When we separated light into its constituent lines, or binned scintillation photons according to their energy, we were performing spectral analysis. Since light and gamma rays most commonly act as waves, what we have been doing is deconstructing a complex electromagnetic wave into many simple sine waves. The same method can be applied to the analysis of any type of wave, such as sound vibrations, radio signals, or ripples on a pond.
For example, in Figure 103, we saw how we could make a wave with concentrated probability by adding a number of simple sine waves. Figure 107 shows the spectrum of these waves to demonstrate how easy it is to analyze a signal by looking at its sine wave components. This is known as Fourier analysis in honor of French mathematician and physicist Joseph Fourier, who discovered in the 1800s that complex waves could be described mathematically as a series of simpler sine waves. The sine wave components of a signal are thus called its Fourier components.
Figure 107 A signal that varies as a function of time (left) can be represented by the frequencies of the sine wave components required to make that signal (right). The spectrum of the signal shows the power of the sine waves that need to be added to produce that signal.
Now, to be able to represent any of the waves by a spectrum of neat discrete lines like the ones in Figure 107, we must assume that they repeat in the same exact way over and over forever. However, if we can’t analyze the signal for an infinite amount of time (and who has that much time to wait, when there is so much other fun stuff to do?), then we cannot be certain the signal fulfills the requirement of repeating identically forever.
Let’s use an audio signal generator to produce an analog of the quantum wave-packet to explore the way in which the Fourier spectrum changes as we vary the length of time available to observe a simple signal. First, download the free ToneBurst† software by David Taylor from Edinburgh, United Kingdom. You will also need to download the runtime library from his Web site. In addition, download the free Spectrum Lab sound card spectrum analyzer by Wolfgang Buescher.‡
Run both programs on the same PC at the same time. Spectrum Lab should display the spectral content of the sound picked up by your PC’s microphone. Now, turn up the volume on your speakers, and set ToneBurst to produce a 500-Hz tone with a duration of 500 ms (turn “Declick” on). The spectrum should suddenly show a sharp spike at 500 Hz. Once you verify that this works, experiment with the effect of shortening and lengthening the duration of the tone burst from 50 ms (25 cycles at 500 Hz) up to 1 s (500 cycles at 500 Hz). Do you see how the width of the spectrum sharpens as you increase the burst duration? Play with different settings and try to measure the way in which the width of the spectrum doubles each time you half the duration of the tone burst. (Figure 108)
Figure 108 The width of the Fourier spectrum of a truncated sine signal depends on the time available for observing the sine signal. (a) An infinitely long sine signal produces a spectral line of zero width, meaning we have absolute certainty of its frequency. (b) Truncating the wave to an observation time Δt increases the width of the spectrum by 1/Δt on each side of the sine wave’s fundamental frequency fo. (c–e) The width of the spectrum doubles each time we halve the observation time Δt. Please note that the same scale is not used on every graph for the spectrum power (vertical axis, which does not intercept at f = 0).
To understand these results, you must remember that a mathematical sine wave is infinitely long. If you cut off the ends to create a wave packet of finite duration, you have to add many other sine components to cancel the wave beyond the ends of the wave packet. The shorter the segment of the wave you keep, the more additional sine waves you need to cancel the original sine wave outside of the wave packet.
In other words, a sharp function in the time domain yields a very broad function in the frequency domain, and vice versa. This is a form of the Uncertainty Principle and a very important concept in signal analysis—it is not possible to have a very narrow band signal of short duration. In summary, the shorter the wave packet, the broader the spectrum.
The analogy with the time–energy Uncertainty Principle comes from Planck’s relationship E = hf (which is valid for both light and particles, as shown by de Broglie). The uncertainty in the measurement of a particle’s energy is ΔE, which is analogous in our Fourier spectrum model to Δf. The uncertainty in time Δt is the time during which we can observe the wavepacket.
As we had stated before, the time–energy Uncertainty Principle is expressed as:
In our model, the wavepacket duration–spectrum uncertainty relationship is:
We can therefore see that a particle’s energy state that exists only for a limited time cannot have a definite energy. To have a definite energy, the de Broglie frequency of the particle must be accurately defined, which requires the state to remain observable for a very large number of cycles.
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