Summary
The nonrelativistic classical and quantum equations of motion are exactly solved for a harmonic oscillator in the de Sitter and spatially flat Friedmann universes. In the latter case the creation of energy quanta owing to exchange of energy between the oscillator and the gravitational field is discussed and a comparison is made with the analogous relativistic phenomenon of particle creation in expanding space-times. By pursuing the Newtonian interpretation up to its ultimate consequences, it is found that, at the singularity, an infinite amount of creation of energy quanta occurs and that the Hilbert space structure of quantum mechanics becomes too narrow to accommodate the wave function which describes the state of the oscillator.
Riassunto
Le equazioni nonrelativistiche classiche e quantistiche del moto si risolvono esattamente per un oscillatore armonico nell’universo di de Sitter e nell’universo di Friedmann spaziale piatto. Nell’ultimo caso si discute la creazione di quanti di energia dovuti allo scambio di energia tra il campo oscillatorio e quello gravitazionale e si fa un confronto con l’analogo fenomeno relativistico della creazione di particelle negli spazi-tempi in espansione. Seguendo l’interpretazione newtoniana fino alle sue estreme conseguenze, si trova che, in corrispondenza della singolarità, avviene una creazione infinita di quanti di energia e che la struttura dello spazio di Hilbert della meccanica quantistica diviene troppo stretta per comprendere la funzione d’onda che descrive lo stato dell’oscillatore.
Резюме
Для гармонического осциллятора во вселенной де Ситтера и в пространственно плоской вселенной фридмана точно решаются нерелятивистскис классические и квантовые уравнения движения. В случае вселенной фридмана обсуждается образование энергетических квантов, связанное с обменом энергией между осциллятором и гравитапионным полем. Проводится сравнение с аналогичным релятивистским явлением образования частиц в расширяюшемся пространствевремени. Следуя ньютоновской интерпретации, получается, что в сингулярности образуется бесконечное количество энергетических квантов и структура гильбертова пространства квантовой механики становится слишком узкой, чтобы приспособить волновую функцию, которая описывает состояние осциллятора.
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Partially supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnológico, CNPq, Brasil.
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Lemos, N.A., Natividade, C.P. Harmonic oscillator in expanding universes. Nuovo Cim B 99, 211–225 (1987). https://doi.org/10.1007/BF02726583
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DOI: https://doi.org/10.1007/BF02726583