Time-Energy
Uncertainty Relation
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The
Heisenberg Uncertainty Principle (HUP) is defined as an inequality between the
standard deviations (aka uncertainties) of two observables. To be precise the inequality is [1]
If
then
The HUP
is more appropriately referred to as the Heisenberg Uncertainty Theorem since it
can be derived from more basic concepts.
In non-relativistic quantum mechanics time is treated as a parameter, not
as an observable. The quantity Dt
is therefore not the standard deviation of an observable “t”. If one
were to use symbols which more accurately displayed the statistical properties
of observables and the true meaning of Dt
then the HUP for time-energy would appear as
which
more clearly shows the different nature of the uncertainty in E and the
time interval Dt.
As shown below, Dt
represents the amount of time it takes the expectation value of an observable Q
to change by one standard deviation.
The
time derivative of the expectation of an observable Q is
In what follow we choose an observable Q that does not depend on time.
The partial derivative of Q therefore vanishes and the second term
on the right hand side of Eq. (1) vanishes. Calculate the uncertainties in H
and Q, i.e. the standard deviations of H and Q (sH
and sQ)
or
We now
define DE
= sH
and
Upon
substituting into Eq. (5) we obtain
Dt
therefore represents the amount of time it takes the expectation value of Q
to change by one standard deviation.
Important point - Notice that Dt
depends entirely on what observable Q you care to look at. For the very same system the change might be
rapid for one observable and slow for another. But if DE
is small then the rate of change of all observables must be very gradually; or to
put it another way, if any observable changes rapidly the uncertainty
in energy must be large.
For more information of the time-energy uncertainty relation please see the article Time as an Observable, by William Unruh.
[1] Quantum
Mechanics – Volume I, Cohen-Tannoudji, Diu, Laloë,
John Wiley & Sons, (1977) page 287.