xii
Preface
All useful information about the motion of a single charged particle is contained in the integral of the classical Lagrangian function
between two arbitrary points in time. This integral is known historically as Hamilton’s principal function, and alternatively as the
eikonal function. The actual path taken by the particle, chosen
among a multiplicity of mathematically possible paths, is the path
for which this integral has an extremum. In the important special
case where the general electromagnetic potential has no explicit
time dependence, the action integral reduces to a line integral of
the canonical momentum component along the ray path. This is
a considerable simplification in problems where one is only interested in the spatial coordinates of a ray, without the need to know
the arrival time at any given point. The extremum condition is
generally known as the principle of least action, which is expressible in concise and precise mathematical terms.
In quantum mechanics all relevant information about the motion
of a single particle is contained in the wave function, for which
the same action integral in units of Planck’s constant h ¯ is the
phase. It follows that all possible paths in the immediate vicinity
of the classical path interfere constructively. The classical path is
thus the path that maximizes the probability. This clarifies the
particle–wave duality in concise and elegant mathematical terms.
A close analogy exists between Fermat’s principle of light optics
and the principle of least action for a charged particle. The analogy between light optics and charged particle optics is deep, and
is manifested in quite practical ways, including diffraction and interference. These ideas are derived mathematically from first principles.
The literature of this mature field is extensive. Several books are
of particular interest. The three-volume set by Hawkes and Kasper
[43, 44, 45] describes the main principles in precise and comprehensive detail, with reference to the work of many authors over the
decades. There is arguably no better review of the enormous body
of work that brought the field to its present state. Geometrical
Charged-Particle Optics by Rose [75] is both general and compre
Preface
All useful information about the motion of a single charged particle is contained in the integral of the classical Lagrangian function
between two arbitrary points in time. This integral is known historically as Hamilton’s principal function, and alternatively as the
eikonal function. The actual path taken by the particle, chosen
among a multiplicity of mathematically possible paths, is the path
for which this integral has an extremum. In the important special
case where the general electromagnetic potential has no explicit
time dependence, the action integral reduces to a line integral of
the canonical momentum component along the ray path. This is
a considerable simplification in problems where one is only interested in the spatial coordinates of a ray, without the need to know
the arrival time at any given point. The extremum condition is
generally known as the principle of least action, which is expressible in concise and precise mathematical terms.
In quantum mechanics all relevant information about the motion
of a single particle is contained in the wave function, for which
the same action integral in units of Planck’s constant h ¯ is the
phase. It follows that all possible paths in the immediate vicinity
of the classical path interfere constructively. The classical path is
thus the path that maximizes the probability. This clarifies the
particle–wave duality in concise and elegant mathematical terms.
A close analogy exists between Fermat’s principle of light optics
and the principle of least action for a charged particle. The analogy between light optics and charged particle optics is deep, and
is manifested in quite practical ways, including diffraction and interference. These ideas are derived mathematically from first principles.
The literature of this mature field is extensive. Several books are
of particular interest. The three-volume set by Hawkes and Kasper
[43, 44, 45] describes the main principles in precise and comprehensive detail, with reference to the work of many authors over the
decades. There is arguably no better review of the enormous body
of work that brought the field to its present state. Geometrical
Charged-Particle Optics by Rose [75] is both general and compre