A.18 The energy-time uncertainty relationship

As mentioned in chapter 4.5.3, Heisenberg’s formulae

\begin{displaymath}
\Delta{p_x}\Delta{x}\mathrel{\raisebox{-1pt}{$\geqslant$}}\frac12\hbar
\end{displaymath}

relating the typical uncertainties in momentum and position is often very convenient for qualitative descriptions of quantum mechanics, especially if you misread $\raisebox{-.5pt}{$\geqslant$}$ as $\vphantom0\raisebox{1.1pt}{$\approx$}$.

So, taking a cue from relativity, people would like to write a similar expression for the uncertainty in the time coordinate,

\begin{displaymath}
\Delta{E}\Delta{t}\mathrel{\raisebox{-1pt}{$\geqslant$}}\frac12\hbar
\end{displaymath}

The energy uncertainty can reasonably be defined as the standard deviation $\sigma_E$ in energy. However, if you want to formally justify the energy-time relationship, it is not at all obvious what to make of that uncertainty in time $\Delta{t}$.

To arrive at one definition, assume that the variable of real interest in a given problem has a time-invariant operator $A$. The generalized uncertainty relationship of chapter 4.5.2 between the uncertainties in energy and $A$ is then:

\begin{displaymath}
\sigma_E \sigma_A \mathrel{\raisebox{-1pt}{$\geqslant$}}\frac12\vert\langle[H,A]\rangle\vert.
\end{displaymath}

But according to chapter 7.2 $\vert\langle[H,A]\rangle\vert$ is just $\hbar\vert{\rm d}\langle{A}\rangle/{\rm d}{t}\vert$.

So the Mandelshtam-Tamm version of the energy-time uncertainty relationship just defines the uncertainty in time to be

\begin{displaymath}
\Delta t = \sigma_A\Bigg/\left\vert\frac{{\rm d}\big\langle A\big\rangle }{{\rm d}t}\right\vert.
\end{displaymath}

That corresponds to the typical time in which the expectation value of $A$ changes by one standard deviation. In other words, it is the time that it takes for $A$ to change to a value sufficiently different that it will clearly show up in measurements.