DERIVATION OF THE ENERGY-TIME UNCERTAINTY RELATION

A derivation from first principles is given of the energy-time uncerta inty relation in quantum mechanics. A canonical transformation is made in classical mechanics to a new canonical momentum, which is energy E , and a new canonical coordinate T, which is called tempus, conjugate to the energy. Tempus T, the canonical coordinate conjugate to the ene rgy, is conceptually different from the time t in which the system evo lves. The Poisson bracket is a canonical invariant, so that energy and tempus satisfy the same Poisson bracket as do p and q. When the syste m is quantized, we find the energy-time uncertainty relation DELTAEDEL TAT greater-than-or-equal-to HBAR/2. For a conservative system the ave rage of the tempus operator T is the time t plus a constant. For a fre e particle and a particle acted on by a constant force, the tempus ope rators are constructed explicitly, and the energy-time uncertainty rel ation is explicitly verified.