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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/27575

Title: On the Time-Reversal Invariance of the Fundamental Commutation Relation
Authors: Hsiang-Shun Chou
Contributors: NTOU:Institute of Optoelectronic Sciences
國立臺灣海洋大學:光電科學研究所
Date: 2004-06
Issue Date: 2011-10-21T02:32:12Z
Publisher: Chinese Journal of Physics
Abstract: Abstract:We check the time-reversal invariance of the fundamental commutation relation [p, x] = −i¯h. Although it has been checked in the active picture, where the transformation affects the states but not the operators, it is a puzzling issue in the passive picture, where the transformation affects the operators but not the states. It is puzzling because [p0, x0] = [−p, x] = i¯h in the passive picture. At first sight, it seems that the fundamental commutation relation is not time-reversal invariant. In fact, in both older and more recent textbooks on quantum mechanics, the time-reversal invariance of the fundamental commutation relation has never been explicitly clarified in the passive picture. We point out that the claims in standard textbooks concerning the time-reversal invariance of the fundamental commutation relation in the passive picture are misleading. We show that [p, x]0 = [p0, x0] = [−p, x] = i¯h, if the time-reversal operator is unitary. Hence the fundamental commutation relation is not time-reversal invariant if the time-reversal operator is unitary. On the other hand, [p, x]0 = [x0, p0] = [x,−p] = −i¯h, if the time-reversal operator is antiunitary. We conclude that the fundamental commutation relation is time-reversal invariant, provided that the time-reversal operator is antiunitary. The important fact [p, x]0 = [x0, p0], which is the origin of the difficulties, is not appreciated in standard textbooks on quantum mechanics. The present discussion provides a more fundamental justification for taking the time-reversal operator to be antiunitary, which applies to particles with arbitrary spin. PACS numbers: 03.65.-w
Relation: 42(3), pp.243-248
URI: http://ntour.ntou.edu.tw/handle/987654321/27575
Appears in Collections:[光電科學研究所] 期刊論文

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