vi
Contents
2.5.6 Spherical aberration . . . . . . . . . . . . . 77
2.5.7 Field aberrations . . . . . . . . . . . . . . . 80
2.5.8 Chromatic aberration . . . . . . . . . . . . . 85
2.5.9 Intensity point spread function . . . . . . . 88
2.6 Stochastic Coulomb scattering . . . . . . . . . . . . 94
2.6.1 Monte Carlo simulation . . . . . . . . . . . 95
2.6.2 Analytical approximation by Markov’s method
of random flights . . . . . . . . . . . . . . . 99
2.7 Hamilton–Jacobi theory . . . . . . . . . . . . . . . 109
2.7.1 Canonical transformations . . . . . . . . . . 110
2.7.2 Applications of Hamilton–Jacobi theory . . 117
2.7.3 Hamilton–Jacobi theory and geometrical
optics . . . . . . . . . . . . . . . . . . . . . 120
3 Wave optics
123
3.1 Quantum mechanical description of particle motion 125
3.1.1 The postulates of quantum mechanics . . . . 125
3.1.2 Particle motion in a field-free space . . . . . 139
3.1.3 Wave packet propagation and the Heisenberg uncertainty principle . . . . . . . . . . 149
3.1.4 The quantum mechanical analog of Fermat’s
principle for matter waves . . . . . . . . . . 153
3.2 Particle motion in a general electromagnetic potential157
3.2.1 Path integral approach for the time-dependent
wave function . . . . . . . . . . . . . . . . . 157
3.2.2 Series solution for a particle in a general
electromagnetic potential . . . . . . . . . . . 166
3.2.3 Quantum interference effects in electromagnetic potentials . . . . . . . . . . . . . . . . 174
3.2.4 The Klein–Gordon equation and the covariant wave function . . . . . . . . . . . . . . . 179
3.2.5 Physical interpretation of the wave function
and its practical application . . . . . . . . . 184
3.3 Diffraction . . . . . . . . . . . . . . . . . . . . . . . 187
3.3.1 The Fresnel–Kirchhoff relation . . . . . . . . 191
3.3.2 The Fresnel and Fraunhofer approximations 198
3.3.3 Amplitude in the Gaussian image plane . . . 203
Contents
2.5.6 Spherical aberration . . . . . . . . . . . . . 77
2.5.7 Field aberrations . . . . . . . . . . . . . . . 80
2.5.8 Chromatic aberration . . . . . . . . . . . . . 85
2.5.9 Intensity point spread function . . . . . . . 88
2.6 Stochastic Coulomb scattering . . . . . . . . . . . . 94
2.6.1 Monte Carlo simulation . . . . . . . . . . . 95
2.6.2 Analytical approximation by Markov’s method
of random flights . . . . . . . . . . . . . . . 99
2.7 Hamilton–Jacobi theory . . . . . . . . . . . . . . . 109
2.7.1 Canonical transformations . . . . . . . . . . 110
2.7.2 Applications of Hamilton–Jacobi theory . . 117
2.7.3 Hamilton–Jacobi theory and geometrical
optics . . . . . . . . . . . . . . . . . . . . . 120
3 Wave optics
123
3.1 Quantum mechanical description of particle motion 125
3.1.1 The postulates of quantum mechanics . . . . 125
3.1.2 Particle motion in a field-free space . . . . . 139
3.1.3 Wave packet propagation and the Heisenberg uncertainty principle . . . . . . . . . . 149
3.1.4 The quantum mechanical analog of Fermat’s
principle for matter waves . . . . . . . . . . 153
3.2 Particle motion in a general electromagnetic potential157
3.2.1 Path integral approach for the time-dependent
wave function . . . . . . . . . . . . . . . . . 157
3.2.2 Series solution for a particle in a general
electromagnetic potential . . . . . . . . . . . 166
3.2.3 Quantum interference effects in electromagnetic potentials . . . . . . . . . . . . . . . . 174
3.2.4 The Klein–Gordon equation and the covariant wave function . . . . . . . . . . . . . . . 179
3.2.5 Physical interpretation of the wave function
and its practical application . . . . . . . . . 184
3.3 Diffraction . . . . . . . . . . . . . . . . . . . . . . . 187
3.3.1 The Fresnel–Kirchhoff relation . . . . . . . . 191
3.3.2 The Fresnel and Fraunhofer approximations 198
3.3.3 Amplitude in the Gaussian image plane . . . 203