Lī reizinājuma formula

Lī reizinājuma formula (pazīstama arī ar nosaukumiem Lī–Trottera jeb Lī–Trottera–Kato reizinājuma formula) ir sakarība^[1]^[2]

e^{X+Y}=\lim _{n\rightarrow \infty }\left(e^{X/n}e^{Y/n}\right)^{n},

kur X un Y ir N × N reālas vai kompleksas matricas un e apzīmē matricas eksponentfunkciju. Formula nosaukta par godu norvēģu matemātiķim Sofusam Lī, Heilam Troteram (Hale F. Trotter)^[3] un japāņu matemātiķim Tosio Kato.^[4]

Lī reizinājuma formula ir vispārinājums sakarībai

e^{x+y}=e^{x}e^{y},\,

kur x un y ir reāli vai kompleksi skaitļi un e ir parastā eksponentfunkcija. Matricām šāda sakarība izpildās tikai tad, ja tās komutē jeb X Y = Y X. Vispārīgā gadījumā matricām šī sakarība neizpildās jeb

e^{X+Y}\neq e^{X}e^{Y},\,

tāpēc jālieto Lī reizinājuma formula. Šī formula ir nozīmīga kvantu mehānikā^[5]^[6]^[7] un kvantu skaitļošanā.^[8]

Atsauces

↑ Varadarajan, V.S. (1984), Lie Groups, Lie Algebras, and Their Representations, Springer-Verlag, ISBN 9780387909691, 99. lpp.
↑ Hall, Brian C. (2003), Lie groups, Lie algebras, and representations: an elementary introduction, Springer, ISBN 9780387401225, 35. lpp.
↑ Trotter, Hale F. (1959), "On the Product of Semi-Groups of Operators", Proceedings of the American Mathematical Society (American Mathematical Society) 10 (4): 545–551.
↑ Kato, Tosio (1974), "On the Trotter-Lie product formula", Proc. Japan Acad. 50 (9): 694–698, doi:10.3792/pja/1195518790.
↑ Albeverio, Sergio A.; Høegh-Krohn, Raphael J. (1976), Mathematical Theory of Feynman Path Integrals: An Introduction, Lecture Notes in Mathematics, 423 (1st izd.), Berlin, New York: Springer-Verlag, doi:10.1007/BFb0079827, ISBN 978-3-540-07785-5, 2. lpp.
↑ De Raedt, Hans; De Raedt, Bart (1983-12-01), "Applications of the generalized Trotter formula", Phys. Rev. A (American Physical Society) 28 (6): 3575–3580, doi:10.1103/PhysRevA.28.3575.
↑ Leimkuhler, Benedict; Reich, Sebastian (2004), Simulating Hamiltonian dynamics, Cambridge University Press, ISBN 9780521772907, 134. lpp.
↑ Andrew M. Childs, Simulating Hamiltonian dynamics, lekciju materiāli, 2. lpp.

Ārējās saites

Trotter formula, SklogWiki.

[1] Varadarajan, V.S. (1984), Lie Groups, Lie Algebras, and Their Representations, Springer-Verlag, ISBN 9780387909691, 99. lpp.

[2] Hall, Brian C. (2003), Lie groups, Lie algebras, and representations: an elementary introduction, Springer, ISBN 9780387401225, 35. lpp.

[3] Trotter, Hale F. (1959), "On the Product of Semi-Groups of Operators", Proceedings of the American Mathematical Society (American Mathematical Society) 10 (4): 545–551.

[4] Kato, Tosio (1974), "On the Trotter-Lie product formula", Proc. Japan Acad. 50 (9): 694–698, doi:10.3792/pja/1195518790.

[5] Albeverio, Sergio A.; Høegh-Krohn, Raphael J. (1976), Mathematical Theory of Feynman Path Integrals: An Introduction, Lecture Notes in Mathematics, 423 (1st izd.), Berlin, New York: Springer-Verlag, doi:10.1007/BFb0079827, ISBN 978-3-540-07785-5, 2. lpp.

[6] De Raedt, Hans; De Raedt, Bart (1983-12-01), "Applications of the generalized Trotter formula", Phys. Rev. A (American Physical Society) 28 (6): 3575–3580, doi:10.1103/PhysRevA.28.3575.

[7] Leimkuhler, Benedict; Reich, Sebastian (2004), Simulating Hamiltonian dynamics, Cambridge University Press, ISBN 9780521772907, 134. lpp.

[8] Andrew M. Childs, Simulating Hamiltonian dynamics, lekciju materiāli, 2. lpp.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]