39.5.2 Uncertainty

There is another way of thinking about the time-frequency relationship $Δ f Δ t \approx 1$ . Suppose you want to determine when a wave packet arrives at a specific point in space, such as at a detector of some sort. At what instant of time can you say that the wave packet is detected? When the front edge arrives? When the maximum amplitude arrives? When the back edge arrives? Because a wave packet is spread out in time, there is not a unique and well-defined time t at which the packet arrives. All we can say is that it arrives within some interval of time $Δ t$ . We are uncertain about the exact arrival time.

Similarly, suppose you would like to know the oscillation frequency of a wave packet. There is no precise value for $f$ because the wave packet is constructed from many waves within a range of frequencies $Δ f$ . All we can say is that the frequency is within this range. We are uncertain about the exact frequency.

The time-frequency relationship $Δ f Δ t \approx 1$ tells us that the uncertainty in our knowledge about the arrival time of the wave packet is related to our uncertainty about the packet’s frequency. The more precisely and accurately we know one quantity, the less precisely we will be able to know the other.

FIGURE 39.15 shows two different wave packets. The wave packet of Figure 39.15a is very narrow and thus very localized in time. As it travels, our knowledge of when it will arrive at a specified point is fairly precise. But a very wide range of frequencies $Δ f$ is required to create a wave packet with a very small $Δ t$ . The price we pay for being fairly certain about the time is a very large uncertainty $Δ f$ about the frequency of this wave packet.

The figure consists of two parts labeled ("a") and (b) that illustrate two wave packets with different Delta t.

FIGURE 39.15 Full Alternative Text

Figure 39.15b shows the opposite situation: The wave packet oscillates many times and the frequency of these oscillations is pretty clear. Our knowledge of the frequency is good, with minimal uncertainty $Δ f$ . But such a wave packet is so spread out that there is a very large uncertainty $Δ t$ as to its time of arrival.

In practice, $Δ f Δ t \approx 1$ is really a lower limit. Technical limitations may cause the uncertainties in our knowledge of $f$ and t to be even larger than this relationship implies. Consequently, a better statement about our knowledge of a wave packet is

Δ f Δ t \geq 1

(39.23)

The fact that waves are spread out makes it meaningless to specify an exact frequency and an exact arrival time simultaneously. This is an inherent feature of waviness that applies to all waves.