The Double-Slit Experiment

CHAPTER 6 Quantum Theory and the Electronic Structure of Atoms 

Sample Problem 6.1 illustrates the conversion between wavelength and frequency. 

Think About It Make sure your units cancel properly. A common error in this type of problem is neglecting to convert wavelength to meters. 

The wavelength of a laser used in the treatment of vascular skin lesions has a wavelength of 532 nm. What is the frequency of this radiation  

Strategy Wavelength and frequency are related by Equation 6.1 (c = v), so we must rearrange Equation 6.1 to solve for frequency. Because we are given the wavelength of the electromagnetic radiation in nanometers, we must convert this wavelength to meters and use c = 3.00 X lO m/s. 

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If the motion of a particle in the double-slit experiment is to be represented by a wave function, then that wave function must determine the probability density P(x). For mechanical waves in matter and for electromagnetic waves, the intensity of a wave is proportional to the square of its amplitude. By analogy, the probability density P(x) is postulated to be the square of the absolute value of the wave function (x)... 

C. Dewdney, A. Kyprianidis, and J. P. Vigier, Causal non-local interpretation of the double slit experiment and quantum statistics, Epistemol. Lett. 36, 71 (1984). 

Still, the interference patterns from the double slit experiment clearly indicated that light was a wave. How could Einstein reconcile his conclusions about the particle nature of light with the results of the double slit experiment Was it possible that light could act as either particle or wave ... 

FIGURE 2.2. Energy diagram showing interference between vibrational modes. Interference arises because VSFS is a coherent process. The multiple paths are analogous to those in the double slit experiment. 

Deviating from the historical development, we will illustrate these effects by a modification of the double-slit experiment. Let us equip the laser source with a dimmer switch capable of reducing the light intensity by several orders of magnitude, as shown in Fig. 2.4. With each successive filter the diffraction pattern... 

There was never any drawn-out controversy about whether electrons or any other constituents of matter were other than particle-like. Individual electrons produce scintillations on a phosphor screen—that is how TV works. But electrons also exhibit diffraction effects, which indicates that they too have wavelike attributes. An analog of the double-slit experiment using electrons instead of light is technically difficult, but has been done. An electron gun, instead of a light source, produces a beam of electrons at a selected velocity. Then, everything that happens for photons has its analog for electrons, as shown by the diffraction pattern in Fig. 2.7. Diffraction experiments have been more recently carried out for particles as large as atoms and molecules, even for C6o fullerene molecules. 

The interference of microscopic particles leads to a diffraction pattern with deviations with respect to the mere sum of the individual probabilities. The two events are no longer independent. If we wish to state in advance where the next particle will appear, we are unable to do so. The best we can do is to say that the next particle is more likely to strike in one area than another. A limit to our knowledge, associated with the wave-matter duality, becomes apparent. In the double-slit experiment, we may know the momentum of each particle but we do not know an5 hing about the way the particles traverse the slits. Alternatively, we could think of an experiment that would enable us to decide through which slit the particle has passed, but then the experiment would be substantially different and the particles would arrive at the screen with different distributions. In particular, the two slits would become distinguishable and independent events would occur. No interference would be detected, that is, the wave nature of the particle would be absent. In such an experiment, in order to obtain information about the particle position just beyond the slits, we would change its momentum in an unknown way. Indeed, recent experiments have shown that interference can be made to disappear and reappear in a quantum eraser (ref. 6 and references therein). 

The example of the double-slit experiment already shows that the physical information contained in a state function is inherently probabilistic in nature. In the next chapter this feature will be further developed leading to the central concept of this book orbitals. For example, the orbitals of the H atom are wavefunctions (x,y,z) that enable the electron probability density to be known (x,y,z) (x,y,z). The idea of a trajectory (or orbit) is replaced by the idea of a probability distribution. 

This is known as the Heisenberg indeterminacy principle (also called uncertainty principle by some authors). It has to do with precision and not with accuracy. This situation has already been met in Section 1.2 when referring to the double-slit experiment. 

This is why Richard Feynman said the double-slit experiment to be at the heart of quantum mechanics [Feynman 1965], One might argue that another central issue of quantum physics, namely entanglement, is missing in this example. However, it turns out to be an essential ingredient if we consider how which-path information is diffused into the environment, i.e., if we include decoherence, as will be done in Sec. 3 and Sec. 4. 

Modern-day version of the double-slit experiment using a beam of electrons. Even when the electrons are fired one at a time through the screen with the slits, an interference pattern develops on the optical screen. [Reproduced from http //commons.wikimedia.org /wiki/File Two-Slit Experiment Electrons.svg (accessed December 1, 2013).]... 

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