RSS feeds
ÐóññêèéEnglish
Issue 4, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

2007

 / 

December

←  →

Methodological notes


Feynman disentangling îf noncommuting operators and group representation theory


Russian Federation State Scientific Center ‘A.I. Alikhanov Institute of Theoretical and Experimental Physics’, ul. Bolshaya Cheremushkinskaya 25, Moscow, 117259, Russian Federation

Feynman’s method for disentangling noncommuting operators is discussed and applied to nonstationary problems in quantum mechanics, including the excitation of à harmonic oscillator by an external force and/or bó time-varying frequency; spin rotation in à time-varying magnetic field; the disentangling of àn atom (ion) Hamiltonian in à laser field; à model with the hidden symmetry group of the hydrogen atom; and the theory of coherent states. The Feynman operator calculus combined with simple group-theoretical considerations allows disentangling the Hamiltonian and obtaining exact transition probabilities between the initial and final states of à quantum oscillator in analytic form without cumbersome calculations. The case of à D-dimensional oscillator is briefly discussed, in particular, in application to the problem of vacuum pair creation in an intense electric field.

Fulltext pdf (322 KB)
Fulltext is also available at DOI: 10.1070/PU2007v050n12ABEH006401
PACS: 02.20.−a, 03.65.Ca, 03.65.Fd (all)
DOI: 10.1070/PU2007v050n12ABEH006401
URL: https://ufn.ru/en/articles/2007/12/b/
000254795400002
2-s2.0-41949141349
2007PhyU...50.1217P
Citation: Popov V S "Feynman disentangling îf noncommuting operators and group representation theory" Phys. Usp. 50 1217–1238 (2007)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Îðèãèíàë: Ïîïîâ Â Ñ «Ôåéíìàíîâñêèé ìåòîä ðàñïóòûâàíèÿ îïåðàòîðîâ è òåîðèÿ ïðåäñòàâëåíèé ãðóïï» ÓÔÍ 177 1319–1340 (2007); DOI: 10.3367/UFNr.0177.200712f.1319

References (114) ↓ Cited by (18) Similar articles (20)

  1. Feynman R P Rev. Mod. Phys. 20 367 (1948)
  2. Feynman R P, Hibbs A R Quantum Mechanics And Path Integrals (New York: McGraw-Hill, 1965); Feinman R, Khibs A Kvantovaya Mekhanika i Integraly Po Traektoriyam (M.: Mir, 1968)
  3. Feynman R P Phys. Rev. 76 749 (1949); Feynman R P Phys. Rev. 76 769 (1949), sm. takzhe lyuboi uchebnik po KED i teorii polya
  4. Feinman R Kvantovaya Elektrodinamika (M.: Mir, 1964)
  5. Feinman R KED — Strannaya Teoriya Sveta i Veshchestva (Bibliotechka «Kvant», Vyp. 66) (M.: Nauka, 1988)
  6. Feynman R P Phys. Rev. 84 108 (1951); Feinman R Problemy Sovremennoi Fiziki Vyp. 3 (M.: IL, 1955) p. 37
  7. Feynman R P Phys. Rev. 80 440 (1950)
  8. Fujiwara I Prog. Theor. Phys. 7 433 (1952)
  9. Schwinger J Phys. Rev. 91 728 (1953)
  10. Popov V S Zh. Eksp. Teor. Fiz. 35 985 (1958)
  11. Popov V S Phys. Lett. A 342 281 (2005); Popov V S Zh. Eksp. Teor. Fiz. 128 944 (2005); Popov V S, Trusov M A Nauchnaya Sessiya MIFI-2007 T. 5 (M.: MIFI, 2007) p. 196
  12. Wigner E P Group Theory And Its Application To The Quantum Mechanics Of Atomic Spectra (New York: Academic Press, 1959); Vigner E P Teoriya Grupp i Ee Prilozheniya k Kvantovomekhanicheskoi Teorii Atomnykh Spektrov (M.: IL, 1961)
  13. Van-der-Varden B L Metod Teorii Grupp v Kvantovoi Mekhanike (Khar’kov: ONTI NKTP, 1938); van der Waerden B L Group Theory And Quantum Mechanics (Berlin: Springer, 1980)
  14. Veil’ G Teoriya Grupp i Kvantovaya Mekhanika (M.: Nauka, 1986)
  15. Pontryagin L S Nepreryvnye Gruppy 2-e Izd. (M.: Gostekhizdat, 1954)
  16. Petrashen’ M I, Trifonov E D Primenenie Teorii Grupp v Kvantovoi Mekhanike (M.: URSS, 1999)
  17. Vilenkin N Ya Spetsial’nye Funktsii i Teoriya Predstavlenii Grupp (M.: Nauka, 1965)
  18. Fronsdal K Teoriya Grupp i Elementarnye Chastitsy (Per. pod red. D Ivanenko) (M.: Mir, 1967) p. 324
  19. Bateman H Higher Transcendental Functions (Compiled A Erdélyi) Vol. 1, 2 (New York: McGraw-Hill, 1953); Beitmen G, Erdeii A Vysshie Transtsendentnye Funktsii T. 1, 2 (M.: Nauka, 1965-1966)
  20. Abramowitz M, Stegun I A (Eds) Handbook Of Mathematical Functions With Formulas, Graphs, And Mathematical Tables (New York: Dover Publ., 1965); Abramovitsa M, Stigan I (Red.) Spravochnik Po Spetsial’nym Funktsiyam s Formulami, Grafikami i Matematicheskimi Tablitsami (M.: Nauka, 1979)
  21. Schwinger J Phys. Rev. 82 914 (1951); Schwinger J Phys. Rev. 91 713 (1953); Schwinger J Phys. Rev. 91 728 (1953); Schwinger J Phys. Rev. 93 615 (1954); Schwinger J Phys. Rev. 94 1362 (1954)
  22. Shvinger Yu Teoriya Kvantovannykh Polei (M.: IL, 1956)
  23. Landau L D, Lifshits E M Kvantovaya Mekhanika: Nerelyativistskaya Teoriya (M.: Nauka, 1974)
  24. Majorana E Nuovo Cimento 9 43 (1932)
  25. Varshalovich D A, Moskalev A N, Khersonskii V K Kvantovaya Teoriya Uglovogo Momenta (M.: Nauka, 1975)
  26. Di Dzhakomo F, Nikitin E E Usp. Fiz. Nauk 175 545 (2005); Di Giacomo F, Nikitin E E Phys. Usp. 48 515 (2005)
  27. Güttinger P Z. Phys. 73 169 (1931)
  28. Schwinger J Phys. Rev. 51 648 (1937)
  29. Landau L Phys. Z. Sowjetunion 1 88 (1932); Landau L Phys. Z. Sowjetunion 2 46 (1932)
  30. Zener C Proc. R. Soc. London Ser. A 137 696 (1932)
  31. Stückelberg E C G Helv. Phys. Acta 5 369 (1932)
  32. Bloch F, Rabi I I Rev. Mod. Phys. 17 237 (1945)
  33. Abragam A The Principles Of Nuclear Magnetism (Oxford: Clarendon Press, 1961); Abragam A Yadernyi Magnetizm (M.: IL, 1963)
  34. Bargmann V, Michel L, Telegdi V L Phys. Rev. Lett. 2 435 (1959)
  35. Frenkel J Z. Phys. 37 243 (1926)
  36. Berestetskii V B, Lifshits E M, Pitaevskii L P Kvantovaya Elektrodinamika (M.: Fizmatlit, 2001)
  37. Popov V S Zh. Eksp. Teor. Fiz. 38 1584 (1960)
  38. Shapiro I S Dokl. Akad. Nauk SSSR 106 647 (1956)
  39. Chzhou Guan-Chzhao, Zastavenko L G Zh. Eksp. Teor. Fiz. 35 1417 (1958)
  40. Popov V S Zh. Eksp. Teor. Fiz. 37 1116 (1959)
  41. Gel’fand I M, Minlos R A, Shapiro Z Ya Predstavleniya Gruppy Vrashchenii i Gruppy Lorentsa, Ikh Primeneniya (M.: Fizmatgiz, 1958)
  42. Naimark M A Lineinye Predstavleniya Gruppy Lorentsa (M.: Fizmatgiz, 1958)
  43. Husimi K Prog. Theor. Phys. 9 381 (1953)
  44. Lewis H R (Jr.), Riesenfeld W B J. Math. Phys. 10 1458 (1969)
  45. Popov V S, Perelomov A M Zh. Eksp. Teor. Fiz. 57 1684 (1969)
  46. Baz’ A I, Zel’dovich Ya B, Perelomov A M Rasseyanie, Reaktsii i Raspady v Nerelyativistskoi Kvantovoi Mekhanike 2-e izd. (M.: Nauka, 1971)
  47. Dykhne A M Zh. Eksp. Teor. Fiz. 38 570 (1960)
  48. Perelomov A M, Popov V S Teor. Mat. Fiz. 3 377 (1970)
  49. Malkin I A, Man’ko V I, Trifonov D A Phys. Rev. D 2 1371 (1970)
  50. Akhundova E A, Dodonov V V, Man’ko V I Teor. Mat. Fiz. 60 413 (1984)
  51. Malkin I A, Man’ko V I Dinamicheskie Simmetrii i Kogerentnye Sostoyaniya Kvantovykh Sistem (M.: Nauka, 1979)
  52. Popov V S Zh. Eksp. Teor. Fiz. 62 1248 (1972)
  53. Ringwald A Phys. Lett. B 510 107 (2001)
  54. Popov V S Pis’ma ZhETF 74 151 (2001); Popov V S Phys. Lett. A 298 83 (2002)
  55. Alkofer R et al. Phys. Rev. Lett. 87 193902 (2001)
  56. Tajima T, Mourou G Phys. Rev. ST Accel. Beams 5 031301 (2002)
  57. Narozhny N B, Bulanov S S, Mur V D, Popov V S Phys. Lett. A 330 1 (2004); Bulanov S S, Narozhnyi N B, Mur V D, Popov V S Zh. Eksp. Teor. Fiz. 129 14 (2006)
  58. Narozhny N B Laser Phys. 15 1458 (2005)
  59. Mourou G A, Tajima T, Bulanov S V Rev. Mod. Phys. 78 309 (2006)
  60. Sauter F Z. Phys. 73 547 (1931)
  61. Heisenberg W, Euler H Z. Phys. 98 714 (1936)
  62. Schwinger J Phys. Rev. 82 664 (1951)
  63. Eckart C Phys. Rev. 35 1303 (1930)
  64. Nikishov A I Nucl. Phys. B 21 346 (1970); Narozhnyi N B, Nikishov A I Yad. Fiz. 11 1072 (1970)
  65. Parker L Phys. Rev. 183 1057 (1969); Parker L Phys. Rev. D 3 346 (1971)
  66. Zel’dovich Ya B, Starobinskii A A Zh. Eksp. Teor. Fiz. 61 2161 (1971)
  67. Zel’dovich Ya B, Pitaevskii L P, Popov V S, Starobinskii A A Usp. Fiz. Nauk 105 780 (1971); Zel’dovich Ya B, Pitaevskii L P, Popov V S, Starobinskii A A Sov. Phys. Usp. 14 811 (1972)
  68. Keldysh L V Zh. Eksp. Teor. Fiz. 47 1945 (1964)
  69. Wolkow D M Z. Phys. 94 250 (1935); Volkov D M Zh. Eksp. Teor. Fiz. 7 1286 (1937)
  70. Popov V S Usp. Fiz. Nauk 174 921 (2004); Popov V S Phys. Usp. 47 855 (2004)
  71. Manakov N L, Ovsiannikov V D, Rapoport L P Phys. Rep. 141 320 (1986)
  72. Mur V D, Popruzhenko S V, Pozdnyakov S G, Popov V S Phys. Lett. A 316 226 (2003)
  73. Fock V Z. Phys. 98 145 (1935)
  74. Bargmann V Z. Phys. 99 576 (1936)
  75. Perelomov A M, Popov V S Zh. Eksp. Teor. Fiz. 50 179 (1966)
  76. Bander M, Itzykson C Rev. Mod. Phys. 38 330 (1966); Bander M, Itzykson C Rev. Mod. Phys. 38 346 (1966)
  77. Popov V S v Sb. Fizika Vysokikh Energii i Teoriya Elementarnykh Chastits (Otv. red. V P Shelest) (Kiev: Naukova dumka, 1967) p. 702
  78. Thomas L H Nature 117 514 (1926)
  79. Wigner E P Rev. Mod. Phys. 29 255 (1957)
  80. Malykin G B Usp. Fiz. Nauk 176 865 (2006); Malykin G B Phys. Usp. 49 837 (2006)
  81. Ritus V I Usp. Fiz. Nauk 177 105 (2007); Ritus V I Phys. Usp. 50 95 (2007)
  82. Glauber R J Phys. Rev. 84 395 (1951)
  83. Kumar K J. Math. Phys. 6 1923 (1965); Kumar K J. Math. Phys. 6 1928 (1965)
  84. Magnus W, Karrass A, Solitar D Combinatorial Group Theory 2nd ed. (New York: Dover Publ., 1976)
  85. Schrödinger E Naturwissenshaften 14 664 (1926)
  86. Glauber R J Phys. Rev. Lett. 10 84 (1963); Glauber R J Phys. Rev. 130 2529 (1963); Glauber R J Phys. Rev. 131 2766 (1963)
  87. Glauber R Dzh Usp. Fiz. Nauk 176 1342 (2006)
  88. Klauder J R, Sudarshan E C G Fundamentals Of Quantum Optics (New York: W.A. Benjamin, 1968); Klauder Dzh, Sudarshan E Osnovy Kvantovoi Optiki (M.: Mir, 1970)
  89. Fock V Z. Phys. 49 339 (1928)
  90. Fok V A Raboty Po Kvantovoi Teorii Polya (L.: Izd-vo LGU, 1957) p. 9
  91. Bargmann V Commun. Pure Appl. Math. 14 187 (1961); Bargmann V Rev. Mod. Phys. 34 829 (1962)
  92. Bender C M, Wu T T Phys. Rev. 184 1231 (1969); Bender C M, Wu T T Phys. Rev. D 7 1620 (1973)
  93. Brézin E, Le Guillou J-C, Zinn-Justin J Phys. Rev. D 15 1544 (1977); Brézin E, Le Guillou J-C, Zinn-Justin J Phys. Rev. D 15 1558 (1977)
  94. Hioe F T, MacMillen D, Montroll E W Phys. Rep. 43 305 (1978)
  95. Dolgov A D, Popov V S Phys. Lett. B 79 403 (1978); Alliluev S P, Eletsky V L, Popov V S Phys. Lett. A 73 103 (1980); Dolgov A D, Popov V S Zh. Eksp. Teor. Fiz. 75 2010 (1978)
  96. Kazakov D I, Shirkov D V Fortschr. Phys. 28 465 (1980)
  97. Migdal A B, Krainov V P Priblizhennye Metody Kvantovoi Mekhaniki (M.: Nauka, 1966)
  98. Migdal A B Kachestvennye Metody v Kvantovoi Teorii (M.: Nauka, 1975)
  99. Karasev M V, Maslov V P Nelineinye Skobki Puassona. Geometriya i Kvantovanie (M.: Nauka, 1991)
  100. Bloch F Z. Phys. 74 295 (1932)
  101. Kirzhnits D A Trudy FIAN 16 3 (1961)
  102. Kirzhnits D A Polevye Metody Teorii Mnogikh Chastits (M.: Gosatomizdat, 1963)
  103. Dirac P A M The Principles Of Quantum Mechanics (Oxford: Clarendon Press, 1958); Dirak P A M Sobranie Nauchnykh Trudov T. 1 (M.: Fizmatlit, 2002) p. 9
  104. Perelomov A M Teor. Mat. Fiz. 6 213 (1971); Perelomov A M Usp. Fiz. Nauk 123 23 (1977); Perelomov A M Sov. Phys. Usp. 20 703 (1977)
  105. Zel’dovich B Ya, Perelomov A M, Popov V S Preprint ¹ 618 (M.: ITEF, 1968)
  106. Perelomov A M, Popov V S Zh. Eksp. Teor. Fiz. 54 1799 (1968)
  107. Park D Z. Phys. 159 155 (1960)
  108. Stone A P Proc. Cambr. Philos. Soc. 52 424 (1956)
  109. Morse P M, Feshbach H Methods Of Theoretical Physics Vol. 2 (New York: McGraw-Hill, 1953), Sect. 12.3
  110. Rabi I I Phys. Rev. 51 652 (1937)
  111. Stepanov V V Kurs Differentsial’nykh Uravnenii (M.: Gostekhizdat, 1953)
  112. Trusov M A, chastnoe soobshchenie
  113. Schwinger J Philos. Mag. 44 1171 (1953)
  114. Schwinger J J. Math. Phys. 2 407 (1961)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions