Derivation of the energy-time uncertainty relation.

A new derivation from first principle. is given of the energy-time uncertainty relation in quantum mechanics. A canonical transformation is made in clusical mechanic. to & new canonical momentum, which is energy E, and a new canonical coordinate T, which is called tempu, co~ugate to the energy. Temp.. T, the canonical coordinate conjugate to the energy, is conceptually different from the time t in which the system evolves. The Poiuon bracket is a canonical invariant, 80 that energy and tempu satisfy the same Poisson bracket as do p and q. When the system is quantised, we find the energy-time uncertainty relation aE aT ~ ,.,/2. For a conservative system the average of the tempu, operator t is the time t plus a constant. For a free particle and a particle acted on by a constant force, the tempu operators are constructed explicitly, and the energy-time uncertainty relation is explicitly verified. ·On leave o{ absence {rom Insututo de Fi'aica Te6rica UNESP 01405-000 Sio Paulo, SP Brazil, with a grant from FAPESP, Brasil.