Classical limit of quantum mechanics. A paradoxical example

Классический предел квантовой механики. Пример парадокса

Summary

In the context of the general problem of equivalence between classical mechanics and quantum mechanics in the macroscopic limit, we point out that, for the particular case of the one-dimensional Coulomb potential, the quantum-mechanical result in the classical limit, corresponding to a certain superposition of odd- and even-parity energy eigenfunctions, leads to inconsistency with classical mechanics. It is shown that the contradiction persists even if the singularity of the Coulomb potential is treated as the limiting case of a modified Coulomb potential in which the singularity has been smoothed out. The possible implication of this paradoxical finding is briefly discussed.

Riassunto

Nel contesto del problema generale dell'equivalenza tra meccanica classica e quantistica nel limite macroscopico, si evidenzia che, per il caso particolare del potenziale di Coulomb unidimensionale, il risultato quantomeccanico nel limite classico, corrispondente a una certa sovrapposizione di autofunzioni di energia con parità dispari e pari, porta all'incoerenza con la meccanica classica. Si mostra che la contraddizione persiste anche se la singolarità del potenziale di Coulomb è trattata come caso limite di un potenziale di Coulomb modificato nel quale la singolarità è stata facilitata. Si discute brevemente la possibile implicazione di questa scoperta paradossale.

Резюме

В связи с общей проблемой эквивалентности между классической механикой и квантовой механикой в макроскопическом пределе, мы отмечаем, что для частного случая одномерного кулоновского потенциала квантовомеханический результат в классическом пределе, соответствующий определенной суперпозиции четных и нечетных собственных энергетических состояний, приводит к противоречию с классической механикой. Показывается, что это противоречие сохраняется, даже если сингулярность кулоновского потенциала рассматривается, как предельный случай модифицированного кулоновского потенциала, в котором сингулярность сглажена. Вкратце обсуждаются возможные следствия этого парадоксального результата.

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Home, D., Sengupta, S. Classical limit of quantum mechanics. A paradoxical example. Nuovo Cim B 82, 214–224 (1984). https://doi.org/10.1007/BF02732874

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  • PACS. 03.65
  • Quantum theory
  • quantum mechanics