Details of the double-bond isomerization of 1-hexene over H-ZSM-5 were clarified using density functional theory. It is found that the reaction proceeds by a mechanism which involves the Br${o}$nsted acid part of the zeolite solely. According to this mechanism, 1-hexene is first physically adsorbed on the acidic site, and then, the acidic proton transfers to one carbon atom of the double bond, while the other carbon atom of the double bond bonds with the Br${o}$nsted host oxygen, yielding a stable alkoxy intermediate. Thereafter, the Br${o}$nsted host oxygen abstracts a hydrogen atom from the $C_6H_{13}$ fragment and the C-O bond is broken, restoring the acidic site and yielding trans-2-hexene. The calculated activation barrier is 12.65 kcal/mol, which is in good agreement with the experimental value. These results well explain the energetic aspects during the course of double-bond isomerization and extend the understanding of the nature of the zeolite active sites.