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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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