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alive was 50%, but there was no way to predict what would happen when a particular box was opened (a single measurement). The cat problem is not physically realizable and, therefore, it is not a true quantum mechanical problem. A physically real problem that is akin to the cat problem is discussed in a later chapter. The cat problem was intended to introduce the idea that an observation can change a system and that only a probability could be ascertained from a series of experiments. For real systems that are absolutely small, quantum mechanics is the theory that permits the calculation and understanding of the distribution of probabilities that are obtained when measurements are made on many identically prepared systems. How quantum mechanical probability distributions arise and how to think about the nature of the disturbance that accompanies measurements on absolutely small systems are discussed in the following chapters.

3

Some Things About Waves

TO ADDRESS THE NATURE of the inherent disturbance that accompanies a measurement and to understand what can and cannot be measured about an absolutely small quantum mechanical system, first it is necessary to spend some time discussing classical waves and the classical description of light. At the beginning of the twentieth century, a variety of experiments produced results that could not be explained with classical mechanics. The earliest of these involved light. Therefore, we will first discuss an experiment that seemed to show that classical ideas work perfectly. Then, in Chapter 4, we will present one of the experiments that demonstrated that the classical mechanics description could not be correct and, furthermore, that a classical reanalysis of the experiment seemed to work, but actually didn’t. Finally, the correct analysis of the experiment involving light will be given using quantum ideas, which will bring us back to Schrödinger’s Cat.

WHAT ARE WAVES?

There are many types of classical waves, water waves, sound waves, and light waves (electromagnetic waves). All waves have certain common properties, including amplitude, wavelength, speed, and direction of propagation (the direction in which a wave is traveling). Figure 3.1 shows a wave traveling in the x direction. The amplitude of the wave is the “distance” between its positive and negative peaks, the up-to-down distance. The wavelength is the distance along the direction of propagation between two positive or negative peaks. This is the distance over which the wave repeats itself. If you are riding on the wave and you move any integer number of wavelengths forward or backward along the wave, everything looks the same. The wave is traveling with some velocity, V.

WAVES HAVE VELOCITIES AND FREQUENCIES

The velocity depends on the type of wave, and the velocity of a wave needs a little discussion. Imagine you are standing beside the wave in Figure 3.1, but the wave is so long that you cannot see its beginning or end. Still, you can determine its velocity using a timing device. Start timing when a positive peak just reaches you and stop timing when the next positive peak reaches you. You now have enough information to determine the wave’s velocity. The wave has traveled d, a distance, of one wavelength, in time t. The distance equals the velocity multiplied by the time, d = Vt. (If you are in a car going at velocity, V = 60 miles per hour, and you travel for a time, t = 1 hour, then you have traveled a distance, d = 60 miles.) If we take the distance of one wavelength and divide it by the time it took to travel one wavelength, then we know the velocity, V = d/t. Watching the wave go by is like watching a very long train go by. You see boxcar after boxcar pass you. If you know the length of one boxcar and how long it takes that one boxcar to pass by, then you can determine the velocity of the train.

FIGURE 3.1. A wave traveling in the x direction. The black line represents zero amplitude of the wave. The wave undergoes positive and negative oscillations about zero. The distance between the