Vector plane waves definition

We briefly recall the definitions and the basic properties of scalar and vector plane waves [14]. The scalar plane wave is defined by... 

We see from Fig. 2.5 that the Gaussian wave packet has its intensity, F 2, centred on x0 with a half width, W, whereas (k) 2 is centred on k0 with a half width, 1/W. Thus the wave packet, which is centred on x0 with a spread Ax — W, is a linear superposition of plane waves whose wave vectors are centred on k0 with a spread, A = jW. But from eqn (2.8), p = Hk. Therefore, this wave packet can be thought of as representing a particle that is located approximately within Ax = W of x0 with a momentum within Ap = h/W of po = hk0. If we try to localize the wave packet by decreasing W, we increase the spread in momentum about p0. Similarly, if we try to characterize the particle with a definite momentum by decreasing 1/W, we increase the uncertainty in position. 

In the case of plane-wave basis sets the scaling proceeds on the reciprocal-space vectors as G- (1+ ) G, which is seen by the definition aj b< = 6 4, where and b are real- and reciprocal-lattice primitive translation vectors, respectively. Thus one finds the derivative of reciprocal-space vectors given by... 

Now consider the quantum dynamics of the same system. If the discs scatter a plane wave of wave vector k, the same matrix describes the scattering processes in the classical and semiclassical descriptions, but in the latter the parameters must be fixed as follows z = -ik,r = -l for Dirichlet boundary conditions,and P = 1/2 because the square of the wave function is the density of probabilityTherefore, we can analyze both the classical and the quantum dynamics of this model system with the same formalism, keeping in mind the different definitions of the parameters z, P and r. In the following we shall consider only the case lrl = l. 

Figure 9.7 Definition of the plane of incidence (p plane) and the incidence angle 4>o through the wave vectors of the incident and emerging (reflection set up) plane waves. Ap, A, Bp, and 6, denote the complex amplitudes of the p and s modes...

This is the definition of the Bragg plane it is formed by the tips of all the vectors q which satisfy Eq. (3.28) for a given G. This relation determines all vectors q that lead to constructive interference. Since the angle of incidence and the magnitude of the wave-vector q can be varied arbitrarily, Eq. (3.28) serves to identify all the families of planes that can reflect radiation constructively. Therefore, by scanning the values of the angle of incidence and the magnitude of q, we can determine all the G vectors, and from those all the R vectors, i.e. the Bravais lattice of the crystal. 

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