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This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.

This chapter reviews balance laws and relevant solutions to Maxwell’s equations in connection with the time-harmonic dependence. The wave equation becomes the Helmholtz equation and, for it, Green’s functions and Poynting’s theorem are considered. Wave solutions are investigated by letting the material be dissipative. Huyghens principle is re-examined and the properties of radiating solutions are established. Superposition of waves, dispersion effects, the Doppler effect, and group and signal velocity are reviewed.

This chapter is the first of two dealing with the dynamical diffraction of incident spherical waves. It makes use of Kato's theory, which is based on a Fourier expansion of the spherical wave. The transmission and reflection geometries are handled separately. Two methods of integration are given — direct integration and integration by the stationary phase method. The amplitude and intensity distributions of the reflected and refracted waves on the exit surface are calculated. It is shown that equal-intensity fringes are formed within the Borrmann triangle (Pendellösung fringes) that can be interpreted as due to interferences between the waves associated with the two branches of the dispersion surface. The integrated intensity is calculated and the influence of the polarization of the incident wave discussed. The last section describes the diffraction of ultra-short pulses of plane-wave X-rays such as those emitted by a free-electron laser and which can be handled by considering their Fourier expansion in frequency space.

This chapter discusses basic techniques from the theory of stationary phase. After giving an overview of the method of stationary phase, the chapter moves on to a discussion of pseudodifferential operators, by going over the basics from the calculus of pseudodifferential operators and their various microlocal properties, in the process obtaining an equivalent definition of wave front sets, before defining pseudodifferential operators on manifolds and going over some of their properties. The chapter then lays out the propagation of singularities as well as Egorov's theorem, which involves conjugating pseudodifferential operators. Finally, this chapter describes the Friedrichs quantization, and differentiates it from the Kohn-Nirenberg quantization presented earlier in the chapter.

Phase integral approximations express the quantum wavefunction in terms of the classical action integral, S(x)=∫p(x)dx.The main aim of the chapter is to highlight the strengths and weaknesses of the standard JWKB expansion for S(x) in powers of Planck’s constant. Remarkably, the catastrophic weakness at a classical turning point is shown to be eliminated by use of an alternative Airy uniform approximation, which depends only on the primitive JWKB phase information. The discussion brings out an important primitive semiclassical connection between the JWKB function and the stationary phase approximation. The analysis also leads naturally to Maslov phase contributions to the Bohr–Sommerfeld quantization condition and to an equivalent formula for the semiclassical phase shift. An alternative type of general uniform theory, again dependent on the primitive JWKB information, is extended to a family of multi-turning-point problems. Finally, higher-order corrections to the JWKB approximation are derived and tested against the pathological quartic oscillator problem.

Two types of matrix element approximation are adopted according to whether the wavefunctions are taken in angle–action or normalized JWKB forms. The former gives the Heisenberg correspondence between matrix elements and classical Fourier components. The latter approximation is appropriate to situations for which the dominant contribution to the integral comes from stationary phase or ‘Condon’ points, at which both coordinates and momenta are conserved between the two states. The presence of a single such point leads to a ‘Condon reflection’ pattern such that the energy variation of the matrix element mimics the nodal pattern of the parent wavefunction. Complications arising from multiple Condon points are discussed.