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Journal of Mathematical Physics

Mathematical analysis of the Mandelstam–Tamm time-energy uncertainty principle

John Edward Gray, Andrew Vogt

Published Apr 23, 2005• 18 citations
Publisher

Abstract

In the Mandelstam–Tamm version of the time-energy uncertainty principle Δt denotes the infimum of time intervals that elapse before the change in the mean of any observable has the same magnitude as its standard deviation. We clarify this interpretation, and show that the infimum is achieved for certain observables and thus that this famous inequality is actually an equality.

DOI
10.1063/1.1897164

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In the Mandelstam–Tamm version of the time-energy uncertainty principle Δt denotes the infimum of time intervals that elapse before the change in the mean of any observable has the same magnitude as its standard deviation. We clarify this interpretation, and show that the infimum is achieved for certain observables and thus that this famous inequality is actually an equality.
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 ℓ1 norm of energy coherence C is invariant with respect to the unitary evolution of the quantum state. Thus, the ℓ1 norm of energy coherence 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 ℓ1 norm of energy coherence, 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 ℓ1 norm of energy coherence, 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 Schrödinger equation is a scalar variable that commutes with the quantum Hamiltonian and is not subject to statistical variance.
Georgiev's analysis of a freely precessing spin correctly emphasizes that the Mandelstam-Tamm quantity ΔT=ΔA^/∣d⟨A^⟩/dt∣ΔT=Δ\hat A/|d\langle\hat A\rangle/d t|ΔT=ΔA^/∣d⟨A^⟩/dt∣ is not the variance of a time operator. He concludes, however, that its divergence near the extrema of the clock signal is devoid of physical meaning and that ΔTΔTΔT does not disclose how well the system measures the passage of time. For the same projective σ^x\hatσ_xσ^x​ readout, we show that ΔTΔTΔT equals the inverse square root of the classical Fisher information in one binary outcome at regular operating points. Its divergence at submaximal-coherence extrema therefore identifies a loss of local sensitivity in the selected readout, not an infinite variance of physical time; a complementary σ^y\hatσ_yσ^y​ measurement remains informative. This interpretation also separates Georgiev's exactly known tick--tock period from the statistical precision with which a tick can be located. It preserves his warning against a time-operator interpretation while qualifying his stronger conclusion that the Mandelstam--Tamm scale lacks physical content.
In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation τ F Δ H ≥ ℏ / 2 provides a general lower bound to the characteristic time τ F = Δ F / | d ⟨ F ⟩ / d t | with which the mean value of a generic quantum observable F can change with respect to the width Δ F of its uncertainty distribution (square root of F fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty Δ H (square root of energy fluctuations). Here we show that when unitary evolution is complemented with a steepest-entropy-ascent model of dissipation, the resulting nonlinear master equation entails that these lower bounds get modified and depend also on the entropy uncertainty Δ S (square root of entropy fluctuations). For example, we obtain the time–energy-and–time–entropy uncertainty relation ( 2 τ F Δ H / ℏ ) 2 + ( τ F Δ S / k B τ ) 2 ≥ 1 where τ is a characteristic dissipation time functional that for each given state defines the strength of the nonunitary, steepest-entropy-ascent part of the assumed master equation. For purely dissipative dynamics this reduces to the time–entropy uncertainty relation τ F Δ S ≥ k B τ , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty Δ S .