Circular standing wave

FIGURE 4.18 A circular standing wave on a closed loop. The state shown has n = 7, with seven full wavelengths around the circle. 

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 ... 

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. 

Fig. 13.21 shows another example of oscillatory burning of an RDX-AP composite propellant containing 0.40% A1 particles. The combustion pressure chosen for the burning was 4.5 MPa. The DC component trace indicates that the onset of the instability is 0.31 s after ignition, and that the instability lasts for 0.67 s. The pressure instability then suddenly ceases and the pressure returns to the designed pressure of 4.5 MPa. Close examination of the anomalous bandpass-filtered pressure traces reveals that the excited frequencies in the circular port are between 10 kHz and 30 kHz. The AC components below 10 kHz and above 30 kHz are not excited, as shown in Fig. 13.21. The frequency spectrum of the observed combustion instability is shown in Fig. 13.22. Here, the calculated frequency of the standing waves in the rocket motor is shown as a function of the inner diameter of the port and frequency. The sonic speed is assumed to be 1000 m s and I = 0.25 m. The most excited frequency is 25 kHz, followed by 18 kHz and 32 kHz. When the observed frequencies are compared with the calculated acoustic frequencies shown in Fig. 13.23, the dominant frequency is seen to be that of the first radial mode, with possible inclusion of the second and third tangential modes. The increased DC pressure between 0.31 s and 0.67 s is considered to be caused by a velocity-coupled oscillatory combustion. Such a velocity-coupled oscillation tends to induce erosive burning along the port surface. The maximum amplitude of the AC component pressure is 3.67 MPa between 20 kHz and 30 kHz. - ...

In eqn. 2.7 the number n is a quantum number5. It is in fact related to the number of nodes in the wavefunction and must in this case be a positive integer (n = 1, 2, etc.). This would apply to a wave which follows one direction only. Since real space is three-dimensional a standing wave must be defined by three quantum numbers. The motions of electrons around nuclei are essentially circular, so that the use of polar coordinates is preferable and the three quantum numbers are ... 

If an electron behaves as a wave as it moves in a hydrogen atom, a stable orbit can result only when the circumference of a circular orbit contains a whole number of waves. In that way, the waves can join smoothly to produce a standing wave with the circumference being equal to an integral number of wavelengths. This equality can be represented as... 

An example of a one-port device is the bulk resonator shown in Figure 6.1, which has a single, planar electrode on each side of a slab of piezoelectric material (these two electrodes together comprise a single port). Most often, the material takes the form of a disk and the electrodes are circular, covering less than the entire surface of the disk. Connection to an external circuit is typically made via a coaxial cable, with one of the two electrodes connected to the shield and the other to the center conductor. This device is known as a resonator because an external circuit (see Section 6.3.3.2) excites the piezoelectric substrate in such a way that a standing wave is set up in the crystal, which thus resonates. 

The hydrogen electron visualized as a standing wave around the nucleus. The circumference of a particular circular orbit has to correspond to a whole number of wavelengths, as shown in (a) and (b), or else destructive interference occurs, as shown in (c). This model is consistent with the fact that only certain electron energies are allowed the atom is quantized. (Although this idea has encouraged scientists to use a wave theory, it does not mean that the electron really travels in circular orbits.)... 

Saylor JR, Szeri AJ, and Foulks GP (2000) Measurement of surfactant properties using a standing circular capillary wave field. Experiment in Fluids (in press)... 

When the microwave energy is coupled with a gas stream, a microwave or cavity resonator is required. It is a closed metal tube of rectangular or circular cross-section, and its inner dimensions must be such that standing waves can arise. 

Figure 15.2. Horizontal standing waves in a circular cavity. If the circle is one half wavelength wide, there can be one standing wave. If the circle is one wavelength wide, there can be three standing waves. If the circle is two wavelengths wide there can be seven standing waves, etc.

The spin orientation order in the sample is created and monitored using standing wave optical pumping of the ground state hyperfine levels of the Tm" 2 ion (see Fig. 1). These levels have a magneto-optic susceptibility which depends on the electron spin orientation. The absorption constant for right or left circularly polarized light can be written as... 

A weak cw circularly polarized probe beam is applied to the sample and together with its reflection off the mirror creates a standing wave probe field which monitors and when the probe wavelength is matched to the pump wavelength. Solving Maxwell s Equations for the probe field inside the sample volume with a modulated adsorption constant and index of refraction, for the case where yields a coupled wave equation for the inci-... 

Equation (1-30) gives the wavelength of the light waves or electron waves. For an electron traveling in a circular Bohr orbit, there must be an integral number of wavelengths in order to have a standing wave (see Fig. 1-2), or... 

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