# What is $\Delta t$ in the time-energy uncertainty principle?

Let a quantum system with Hamiltonian $$H$$ be given. Suppose the system occupies a pure state $$|\psi(t)\rangle$$ determined by the Hamiltonian evolution. For any observable $$\Omega$$ we use the shorthand $$\langle \Omega \rangle = \langle \psi(t)|\Omega|\psi(t)\rangle.$$ One can show that (see eq. 3.72 in Griffiths QM) $$\sigma_H\sigma_\Omega\geq\frac{\hbar}{2}\left|\frac{d\langle \Omega\rangle}{dt}\right|$$ where $$\sigma_H$$ and $$\sigma_\Omega$$ are standard deviations $$\sigma_H^2 = \langle H^2\rangle-\langle H\rangle^2, \qquad \sigma_\Omega^2 = \langle \Omega^2\rangle-\langle \Omega\rangle^2$$ and angled brackets mean expectation in $$|\psi(t)\rangle$$. It follows that if we define $$\Delta E = \sigma_H, \qquad \Delta t = \frac{\sigma_\Omega}{|d\langle\Omega\rangle/dt|}$$ then we obtain the desired uncertainty relation $$\Delta E \Delta t \geq \frac{\hbar}{2}$$ It remains to interpret the quantity $$\Delta t$$. It tells you the approximate amount of time it takes for the expectation value of an observable to change by a standard deviation provided the system is in a pure state. To see this, note that if $$\Delta t$$ is small, then in a time $$\Delta t$$ we have $$|\Delta\langle\Omega\rangle| =\left|\int_t^{t+\Delta t} \frac{d\langle \Omega\rangle}{dt}\,dt\right| \approx \left|\frac{d\langle \Omega\rangle}{dt}\Delta t\right| = \left|\frac{d\langle \Omega\rangle}{dt}\right|\Delta t = \sigma_\Omega$$

The time-energy uncertainty relation (and other time-"observable" uncertainty relations that can be constructed) is (considered) not to have same meaning as canonical uncertainty relations. Meaning uncertainty relations costructed from canonical dynamical variables/observables (in the Hamiltonian sense), like position and momentum, since time parameter is not an observable and also not an operator in QM/QFT formalisms.

In fact, there are various approaches and interpretations of time-energy uncertainty. For example:

1. Energy-dispersion ($\Delta E$) of a state and lifetime ($\Delta t$ or $\tau_s$) of the state itself.

2. Energy exchange ($\Delta E$) and time-frame ($\Delta t$) during which this can happen.

3. Energy measurement ($\Delta E$) and time ($\Delta t$) it needs for accuracy (although this is rigorously disputed, see below )

4. ..other similar or specialised formulations of the above

In L. Mandelstam and I. Tamm, "The uncertainty relation between energy and time in nonrelativistic quantum mechanics", J Phys (USSR) 1945, they show how one can derive time-observable uncertainty relations for any observable $A$ with

$$\Delta t = \tau_A = \frac{\Delta A}{d\left<A\right> /dt}$$

Time and time-energy uncertainty is used heavily in (quantum/mixed) statistical mechanics of systems since it relates half-times and life-times of states and transitions (will have to find some references)

An analysis of various formulations of time-energy uncertainty relations can be found in:

Jan Hilgevoord, The uncertainty principle for energy and time I

and

Jan Hilgevoord, The uncertainty principle for energy and time II

Summary:

The uncertainty principle for energy and time is not a canonical uncertainty relation because it is not based/produced by canonical hamiltonian variables, instead it expresses dispersion and lifetime of a state. There is a confusion of a cartesian space-time $x, t$ (used as parameters) and canonical position and momenta ($q, p$) which are functions of these parameters (however simple in some cases, like $q=x$)

The time-energy uncertaintly relation has a different interpration and derivation than the uncertaintly relation for non-commuting operators. Try John Baez for an explanation, but, roughly speaking $\delta t$ measures the time it takes for the expectation value of some operator to change noticeably.