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 As discussed in the previous pages, light, electrons and other substance particles possess both the natures of particle and wave. These two natures are the essence of "existences". Then, where does the particle nature appear? And where does the wave nature manifest itself?

 [When and Where do the Particle and Wave Natures Appear?] The interference fringes (interference pattern) are the evidence of the wave nature. However, the eduction of individual silver atom on a photographic film (or plate) is yielded by dissociation of a halogenated silver molecule that is induced by the absorption of a particle of light (photon). This educed silver atom is the mark or print of photon. These huge number of marks are collected to be a photographic picture of interference fringes. This is nothing else than appearance of the particle nature of light. It must be determined, according to probability, to which halogenated silver molecule a photon collides or at what position the mark of a photon is left. A larger number of marks of photons are left at the area of higher probability, and fewer marks at a lower probability area. This light-and-shade pattern would be the interference fringes. Thus, when we measure light (or photons), for example as taking a photograph, the particle nature manifests itself. In the case of the photoelectric effect or the Compton effect, light interacts with other substances like atoms in a metal or electrons in a crystal as "particles". In such a case as light interacts with other substances, it behaves as "particles" and shows the particle nature. In cavity radiation, light is absorbed or emitted by the matter composing the wall of the cavity as "particles" and manifests itself as light particles (photons).

 [Let's Calculate the Particle and the Wave Nature] In order to understand the relation between the particle and wave natures, let us reproduce a double-slit experiment like Young's experiment on a PC (personal computer) considering the wave functions in quantum mechanics.

 Fig. (A): An electron beam comes from the distant left, passes through two slits, S1 and S2 and exposes the photographic film on the screen F.

 Fig. (B): A rough sketch (sectioned drawing) of the wave functions in the above experiment.

 A schematic drawing of our experiment is shown in Fig. (A). An electron beam comes from the distant left, passes through two slits, and , and exposes the photographic film on the screen F. A rough sketch (sectioned drawing) of the wave functions in the above experiment Fig. (A) is shown in Fig. (B). The wave function would be separated into two spherical waves (precisely speaking, spherical-wave-like waves) after passing the slits and they are superposed to interfere to each other. Now let us execute the double-slit experiment as shown in Fig. (A) on a PC. It is assumed that an electron beam (cathode rays) coming from the distant left to pass through the two slits, and , and exposes the photographic film F. Suppose the intensity of the beam, which is the number of electrons passing through a unit cross section par a unit time, be enough weak to avoid a meaningless interaction between electrons. The sectioned drawing of the feature of the wave functions is shown in Fig. (B). Since the incident beam is free particles, the corresponding wave is a (right-going) plane wave. After passing through the two slits, it separates into two waves, and , each of which is rather similar to a spherical wave. (Because the slits are of rectangular form, they might be similar to a cylindrical wave.) After passing through the slits, the total wave function would be a superposition of these waves, and , as , and this wave function reaches the film F on the screen. These wave functions are of a constant energy, i.e. of a constant frequency. Accordingly, they are described as stationary states. The functional forms of the wave functions and are approximately plane wave or spherical wave These functional forms are explained on the page, 2-2-A: Plane Wave and Spherical Wave. You may understand the forms of the wave function in the double-slit experiment shown in Fig. (A) and (B). Using the result we can approximately calculate the probability that an electron will be found at any point on the film F. When an electron hits some point on the film, an image (a mark) is made at that point. If the wave function is a plane wave, the probability is uniformly distributed everywhere. When electrons are repeatedly ejected many times, the distribution of their images would be completely random. Generating uniform random numbers with a computer program, we can make such a random distribution of images on a PC. Multiplying the probability to the uniformly random distribution, we can make a picture of a distribution of images or prints of electrons hit on the film F. The photograph-like picture below was made in this way. Assuming the number of the incident electrons in a second (s) to be about 10, we made pictures of the electron distribution varying the "exposure time" from 1 s to 256 s. While, for a very short "exposure time", the distribution of the images is very sparse, it would become dense for a longer exposure time and the interference fringes would be seen clearly. This is an accurate simulation of the interference pattern of the electrons and it conforms to the time-dependent behavior observed in actual experiments. Looking at these results, you may understand that the particle nature and the wave nature of electrons are beautifully mixed and unified within quantum mechanics. Although the interference fringes in the following picture have been made by numerical calculations using quantum mechanics, a very similar photograph was taken in an actual experiment by A. Tonomura (Hitachi, Japan) (1989).

 Double-slit experiment on a personal computer According to quantum mechanics, the double-slit experiment was simulated by calculating the probability that an electron would be found. The number of the incident electrons is assumed to be about 10 par one second (s). The "exposure time" was varied from 1 s to 256 s.
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