Summary
The construction of a consistent multidimensional perturbation scheme is investigated in terms of a collective variable, which is a parameter that traces the classical path, and other variables which describe the fluctuations away from this path. The perturbation scheme is considered in the context of classical and quantum mechanics, and the corresponding fluctuation and Schrödinger equations are considered with respect to either a fixed frame of reference or a moving frame which travels along the classical path. The role played byDirac constraints in the canonical transformations is examined. Finally the lowest-order quantum corrections to the classical energy are calculated.
Riassunto
Si studia la costruzione di uno schema di perturbazione multidimensionale consistente in termini di una variabile collettiva, che è un parametro che traccia il percorso classico, e altre variabili che descrivono le fluttuazioni lontano da questo percorso. Lo schema di perturbazione è considerato nel contesto della meccanica classica e quantistica, e la corrispondente fluttuazione e le equazioni di Schrödinger sono considerate sia rispetto ad un sistema fisso di riferimento che ad uno in movimento che viaggia lungo il percorso classico. Si esamina il ruolo giocato dai vincoli di Dirac nelle trasformazioni canoniche. Infine si calcolano le correzioni quantiche all’energia classica d’ordine inferiore.
РЕжУМЕ
РАссМАтРИВАЕтсь кОН стРУИРОВАНИЕ сОглАсОВАННОИ МНОгО МЕРНОИ схЕМы тЕОРИИ ВОжМУЩЕНИИ В т ЕРМИНАх кОллЕктИВНО И пЕРЕМЕННОИ, кОтОРАь ь ВльЕтсь пАРАМЕтРОМ, ОпИсыВАУЩИМ клАссИЧ ЕскУУ тРАЕктОРИУ, И ДР УгИх пЕРЕМЕННых, кОтОРыЕ ОпИсыВАУт ФлУктУАцИ И ОтНОсИтЕльНО ЁтОИ т РАЕктОРИИ. схЕМА ВОжМУЩЕНИИ РАс сМАтРИВАЕтсь В кОНтЕкстЕ с клАссИЧ ЕскОИ И кВАНтОВОИ МЕх АНИкОИ. сООтВЕтстВУУЩИЕ ФлУктУАцИИ И УРАВНЕН Иь шРЕДИНгЕРА АНАлИж ИРУУтсь лИБО ОтНОсИтЕльНО ФИксИР ОВАННОИ сИстЕМы ОтсЧЕтА, лИБО ДВИжУЩЕИсь сИстЕМы О тсЧЕтА, кОтОРАь пЕРЕМЕЩАЕтс ь ВДОль клАссИЧЕскОИ т РАЕктОРИИ. ИсслЕДУЕт сь РОль ОгРАНИЧЕНИИ ДИРАкА, к ОтОРУУ ОНИ ИгРАУт В кАНОНИЧЕ скИх пРЕОБРАжОВАНИь х. В жАклУЧЕНИЕ В НИжшЕМ п ОРьДкЕ ВыЧИсльУтсь кВАНтОВ ыЕ пОпРАВкИ к клАссИЧ ЕскОИ ЁНЕРгИИ.
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Müller-Kirsten, H.J.W., Wiedemann, A. Collective co-ordinates, dirac constraints and quantization of systems with many degrees of freedom. Nuov Cim A 78, 61–81 (1983). https://doi.org/10.1007/BF02832099
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DOI: https://doi.org/10.1007/BF02832099