Simple Nature - Light and Matter

14.8 Energy-time uncertainty

14.8.1 Classical uncertainty relations
Consider the following classical system of analogies.
space x k ∆x∆k& 1
time t ω ∆t∆ω& 1
Here the quantityk= 2π/λ is called the wavenumber. The in-
equality ∆x∆k&1 is a kind of classical uncertainty relation that is
closely related to the Heisenberg uncertainty principle. Its classical
nature is immediately apparent because it doesn’t involve Planck’s
constant. If you look back at the argument given on p. 901 to jus-
tify the Heisenberg uncertainty principle, you will see that it carries
through equally well if we simply omit the quantum-mechanical in-
gredients and use it to put a bound on ∆x∆kinstead of ∆x∆p.
Once we’ve established the bound on ∆x∆k, the one on ∆x∆pfol-
lows immediately becausep=h/λ=~k.
The second line of the table is in strict analogy to the first line.
A good practical example is the high-speed transmission of digital
data over transmission lines such as fiber-optic cables. Suppose that
we wish to send a string of 0’s and 1’s, and a 1 is to be represented by
a square pulse. If we want to transmit the data at high speed, then
we need the duration ∆tof this pulse to be short, perhaps in the
microsecond or even nanosecond range. This cannot be done if the
signal consists only of a single frequency. A signal that only contains
a single, pure frequency is just a sinusoidal wave that has existed
infinitely far back in the past and will exist infinitely far into the
future. Such a wave carries no information at all. Out frequency-
time uncertainty relation tells us that if the duration of a pulse is to
be, say, a microsecond, then the signal’s spread in frequency much
be at least on the order of 1 MHz. This is why we use the term
“bandwidth” to describe the speed of a communication channel.

14.8.2 Energy-time uncertainty

In a quantum-mechanical context, we haveE=~ω, so there is

an energy-time uncertainty relation,

∆E∆t&~.

As with the Heisenberg uncertainty principle for momentum and

position, the symbol&means that we leave out a numerical factor,

which can only be precisely defined if we fix some specific statistical

definition of ∆, e.g., a standard deviation.

The interpretation of the energy-time uncertainty relation is a

little tricky, because although the classical analogy between space

and time is exact, the quantum-mechanical analogy breaks down.

This is because time in nonrelativistic quantum mechanics, unlike

position, is not an observable (example 9). Time in this theory

Section 14.8 Energy-time uncertainty 995