Now, this brings up an interesting question: if a sound wave is a vibration of matter, and a photon is a vibration of electric and magnetic fields, what exactly vibrates when matter acts as a wave? The upsetting answer is that there is no directly measurable quantity to correspond to the matter wave itself. This is what physicists call the lack of an experimental observable.
That is, while you can use a microphone to measure the vibrations of air pressure that constitute a sound wave, and you can use a radio receiver to measure the electromagnetic oscillations of light waves (remember our microwave interference experiments using a Gunnplexer?), there is no device that can measure a matter wave directly.
Let’s think back to the single-electron interference experiments performed by Tonomura (Figure 101). At any one time, we can only detect the fluorescence produced by a single electron hitting the electron microscope’s screen. We can’t know where the next electron will fall on the screen. We only know that when a large number of electron hits are accumulated, they will build up an interference pattern. That is, for any one electron that hasn’t yet reached the screen, all we can tell is the probability that the electron will hit a certain part of the screen.
Figure 101 We can’t know where the next electron will hit the screen in a single-electron, double-slit experiment. We only know the probability of each electron hitting the screen, such that the interference pattern emerges after accumulating many hits. Inset (a–e): the individual hits in Tonomura’s results come at random, showing no apparent pattern at the beginning of the experiment. However, the interference pattern builds up as single electrons arrive more often at locations where the square of the wavefunction peaks than where it dips. Integrated numbers of electrons detected by the detector are (a) 10, (b) 200, (c) 6,000, (d) 40,000, and (e) 140,000. The inset for this image was obtained from Wikimedia Commons at http://commons.wikimedia.org/wiki/Image:Double-slit_experiment_results_Tanamura_2.jpg. It was released to the public domain with kind permission of Dr. Tonomura. This insert is under the terms of the GNU Free Documentation License.
As uncomfortable as this may sound, a matter wave is actually a probability wave. It is not directly measurable, and it only allows us to determine where there is a high or a low probability of finding the particle. Probability is proportional to the square of the wave’s amplitude, but measuring its square is not the same as measuring the wave itself.*
A matter wave may not be measurable, but as we will see later, quantum physics allows us to calculate its properties for a particle in a physical apparatus. We can then square the mathematical function that defines the matter wave to find the probability of finding the particle at a given place in the apparatus.
We represent the mathematical function for a matter wave by Ψ—the Greek letter psi—and call it the wavefunction. Thus, |Ψ|2 is the probability of finding the particle for which we are calculating the wavefunction Ψ at a certain time and position (given by time t and coordinates x, y, and z). For example, in the simple case of the double-slit experiment of Figure 101, |Ψ|2 can be thought of as being dependent only on the position of the screen along the horizontal axis x.
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