Summary
A widely discussed paradox in quantum mechanics is due to the failure of the matrices representing linear operators to satisfy the algebraic relations of the operators. Several examples involving boundary conditions on the domains of differential operators show how these failures arise, and sufficient conditions are given to prevent them. It is found that in certain cases the time-dependence of the matrix elements of an observable should be calculated via the commutator of the infinite matrices of the observable and the Hamiltonian, and not in terms of the formal commutator of these operators.
Riassunto
Un paradosso molto discusso nella mccoanica quantistica è dovuto all’insuccesso delle matrici che rappresentano operatori lineari a soddisfare le relazioni algebriche degli operatori. Parecchi esempi che coinvolgono condizioni al contorno nei domini degli operatori differenziali mostrano come nascono questi insuccessi, e si dànno le condizioni sufficienti per prevenirli. Si trova che in alouni casi la dipendenza dal tempo degli elementi di matrice di un osservabile dovrebbe essere calcolata per mezzo del commutatoro delle matrici infinite dell’osservabile e dell’hamiltoniano, c non in termini del commutatore formale di questi operatori.
РЕжУМЕ
шИРОкО ИсслЕДОВАННы И пАРАДОкс В кВАНтОВО И МЕхАНИкЕ ОБУслОВлЕН НЕВОжМОжНОстьУ УДОВ лЕтВОРИть АлгЕБРАИЧ ЕскИМ сООтНОшЕНИьМ Дль ОпЕ РАтОРОВ с пОМОЩьУ МАтРИц, пРЕДс тАВльУЩИх лИНЕИНыЕ О пЕРАтОРы. НЕкОтОРыЕ пРИМЕРы, ВклУЧАУЩИЕ гРАНИЧНы Е УслОВИь НА ОБлАстьх ДИФФЕРЕНцИАльНых Оп ЕРАтОРОВ, пОкАжыВАУт, кАк ВОжНИкАЕт тАкАь Н ЕВОжМОжНОсть. пРИВОД ьтсь ДОстАтОЧНыЕ УслОВИь, ЧтОБы УкАжАННАь НЕВО жМОжНОсть НЕ ВОжНИкА лА. пОлУЧАЕтсь, ЧтО Дль НЕ кОтОРых слУЧАЕВ ВРЕМЕННАь жА ВИсИМОсть МАтРИЧНых ЁлЕМЕНтОВ Дль НАБлУДАЕМых ВЕлИ ЧИН ДОлжНА Быть ВыЧИслЕН А ЧЕРЕж кОММУтАтОР БЕ скОНЕЧНых МАтРИц НАБлУДАЕМОИ В ЕлИЧИНы И гАМИльтОНИАНА, А НЕ Ч ЕРЕж ФОРМАльНыИ кОММ УтАтОР ЁтИх ОпЕРАтОРОВ.
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Based on work performed under the auspices of the U.S. Atomic Energy Commission and supported in part under NSF grants GP 8734 and GP 21052.
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Eckstein, S.G., Varga, B.B. & Aks, S. The matrix equations of quantum mechanics. Nuov Cim B 8, 451–460 (1972). https://doi.org/10.1007/BF02743672
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DOI: https://doi.org/10.1007/BF02743672