformations of Heisenbergand GL(N) algebras.D.B. Fairlie, C. K. Zachos. ANL - HEP - CP -91 - 28, DTP - 91 - 19, Jan 1991. 12pp. Invited talk given at MiamiNATO ARW, Coral Gables, FL, Jan 7 - 12, 1991. Published inCoral Gables QFT 1991:81 - 92 (QC174. 45: N21: 1991) Also inDiff. Geom. Meth. 1991: 426 - 438 (QC20: I35: 1991)

12. Apparent consistency of Rutherford’s hypothesis on the neutronstructure via the hadronic generalization of quantum mechanics.1. Nonrelativistic treatment. R. M. Santilli. IBR - TP - 903, IC- 91 - 47, (1990). 81pp.

13. On the Q - Oscillator and the Quantum Algebra Suq (1,1). KulishPp, Damaskinsky Ev. Journal of Physics A Volume: 23 Issue: 9(1990) Pages: L415 - L419

14. Remarks on Lie - Isotopic Lifting and the Noncanonical Commu-tation Relations. Nishioka M. Let. Nuovo Cim. 39, (1984) 268,

14

15. Lie Isotopic Lifting Of Unitary Symmetries And Of Wigner’s The-orem For Extended Deformable Particles. R. M. Santilli. TP - DE- 83 - 5, (1983). 17pp. Lett.Nuovo Cim.38: (1983). 509.

ErgasÐa 15 1. q - parameter dependence of a gas in equilibrium witha deformed solid. DeFalco, L., Mignani, R., Scipioni, R. PhysicsLetters, Section A: 223 (3), (1996) pp. 155 - 156

2. Thermodynamics of a Weakly Deformed Crystal - Lattice with Q- Complex Parameter. De Falco L, Mignani R. Modern PhysicsLetters B Volume: 9 Issue: 23 Pages: 1521 - 1525 (1995)

ErgasÐa 16 1. Propagator for the time - dependent charged oscillatorvia linear and quadratic invariants. Abdalla, M.S., Choi, J. - R.Annals of Physics 322 (12), (2007) pp. 2795 - 2810

2. Lie algebraic treatment of a charged oscillator in the presence of aconstant magnetic field. Abdalla MS. Nuovo Cimento B Volume:112 Issue: 11 (1997) Pages: 1549 - 1554

3. An Anisotropic Charged Oscillator in the Presense of a ConstantMagnetic - Field. Abdalla Ms. Nuovo Cimento B Volume: 105Issue: 10 (1990) Pages: 1119 - 1129

4. Charged - Particle in the Presence of a Variable Magnetic - Field.Abdalla Ms. Physical Review A Volume: 37 Issue: 10 (1988)Pages: 4026 - 4029.

Anaforèc apì biblÐa

ErgasÐa 1 1. Statistical mechanics of quarks and hadrons: proceedingsof an international symposium held at the University of Bielefeld,F.R.G., August 24-31, 1980 by H Satz

ErgasÐa 7 1. Density matrices and Wigner functions of semi classicalquantum systems. Dodonov V. Manko V. Akademia hayk CCCPVol. 167 (1986)

ErgasÐa 8 1. Classical and Quantum Dissipative Systems: dissipativesystems. Mohsen Razavy. Imperial College Press. (2006).

2. Path Integrals in Quantum Mechanics, Statistics, Polymer Physi-cs, and Financial Markets. Hagen Kleinert World Scientific (2004)

15

3. Theory of Nonclassical States of Light.

Dodonov, V. V. Manko V.I. CRC Press (2003)

4. Handbook of Feynman path integrals - Introduction.

Grosche C, Steiner F. Volume: 145 Pages: 1 - + (1998)

ErgasÐa 10 1. V. Dodonov V. Manko, Academy of Sciences U.S.S.R.Lebedev Physical Institute Vol. 183 Moscow 1987

ErgasÐa 11 1. Transport in Nanostructures. David K. Ferry, StephenM. Goodnick, Cambridge University Press, (1997)

2. V. Dodonov V. Manko, Academy of Sciences U.S.S.R. LebedevPhysical Institute Vol. 183 Moscow 1987

ErgasÐa 13 1. Stochastic Quantum Mechanics and Quantum Space -Time. Prugovecki E, D. Reidel Pub. Com. (1984)

ErgasÐa 14 1. Commuting Elements in q - Deformed Heisenberg Al-gebras. Lars Hellstrom, Sergei D. Silvestrov. World Scientific.(2000).

2. On Klauder’s Path: A Field Trip. John R. Klauder, Gerard G.Emch, G. C. Hegerfeldt, L. Streit. World Scientific (1994).

3. Hadronic Mechanics and nonpotential Interactions. Part 2 Physi-cs. Proceedings of the Fifth International Conference on HadronicMechanics. Edited by Hyo Chul Myung. Nova Science PublishersInc. New York. (1992).

Oi phgèc twn anafor¸n

• http://apps.isiknowledge.com/WOS GeneralSearch.do

• http://books.google.com

• http://scholar.google.gr/

• http://www.scopus.com/scopus/home.url

16

Perilhyh twn dhmosieÔsewn

1. Relativistic Wigner Operator and Its Distribution.Sqetikistikìc telest c tou BÐgkner kai h katanom tou.

'Eqei apodeiqjeÐ ìti o telest c BÐgkner pou emfanÐzetai sthn jemelÐwshthc Kbantomhqanik c apì touc Mpop kai KoÔmpo pnw ston q¸ro twn f-sewn, èqei idiotimèc thn diafor twn idiotim¸n dÔo isodunmwn exis¸sewn touSrèntingker kai san idiosunart seic thn gnwst katanom tou BÐgkner.[

H(P , Q)−H(P ∗, Q∗)]F (p, q) = i~

∂

∂ tF (p, q)

P = p− i~2

∂

∂qQ = q +

i~2

∂

∂pP ∗ = p+

i~2

∂

∂qQ∗ = q − i~

2

∂

∂p

Sthn ergasÐa aut efarmìzoume thn mèjodo aut sthn sqetikistik perÐ-ptwsh kai apodeiknÔoume ìti h idiotimèc eÐnai pli h diafor dÔo isodunmwnexis¸sewn tou Ntirk.

H idiosunrthsh eÐnai ènac 4 × 4 pÐnakac me stoiqeÐa katanomèc tÔpouBÐgkner, dhlad

Fij(p, q) =

∫ −∞∞

exp (i/~)pq′Ψi(q − q′/2)Ψ∗j(q + q′/2)dq′

ìpou Ψi(q) eÐnai ta spÐnor tou Ntirk.

2. On the Equation of MotionParathr seic gÔrw apì thn exÐswsh thc kin sewc

BrÐskoume thn exÐswsh kin sewc gia èna fusikì sÔsthma sthn mh - kano-nik kbantomhqanik ìpou o metajèthc twn telest¸n twn suntetagmènwn kaitwn orm¸n eÐnai ènac autosuzug c grammikìc telest c c.

dq

dt= i[H, q] = p,

dp

dt= i[H, p] = −f(q), i[p, q] = c

ìpou f(q) eÐnai mia analutik sunrthsh.

17

3. Some Properties of Commutators and the Equations of MotionKpoiec idiìthtec twn metajet¸n kai oi exis¸seic thc kin sewc

Sthn ergasÐa aut upologÐzoume touc metajètec sunart sewn telest¸ngia mh metajetoÔc telestèc pou an koun ston daktÔlio tou Qizenmpergk.O metajèthc dÔo telest¸n mporeÐ na ekfrasteÐ me thn bo jeia twn basik¸ntelest¸n kai twn parag¸gwn touc. Qrhsimopoi¸ntac to je¸rhma thc para-gwg sewc miac sunart sewc telest¸n wc proc kpoia parmetro, mporoÔmena broÔme thn exÐswsh kin sewc gia èna sÔsthma, sthn kanonik kai mh -kanonik kbantomhqanik . BrÐskoume epÐshc me thn Ðdia mèjodo thn exÐswshkin sewc sthn parstash tou BÐgkner.

4. Wigner Operators of Angular Momentum in Phase SpaceOi telestèc BÐgkner thc stroform c ston q¸ro twn fsewn

OrÐzoume dÔo eÐdh telest¸n BÐgkner gia thn stroform , pou eÐnai to -

jroisma kai h diafor twn sunhjismènwn telest¸n thc stroform c ~L kai ~L∗

sthn parstash tou Bèul.Oi telestèc autoÐ orÐzontai wc akoloÔjwc:

~W+ = 2(~q × ~p) +~2

2~∇p × ~∇q, ~W− = −i~

(~p× ~∇p + ~q × ~∇q

)BrÐskoume tic idiotimèc kai tic idiosunart seic twn telest¸n aut¸n kai

apodeiknÔoume ìti oi telestèc W−i W

+i kai 1

2(W−

i W−i + W+

i W+i ) eÐnai oi dÔo

anexrthtoi KazimÐr analloÐwtoi, thc Lh lgebrac twn BÐgkner telest¸n thcstroform c.

5. On Some Properties of Weyl RelationParathr seic gÔrw apì orismènec idiìthtec thc sqèshc tou Bèul

GenikeÔoume thn akìloujh sqèsh tou Bèul gia tic analutikèc sunart seicF (A, B) kai G(A, B) twn telest¸n A kai B, ìpou [A, B] = iC, C ∈ R.

exp itF exp −isG exp −itF = exp −isG(t)

G(t) = exp itFG exp −itFMeletoÔme aut thn genik sqèsh tou Bèul gia thn perÐptwsh pou h su-

nrthsh telest¸n sumpÐptei me thn Qamiltonian enìc kbantikoÔ sust matocF = H(p, q).

18

6. Ordering of the Exponential of Quandratic Forms in Boson Operatorsand Some Applications

H ditaxic thc ekjetik c sunrthshc tetragwnik¸n morf¸nme mpozonikoÔc telestèc kai orismènec efarmogèc

Sthn ergasÐa aut brÐskoume thn kanonik diatetagmènh morf thc akì-loujhc genik c ekjetik c sunrthshc, twn Mpozonik¸n telest¸n a kai a+

ìpou [a, a+] = 1,

exp αa2 + β(a+)2 + γ(a+a+ aa+) + δa+ εa+

qrhsimopoi¸ntac thn parametrik parag¸gish.Me thn bo jeia twn kanonik¸n morf¸n pou upologÐzoume, brÐskoume thn

m tra puknìthtac kai thn sunrthsh katanom c tou BÐgkner tou hlektroma-gnhtikoÔ pedÐou me ènan eujÔ trìpo.

7. Quantum Friction in a Uniform Magnetic FieldH kbantik trib se omogenèc magnhtikì pedÐo

H kbntwsh twn susthmtwn apìsbeshc brÐsketai sthn koruf tou je-wrhtikoÔ endiafèrontoc. MÐa apì thc mejìdouc sunÐstatai sthn qr sh Qa-miltonian¸n pou exart¸ntai analutik apì ton qrìno. Sthn perÐptwsh pou htrib eÐnai grammik anlogh thc taqÔthtac me stajer analogÐac Ðsh me γ, hQamiltonian sthn kanonik parstash èqei thn morf :

H =1

2m

(~p+

e

ceγt ~H × ~q

)2e−γt + eγtV (~q)

MeletoÔme èna sÔsthma mèsa se magnhtikì pedÐo me dianusmatikì dunamikì~A = (−1

2Hq2,

12Hq1, 0) kai bajmwtì dunamikì V (~q) = 1

2mω2 (q21 + q22 + q23).

BrÐskoume tic idiosunart seic kai tic idiotimèc thc antÐstoiqhc exÐswshctou Srèntingker.

8. Propagator with friction in Quantum MechanicsDiadot c me trib sthn Kbantik Mhqanik

Sthn ergasÐa aut upologÐzoume touc diadotèc gia kpoia kbantomhqaniksust mata me trib . H trib eÐnai mia grammik sunrthsh thc taqÔthtac me

19

stajer trib c Ðsh me γ kai to sÔsthma moizei me èna sÔsthma me qronikexart¸menh mza thc morf c m→ meγt.

Me thn bo jeia tou akìloujou tÔpou

K(q′′, t′′, q′, t′) =

∑n

Ψ∗(q′, t′)Ψ(q′′, t′′))

mporoÔme na upologÐsoume akrib¸c touc diadotèc gia kpoia sust mata metetragwnikèc Qamiltonianèc.

Idiaitèrwc meletoÔme ton exanagkasmèno kai aposbennumèno armonikì ta-lantwt mèsa se omogenèc hlektromagnhtikì pedÐo.

9. The Lie algebra of Wigner operators of angular momentumH Lh lgebra twn telest¸n BÐgkner thc stroform c

Sthn ergasÐa aut meletoÔme touc telestèc thc stroform c ston q¸rotwn fsewn, pou lambnoume apì touc gnwstoÔc telestèc thc stroform c sansunèpeia thc anaparstashc tou BÐgkner. Kataskeuzoume thn Lh lgebratwn telest¸n aut¸n kaj¸c epÐshc kai thn antÐstoiqh Lh omda, kai exetzoumedÔo anaparastseic aut c thc omdac. BrÐskoume epÐshc tic idiosunart seickai tic idiotimèc twn telest¸n aut¸n.

10. Fermi Dirac statistics for free electrons in uniform electricand magnetic fields

Statistik twn Fèrmi - Ntirk gia eleÔjera hlektrìnia mèsa se omogenèchlektrikì kai magnhtikì pedÐa

Sthn ergasÐa aut meletoÔme to fainìmeno Nte Qac - Ban 'Alfen parou-sÐa hlektrikoÔ pedÐou. ApodeiknÔoume ìti gia arket asjenèc hlektrikì pe-dÐo, ìpou oi sunj kec eÐnai katllhlec gia thn kbntwsh thc enèrgeiac, heleÔjerh enèrgeia eÐnai mia hmi - periodik sunrthsh wc proc ta pedÐa. Sansunèpeia brÐskoume, gia thn magnhtik epidektikìthta periodikèc ekfrseicpou eÔkola ekfulÐzontai stic gnwstèc ekfrseic tou fainomènou Nte Qac -Ban 'Alfen ìtan to hlektrikì pedÐo mhdenisjeÐ. Mèsa sta plèsia thc sta-tistik c twn Fèrmi - Ntirk upologÐzoume thn eleÔjerh enèrgeia an mondaìgkou apì thn sqèsh F − nζ = −2kT

∑i

(1 + e(ζ−εi)/kT

). QrhsimopoioÔme

thc prokÔptousa sqèsh gia na upologÐzoume thn magnhtik epidektikìthta sedÔo endiafèrousec oriakèc peript¸seic.

20

1.EustajeÐc katastseicìpou

√α/ζ << 1, α = ~2e2(E2

1 + E22)/8m(µH)2. ApodeiknÔoume ìti

ìtan to hlektrikì pedÐo eÐnai polÔ asjenèc E/H << 10−7, h suneisforperiorÐzetai se mia mikr metjesh sto ìrisma twn periodik¸n ìrwn. Pnw apìaut thn tim prostÐjetai ènac nèoc periodikìc ìroc, mèsa apì thn sunrthshJ0 tou Mpèsel kai ofeÐletai sthn allhlepÐdrash tou magnhtikoÔ pedÐou methn egkrsia sunist¸sa tou hlektrikoÔ pedÐou. Ktw apì kpoiec orismènecsunj kec oi upologismoÐ mac, dÐnoun thn akìloujh magnhtik epidektikìthta.

χ =m3/2

3π2~3µ2(2ζ)1/2

1− 3πkT

µH

(ζ

µH

)1/2 ∞∑r=1

(−1)r

r1/2 sinh (rπ2 kT/µH)×[

J0

(2rπ

µH

√ζα

)sin

(ζ + α

µHrπ − 1

4π

)]Gia mhdenikì hlektrikì pedÐo α → 0, xanabrÐskoume ton gnwstì tÔpo toufainomènou twn Nte Qac Ban 'Alfen.

2. AstajeÐc katastseicìpou

√α/ζ >> 1. 'Otan ta pedÐa gÐnoun isqurìtera, µH >> kT kai

vF << c√

(E21 + E2

2)/H2 ìpou vF eÐnai h taqÔthta Fèrmi, h magnhtik epide-ktikìthta metatrèpetai se mÐa diafor dÔo hmi - periodik¸n ìrwn. Ktw apìkpoiec orismènec sunj kec oi upologismoÐ mac, dÐnoun t¸ra thn akìloujhmagnhtik epidektikìthta.

χ =m3/2

3π2~3µ2(2ζ)1/2

1 +9kT

µH

(α

ζ

)5/4 ∞∑r=1

(−1)r

r sinh(rπ2√α/ζ kT/µH

)×[cos

((√ζ +√α)2

rπ

µH

)− sin

((√ζ −√α)2

rπ

µH

)]Ta ajroÐsmata pou emfanÐzontai sthn parapnw sqèseic sugklÐnoun taqÔ-

tata wc proc to r.

11. Wigner Representation of Bloch Electrons in Uniform FieldsH anaparstash BÐgkner gia hlektrìnia Mploq mèsa se omogen pedÐa

Sthn ergasÐa aut upologÐzoume thn BÐgkner sunrthsh katanom c kai thnsunrthsh diamerismoÔ gia ta hlektrìnia Mploq mèsa se omoiìmorfo hlektri-

21

kì kai magnhtikì pedÐo me thn bo jeia thc paraktw energ c Qamiltonian c.

H(~k, ~q) = E(~k − e

~c~A(~q)

)− e ~E · ~q.

UpologÐzoume thn magnhtik kai thn hlektrik epidektikìthta. Qrhsi-mopoi¸ntac thn basik teqnik thc ditaxhc twn telest¸n, ta megèjh upo-logÐzontai me ènan tètoio trìpo ¸ste na faÐnetai h pl rhc suneisfor touhlektrikoÔ pedÐou sthn magnhtik epidektikìthta.

BrÐskoume epÐshc thn sunrthsh diamerismoÔ twn hlektronÐwn Mploq mèsase omogenèc hlektrikì pedÐo, sthn perÐptwsh enìc aploÔ kubikoÔ krustlloume ton akìloujo nìmo diasporc

E(~k) = ε [cos (a1k1) + cos (a2k2) + cos (a3k3)].BrÐskoume

Z(b) =3∏i=1

I0

(εsinh (beEiai/2)

eEiai/2

)I0 eÐnai h tropopoihmènh mhdenik c txewc sunrthsh tou Mpèsel.

12. Some Remarks on the Caldirola - Montaldi EquationParathr seic gia thn exÐswsh twn Kalntirìla - Montlntu

Se mia prìsfath ergasÐa oi Kalntirìla kai Montlntu qrhsimopoÐhsankpoia exÐswsh diafor¸n gia na perigryoun èna kbantikì aposbennumènosÔsthma. Mia apì tic exis¸seic autèc eÐnai h akìloujh

i~τ

[ψ(t+ τ)− ψ(t− τ)] = H(p, q, t)ψ(q, t)

Sthn shmeÐwsh aut perigrfoume kpoiec endiafèrousec proseggistikècmejìdouc sqetik me tic exis¸seic autèc. BrÐskoume ènan nèo Qamiltonianìtelest pou perigrfei to sÔsthma, pou ìmwc den eÐnai autosuzug c.

13. Some Remarks on the Nonnegative Quantum - MechanicalDistribution Function

Orismènec parathr seic gia thn mh arnhtik sunrthsh katanom cthc kbantik c mhqanik c

Katanomèc tÔpou BÐgkner emfanÐzontai se poll fusik probl mata, a-kìma kai sthn sqetikistik kbantik mhqanik . H sunrthsh katanom c tou

22

BÐgkner eÐnai pragmatik , all epeid eÐnai dunatìn na prei kai arnhtikèctimèc, den mporeÐ na jewrhjeÐ san mia sunrthsh puknìthtac pijanìthtac.

Sthn shmeÐwsh aut exetzoume tic sunj kec gia thn monadikìthta kaikpoiec idiìthtec gia thn mh arnhtik sunrthsh katanom c thc kbantik cmhqanik c. IdiaÐtera meletoÔme thn akìloujh katanom

FS(q, p, a, t) = exp

a

4

∂2

∂q2+

~2

4a

∂2

∂p2

F (q, p, t)

ìpou F (q, p, t) eÐnai h sunhjismènh katanom tou BÐgkner.

14. Foundation of the Lie - Admissible Fock Space of theHadronic Mechanics

H jemelÐwsh tou Lh - apodektoÔ q¸rou Fwk thc Adronik c Mhqanik c

Sthn paroÔsa ergasÐa meletoÔme thn perÐptwsh suzeugmènwn armonik¸ntalantwt¸n sthn Adronik Mhqanik . Oi mh kanonikèc sqèseic antimetajèse-wc twn telest¸n thc jèshc kai thc orm c metatrèpontai sthn anaparstashtou Fok, stic gnwstèc sqèseic thc Q− lgebrac.

Sthn genik perÐptwsh miac Lh - apodekt c lgebrac, èqoume

[ A, A+ ] = AA+ − A+QA,

ìpou Q eÐnai ènac telest c.MporoÔme na orÐsoume nèouc telestèc dhmiourgÐac kai exafnishc tou Fok,

pou na perigrfoun kpoia swmatÐdia ktw apì kpoiec kajorismènec sunj -kec pou prèpei na ikanopoioÔntai apì ton telest Q. 'Otan èqoume ènan aplìadronikì armonikì talantwt , o Q eÐnai èna monìmetro mègejoc mikrìtero apìthn monda kai èqoume energhtikì koresmì sto fsma twn idiotim¸n. SthnperÐptwsh aut isqÔei h genikeumènh sqèsh abebaiìthtac tou Qizenmpergksthn jewrÐa tou SantÐllu.

Telik, dÐnontai oi sÔmfwnec katastseic tou telest exafnishc A kaigenikeÔetai o telest c metjeshc tou Bèul sthn Q− lgebra.

15. Statistical properties of one - demensional noncanonical oscillatorsStatistikèc idiìthtec twn monodistatwn mh kanonik¸n talantwt¸n

MeletoÔme tic statistikèc idiìthtec mh kanonik¸n armonik¸n talantwt¸nse mÐa distash. Gia ton upologismì thc sunrthshc diamerismoÔ gia q > 1

23

jewroÔme tic peript¸seic twn qamhl¸n kai uyhl¸n jermokrasi¸n kai thn je-wrÐa diataraq¸n gia q = 1 + ε, ìpou ε << 1. Ta apotelèsmata pou brÐskoumegia thn eidik jermìthta eÐnai diaforetik apì aut pou dÐnei h antÐstoiqhkanonik jewrÐa.

16. Boltzmann statistics of quantum frictionStatistik Mpìltsmann gia thn kbantik trib

Sthn ergasÐa aut ereunoÔme tic jermodunamikèc idiìthtec apl¸n kbanto-mhqanik¸n susthmtwn parousÐa trib c. Me thn bo jeia twn diadot¸n pouupologÐzoume gia aut ta apl montèla, brÐskoume tic antÐstoiqec sunart seicthc statistik c tou Mpìltsman.

Sthn perioq twn qamhl¸n jermokrasi¸n h antÐstoiqec autèc sunart seicparousizoun an¸malh sumperifor.

17. Deformed Harmonic Oscillator for non - Hermitian operatorand the Behavior of PT− and CPT− Symmetries

O paramorfwmènoc armonikìc talantwt c gia mh ermhtianoÔctelestèc kai h sumperifor twn summetri¸n PT kai CPT .

Sthn paroÔsa ergasÐa meletoÔme ton paramorfwmèno armonikì talantwt gia ton akìloujo mh ermhtianì telest tou Qmilton, ìtan oi parmetroi α,β, m eÐnai sthn genik perÐptwsh migadikèc

H =α

2m

(p1 +

λ

2hq2

)2

+βmω2

2

(q1 −

θ

2~p2

)2

ta λ, θ eÐnai pragmatikèc parmetroi.Gia thn perÐptwsh pou α = 1, β = 1 kai h mza m eÐnai pragmatik ,

brÐskoume tic idiosunart seic kai tic idiotimèc thc enèrgeiac, tic sÔmfwneckatastseic, thn qronik exèlixh twn telest¸n qj, pj kai tic sqèseic abebaiì-thtac sthn eikìna tou Qizenmpergk. Sthn perÐptwsh aut o telest c H eÐnaiermhtianìc kai ikanopoieÐ thn summetrÐa PT .

EpÐshc gia thn perÐptwsh pou to m eÐnai migadikì kai α = 1, β = 1,o telest c H eÐnai mh ermhtianìc kai den ikanopoieÐ thn summetrÐa PT .IkanopoieÐ ìmwc thn CPT summetrÐa kai èqei pragmatikì kai diakritì fsma.

Sthn genik perÐptwsh pou ta α, β, m eÐnai migadik gia ton mh ermhtianìtelest H, brÐskoume migadikì fsma. Gia kpoia idik tim twn migadik¸n pa-

24

ramètrwn α, β to fsma eÐnai pragmatikì, diakritì kai jetikì kai h summetrÐaCPT eÐnai paroÔsa.

To genikì prìblhma tou paramorfwmènou talantwt gia mh ermhtianoÔctelestèc, brÐskei efarmogèc sthn fusik thc sterec katstashc.

18. Harmonic Oscillator in non commuting two dimensional spaceArmonikìc Talantwt c se ènan mh metajetì

q¸ro dÔo diastsewn

Sthn ergasÐa aut meletoÔme ton disdistato armonikì talantwt se sta-jerì magnhtikì pedÐo se èna q¸ro mh metajetì. QrhsimopoioÔme thn akìloujhQamiltonian .

H =1

2m

(p21 + p22

)+

1

2mω2

1 q21 +

1

2mω2

2 q22

Oi basikoÐ telestèc ikanopoioÔn tic akìloujec sqèseic metajèsewc

[q1, q2] = iθ, [p1, p2] = iλ, [q1, p1] = i~, [q2, p2] = i~

ìpou λ, θ eÐnai pragmatikèc jetikèc parmetroi. H parmetroc λ ekfrzeithn parousÐa enìc magnhtikoÔ pedÐou. To sÔsthma den mporeÐ na entopisteÐmèsa se èna embadìn mikrìtero apì to θ lìgw thc nèac sqèshc abebaiìthtac∆q1∆q2 ∼ θ pou sunepgetai apì thn mh antimetajetikìthta twn suntetagmè-nwn q1 kai q2 tou q¸rou.

ApodeiknÔoume pr¸ta ìti to sÔsthma eÐnai isodÔnamo me èna disdistatosÔsthma ìpou oi telestèc thc orm c kai thc jèshc thc deÔterhc distashcikanopoioÔn mia paramorfwmènh sqèsh metajèsewc. An sumbolÐsoume me ke-falaÐa grmmata touc nèouc autoÔc telestèc brÐskoume

[Q1, Q2] = 0, [P1, P2] = 0, [Q1, P1] = i~, [Q2, P2] = i~µ

H parmetroc paramìrfwshc µ = 1 − λθ/~2 eÐnai anexrthth apì thnQamiltonian . Ta dÔo enallassìmena parathr sima megèjh tou sust matoceÐnai τ1 = q1 kai τ2 = q2 − (θ/~)p1.

Grfoume met ton telest thc qronik c exelÐxewc se mia katllhla dia-tetagmènh morf ¸ste mporoÔme na upologÐzoume ton akrib diadot me ènaneujÔ trìpo. ApodeiknÔoume ìti oi gnwstec sunart seic mporoÔn na brejoÔnme thn bo jeia twn lÔsewn tou isodÔnamou klassikoÔ sust matoc. BrÐskoume

25

epÐshc thn qronik exèlixh twn telest¸n twn suntetagmènwn kai twn orm¸n.H mèjodoc mporeÐ na efarmosjeÐ eÔkola kai gia thn perÐptwsh pou oi suqnì-thtec ω1, ω2 h mza m exart¸ntai analutik apì ton qrìno.

EreunoÔme tic jermodunamikèc idiìthtec tou sust matoc sthn statistik tou Mpìltsmann. BrÐskoume thn statistik m tra thc puknìthtac kai thnsunrthsh diamerismoÔ. To sÔsthma eÐnai isodÔnamo me ènan disdistato ar-monikì talantwt me tic ex c dÔo paramorfwmènec suqnìthtec ω1 kai ω2.

ω1,2 =1

2

√(ω1 + ω2)2 + (−ω1ω2θ + λ)2 ± 1

2

√(ω1 − ω2)2 + (ω1ω2θ + λ)2

Telik spoudzoume thn perÐptwsh pou o fasikìc q¸roc thc deÔterhcdistashc gÐnetai klassikìc, dhlad [Q2, P2] = 0. Autì sumbaÐnei ìtan µ =0. Sthn perÐptwsh aut èqoume èna epiplèon enallassìmeno parathr simomègejoc to π2 = p2 + (λ/~)q1. To telikì apotèlesma exarttai t¸ra apì ticakìloujec suqnìthtec

ω1 =1

λ

√(λ2 + ω2

2)(λ2 + ω21) και ω2 = 0

Sthn perÐptwsh enìc eleujèrou sust matoc brÐskoume ω1 = λ.

19. Deformed damped harmonic oscillatorO paramorfwmènoc aposbennumènoc armonikìc talantwt c

Sthn ergasÐa aut meletoÔme ton aposbennumèno armonikì talantwt pa-rousÐa trib c. QrhsimopoioÔme thn akìloujh Qamiltonian pou exarttai apìton qrìno.

H = e2γtα

2m

(p1 +

λ

2hq2

)2

+ e−2γtβmω2

2

(q1 −

θ

2~p2

)2

ìpou λ, θ eÐnai pragmatikèc parmetroi.Qamiltonianèc autoÔ tou tÔpou èqoun qrhsimopoihjeÐ gia thn melèth thc

trib c sthn Kbantik Mhqanik . EÐnai h gnwst Qamiltonian twn Kalnti-rìla - Kanu se mÐa distash ston q¸ro twn fsewn P1 = p1 + λ

2hq2 kai

Q1 = q1 − θ2~ p2, ìpou èqoume eisgei mÐa epÐ plèon distash q2, p2. Oi te-

lestèc ikanopoioÔn tic akìloujec sqèseic antimetajèsewc thc mh metajet cgewmetrÐac, dhlad

[P1, p2] = iλ/2 [Q1, q2] = iθ/2 [q2, p2] = i~ [Q1, P1] = i~µ

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ìpou µ = 1 + λθ/(4~) eÐnai h stajer paramìrfwshc. H deÔterh sqèshantimetajèsewc sunepgetai thn nèa sqèsh abebaiìthtac tou Qizenmpergk∆Q1∆q2 ∼ θ.

BrÐskoume akrib¸c ton diadot tou sust matoc. BrÐskoume epÐshc thnqronik exèlixh twn telest¸n twn suntetagmènwn kai twn orm¸n. Apì toapotèlesma pou brÐskoume gÐnetai fanerì ìti to sÔsthma eÐnai ènac nèoc ta-lantwt c me suqnìthta Ω =

√ab ω2µ2 − γ2.

Telik ereunoÔme tic jermodunamikèc idiìthtec tou sust matoc sthnstatistik tou Mpìltsman. BrÐskoume thn statistik m tra puknìthtac kaithn sunrthsh diamerismoÔ Z(β).

Z(β) =S

θ

2µ√1− cos (γ~β) cosh (Ω~β)− (γ/Ω) sin (γ~β) sinh (Ω~β)

ìpou β = 1/kT . Me ton ìro S sumbolÐzoume to embadìn miac mikr c perioq ctou q¸rou kai o pargontac S/θ ofeÐletai sthn mh antimetajetikìthta touq¸rou.

H eidik jermìthta tou sust matoc emfanÐzei gia asjeneÐc tribèc èna pei-ro arijmì anwmali¸n kai gia isqurèc tribèc emfanÐzontai epiplèon kai merikèctimèc pou mhdenÐzetai.

20. Quantum damped harmonic oscillator on non - commuting planeO kbantikìc aposbennumènoc armonikìc talantwt c

sto mh metajetì epÐpedo

MeletoÔme ton aposbennumèno armonikì talantwt parousÐa trib c.QrhsimopoioÔme thn akìloujh Qamiltonian .

H(p1, q1, t) = e−2γt1

2mp21 + e2γt

mω2

2q21

H apìsbesh proèrqetai apì thn allhlepÐdrash metaxÔ tou sust matocpou parathroÔme kai enìc llou sust matoc pou suqn onomzetai apoj kh loutrì, proc to opoÐo h enèrgeia rèei me ènan trìpo mh antistreptì. MÐamèjodoc pou qrhsimopoi jhke pr¸ta apì ton H. Bateman gia na efarmì-soume thn sunhjismènh mèjodo thc kbntwshc, basÐzetai se mia diadikasÐakat thn opoÐan diplasizoume touc bajmoÔc eleujerÐac ètsi ¸ste t¸ra naqeirizìmaste èna ousiastik apomonwmèno sÔsthma.

Sthn ergasÐa aut qrhsimopoioÔme thn akìloujh Qamiltonian (m = 1):

H(p1, q1, t)−H(p2, q2,−t) =1

2e2γt

(p21 − ω2

2 q22

)− 1

2e−2γt

(p22 − ω2

1 q21

)27

pou eÐnai ousiastik ènac aposbennhmènoc armonikìc talantwt c suzeugmè-noc me to qronik sumplhrwmatikì tou eÐdwlo. Oi dÔo Qamiltonianèc denenallssontai kai oi basikoÐ telestèc ikanopoioÔn tic akìloujec sqèseic an-timetajèsewc thc mh metajet c gewmetrÐac.

[p1, p2] = iλ [q1, q2] = iθ [q1, p1] = i~ [q2, q2] = i~

ìpou ta λ, θ eÐnai pragmatikèc parmetroi.UpologÐzoume ton telest qronik c exèlixhc kai brÐskoume analutik ton

diadot tou sust matoc.O diadot c pou prokÔptei exarttai apì thn parmetro paramìrfwshc µ

kai eÐnai mÐa disdistath sunrthsh katanom c tÔpou Gkouc twn enallasso-mènwn parathrhsÐmwn megej¸n τ1 = q1 kai τ2 = q2 − (θ/~)p1. Oi ìroi poutalantoÔtai exart¸ntai apì tic suqnìthtec Ω1 kai Ω2.

ω1,2 =1

2

√(ω1 + ω2)2 − (λ+ ω1ω2θ)2/~2 ±

1

2

√(ω1 − ω2)2 − (λ− ω1ω2θ)2/~2

Ω1 =√ω21 − γ2, Ω2 =

√ω22 − γ2 µ = 1− λθ

~2=ω1

ω1

ω2

ω2

EreunoÔme tic jermodunamikèc idiìthtec tou sust matoc qrhsimopoi¸ntacthn basik kanonik m tra puknìthtac. BrÐskoume thn statistik sunrthshkatanom c kai thn sunrthsh diamerismoÔ.

UpologÐzoume thn eidik jermìthta gia thn oriak perÐptwsh thc krÐsimhcapìsbeshc, ìpou oi suqnìthtec tou sust matoc mhdenÐzontai Ω1 = 0, Ω2 = 0,pou epitugqnetai an ω1 = ±ω2 = γ/

õ =

√λ/θ.

H eidik jermìthta c tou sust matoc autoÔ parousizei kpoiec anwmalÐ-ec oi opoÐec exafanÐzontai sto klassikì ìrio ~→ 0. H timèc thc jermokrasÐacT ìpou o paronomast c thc sunrthshc diamerismoÔ mhdenÐzetai brÐskoume ìtieÐnai:

Tn ∼=~γ2k

1

1.16556 + nπ, n = 0, 1, 2, · · · Tn =

~γ2k

1

nπ, n = 1, 2, · · ·

ìpou k eÐnai h stajer tou Mpìltsman kai T0 = ~kγ = ~

k

√λµθ.

Telik meletoÔme thn perÐptwsh pou h parmetroc thc paramìrfwshc µmhdenÐzetai. O diadot c eÐnai pli mia katanom tÔpou Gkouc all t¸ra eÐnaimia sunrthsh twn tri¸n enallassomènwn parathrhsÐmwn megej¸nτ1 = q1, τ2 = q2 − (θ/~)p1 kai π2 = p2 + (λ/~)q1.

28

O diadot c exarttai apì tic akìloujec paramètrouc kai suqnìthtec.

~ =√λθ, σ =

√λ

θ, ω =

1

σ

√(σ2 − ω2

2)(ω21 − σ2), Ω =

√ω2 − γ2

H eidik jermìthta èqei kpoiec anwmalÐec kai epÐshc mhdenÐzetai se kpoiashmeÐa gia tic diforec timèc twn paramètrwn.

1. Quantum Deformed Problem of Electrons in the Dirac TheoryTo kbantikì tupo prìblhma twn hlektronÐwn sthn jewrÐa tou Ntirk

Sthn paroÔsa ergasÐa diereunoÔme to tupo prìblhma twn hlektronÐwntou Ntirk sthn mh metajet gewmetrÐa kai sthn Lh apodekt tupopoÐhshthc kbantik c barÔthtac. Eisgontai h qronik paramìrfwsh kai h para-mìrfwsh thc orm c mèsw tou montèlou twn Kalntirìla Montlntu (C.M)kaj¸c epÐshc kai tou montèlou thc pargwgou mikr c apostsewc (S.D.D.).Ta parapnw montèla eÐnai idikèc peript¸seic thn Lh apodekt c jewrÐac.Ta apotelèsmata basÐzontai sthn jewrÐa pou anaptÔqjhke apì touc Gkon-zlez Ntiz pou qrhsimopoÐhsan to montèlo S.D.D. gia na kataskeusounmia tropopoihmènh Lh apodekt exÐswsh twn GouÐler Nte Bit. H ermhneÐathc exÐswshc aut c eÐnai ìti to sÔmpan èqei mia mh mhdenik sunolik enèrgeiah tim thc opoÐac sumpÐptei me thn antÐstoiqh tim enìc armonikoÔ talantwt me mza Plank Ðsh me M∗. EÐnai èna anoiktì sÔsthma pou allhlepidr me ènaeÐdoc exwterikoÔ kìsmou kai dhmiourgeÐtai apì èna eÐdoc fusik c pragmati-kìthtac. EpÐshc h taqÔthta omdac, met thn efarmog twn dÔo montèlwnsthn jewrÐa tou Ntirk, odhgeÐ se mia taqÔthta megalÔterh apì to c kaiikanopoieÐ thn upìjesh tou SantÐllu.

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