Time Spans in a New Class of Time-Energy Uncertainty Relations for Time-Dependent Hamiltonians
Adjunct Professor Tien Kieu
Centre for Quantum and Optical Science, Swinburne University of Technology
Since time is a parameter in quantum mechanics, rather than being represented by an operator, both the derivation and interpretation of time-energy uncertainty relations do not enjoy the transparency and clarity of those of the Heisenberg uncertainty relations of position and momentum. Some recently derived time-energy uncertainty relations for time-dependent Hamiltonians, furthermore, require the evaluation of instantaneous energy eigenvalues and eigenstates. Such an evaluation is hard and may not even be executable in general, and as a consequence it would restrict the applicability of those relations.
We present in this talk a new class of time-energy uncertainty relations which is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Their interpretation is transparent from the mathematical derivations. Moreover, only the initial state, and neither the instantaneous eigenstates nor the full time-dependent wave function at any other times which would demand a full solution for a time-dependent Hamiltonian, is required for the time-energy relations. Explicit results are then presented for particular subcases of interest for time-independent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation.
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