Time-energy uncertainty relation in nonrelativistic quantum mechanics

Abstract: The time-energy uncertainty relation in nonrelativistic quantum mechanics has been intensely debated with regard to its formal derivation, validity, and physical meaning. Here, we analyze two formal relations proposed by Mandelstam and Tamm and by Margolus and Levitin and evaluate their validity using a minimal quantum toy model composed of a single qubit inside an external magnetic field. We show that the $\ell_1$ norm of energy coherence $\mathcal{C}$ is invariant with respect to the unitary evolution of the quantum state. Thus, the $\ell_1$ norm of energy coherence $\mathcal{C}$ of an initial quantum state is useful for the classification of the ability of quantum observables to change in time or the ability of the quantum state to evolve into an orthogonal state. In the single-qubit toy model, for quantum states with the submaximal $\ell_1$ norm of energy coherence, $\mathcal{C}<1$, the Mandelstam-Tamm and Margolus-Levitin relations generate instances of infinite "time uncertainty" that is devoid of physical meaning. Only for quantum states with the maximal $\ell_1$ norm of energy coherence, $\mathcal{C}=1$, the Mandelstam-Tamm and Margolus-Levitin relations avoid infinite "time uncertainty", but they both reduce to a strict equality that expresses the Einstein-Planck relation between energy and frequency. The presented results elucidate the fact that the time in the Schr\"{o}dinger equation is a scalar variable that commutes with the quantum Hamiltonian and is not subject to statistical variance.