Standing waves description

An inevitable consequence of de Broglie s standing-wave description of an electron in an orbit around the nucleus is that the position and momentum of a particle cannot both be known precisely and simultaneously. The momentum of the circular standing wave shown in Figure 4.18 is given exactly hj p = h/, but because the wave is spread uniformly around the circle, we cannot specify the angular position of the electron on the circle at all. We say the angular position is indeterminate because it has no definite value. This conclusion is in stark contrast with the principles of classical physics in which the positions and momenta are all known precisely and the trajectories of particles are well defined. How was this paradox resolved ... 

In the quantum-mechanical description of atoms and molecules, electrons have characteristics of waves as well as particles. In the familiar case of the hydrogen atom, the orbitals Is, 2s, 2p,... describe the different possible standing wave patterns of electron distribution, for a single electron moving in the potential field of a proton. The motion of the electrons in any atom or molecule is described as fully as possibly by a set of wave functions associated with the ground and excited states. 

The observation that the wavelength of light is linked to the particle-like momentum of a photon prompted de Broglie to postulate the likelihood of an inverse situation whereby particulate objects may exhibit wave-like properties. Hence, an electron with linear momentum p could under appropriate conditions exhibit a wavelength A = h/p. The demonstration that an electron beam was diffracted by periodic crystals in exactly the same way as X-radiation confirmed de Broglie s postulate and provided an alternative description of the electronic stationary states on an atom. Instead of an accelerated particle the orbiting electron could be described as a standing wave. To avoid self-destruction by wave interference it is necessary to assume an... 

Unlike the case of the Faraday rotation isolator, even an elementary description of the operational principle of the circulator is not easy, involving as it does the dimensional resonance of the microwave field within the ferrite cylinder. In this context the word resonance does not signify gyromagnetic resonance but a standing-wave resonance determined by the dimensions of the... 

Orbitals are mathematical descriptions of standing waves. Standing waves are a series of harmonics (increasing number of half-wavelengths) and are similar to the MOs for linear pi systems. Shorter wavelength harmonics are higher energy. 

In this chapter, we describe the technique of Fourier transform microwave spectroscopy. We distinguish here two rather different types of sample absorption cells which require somewhat different theoretical descriptions. First, we describe the theory for the relatively broad-band waveguide absorption cell in which the radiation is described as a traveling wave. Second, we describe the narrow-band Fabry-Perot cavity absorption cell in which the radiation is described as a standing wave. 

If confined to a resonator, the wave has to obey the resonance boundary conditions the standing waves in the resonator are called modes. Particular functions U describe the different modes. Because of the nature of laser resonators, which normally are much longer than wide, one may conveniently split the description into longitudinal (often by convention the z-axis) and transverse (the x- and y-axes) components, or modes, with respect to the optical axis of the resonator. As it turns out, to a good approximation, one can use the longitudinal mode components to derive the resonance frequencies in the resonator these then are loosely addressed as longitudinal (frequency ) modes. The transverse mode components can be used to calculate the lateral intensity distribution of a laser beam in general one then speaks of transverse (intensity) modes. 

For a more detailed description of the background, including key references, see Grosehl s 1998 paper [ 1 ]. King provided the first comprehensive calculation of acoustic radiation forces on a small particle within a standing wave in 1934. This theory was developed for rigid spheres in an inviscid fluid and predicts that particles will move toward either the node or antinode of the field depending on the ratio of the fluid and particle densities. When the ratio of the incompressible particle density to the fluid density is... 

The normalization constant is the same for all wavefunctions and does not depend on the quantum number m. Figure 11.9 shows plots of the first few I s. The magnitudes of thes are reminiscent of circular standing waves, and these are also suggestive of de Broglies picture of electrons in a circular orbital. It is only suggestive, and this analogy is not meant to hint that this is a true description of electron motion. 

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