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Теория Купмана — фон Неймана Koopman–von Neumann classical mechanics Mecânica clássica de Koopman-von Neumann
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The Koopman–von Neumann mechanics is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively. As Koopman and von Neumann demonstrated, a Hilbert space of complex, square-integrable wavefunctions can be defined in which classical mechanics can be formulated as an operatorial theory similar to quantum mechanics. A mecânica clássica de Koopman-von Neumann ou mecânica KvN, é uma descrição da mecânica clássica em termos do espaço de Hilbert, introduzida por Bernard Koopman e John von Neumann em 1931 e 1932. Koopman e von Neumann demonstraram que um espaço de Hilbert de funções de onda quadráticas integráveis e complexas pode ser definido de uma forma que a mecânica clássica posa ser formulada como uma teoria operativa semelhante à mecânica quântica. Теорией Ку́пмана — фон Не́ймана (KvN-теорией) в математической физике называется оригинальная переформулировка классической статистической механики, созданная американскими математиками Джоном фон Нейманом и Бернардом Купманом. Формализм механики Купмана — фон Неймана максимально приближен к формализму нерелятивистской квантовой механики: состояние динамической системы в ней описывается при помощи классической волновой функции, являющейся аналогом квантовомеханической волновой функции, классическое уравнение Лиувилля приобретает математическую структуру уравнения Шрёдингера и т. д.
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In classical statistical mechanics, the probability density obeys the Liouville equation with the self-adjoint Liouvillian where denotes the classical Hamiltonian . The same dynamical equation is postulated for the KvN wavefunction thus and for its complex conjugate From follows using the product rule that which proves that probability density dynamics can be recovered from the KvN wavefunction. ;Remark :The last step of this derivation relies on the classical Liouville operator containing only first-order derivatives in the coordinate and momentum; this is not the case in quantum mechanics where the Schrödinger equation contains second-order derivatives. The above axioms to , with the inner product written in the bra–ket notation, are , The expectation value of an observable at time is The probability that a measurement of an observable at time yields is , where . .
dbp:title
Dynamics of the probability density Mathematical form of the operator axioms
dbo:abstract
Теорией Ку́пмана — фон Не́ймана (KvN-теорией) в математической физике называется оригинальная переформулировка классической статистической механики, созданная американскими математиками Джоном фон Нейманом и Бернардом Купманом. Формализм механики Купмана — фон Неймана максимально приближен к формализму нерелятивистской квантовой механики: состояние динамической системы в ней описывается при помощи классической волновой функции, являющейся аналогом квантовомеханической волновой функции, классическое уравнение Лиувилля приобретает математическую структуру уравнения Шрёдингера и т. д. Идеологически KvN-теория диаметрально противоположна представлению Вигнера, в котором сходная идея унификации математического аппарата классической статистической и квантовой физики достигается, наоборот, путём преобразования волновой функции, которая появляется в уравнении Шрёдингера, в функцию Вигнера, определённую в классическом фазовом пространстве. Знаменательно, что обе эти теории были созданы практически одновременно — в 1931—1932 годах. The Koopman–von Neumann mechanics is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively. As Koopman and von Neumann demonstrated, a Hilbert space of complex, square-integrable wavefunctions can be defined in which classical mechanics can be formulated as an operatorial theory similar to quantum mechanics. A mecânica clássica de Koopman-von Neumann ou mecânica KvN, é uma descrição da mecânica clássica em termos do espaço de Hilbert, introduzida por Bernard Koopman e John von Neumann em 1931 e 1932. Koopman e von Neumann demonstraram que um espaço de Hilbert de funções de onda quadráticas integráveis e complexas pode ser definido de uma forma que a mecânica clássica posa ser formulada como uma teoria operativa semelhante à mecânica quântica.
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