### A.18 The energy-time uncertainty relationship

As mentioned in chapter 4.5.3, Heisenberg’s formulae

relating the typical uncertainties in momentum and position is often very convenient for qualitative descriptions of quantum mechanics, especially if you misread as .

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

The energy uncertainty can reasonably be defined as the standard deviation 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 .

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

But according to chapter 7.2 is just .

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

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