Massimo Rudan - Physics of Semiconductor Devices

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Part I
A Review of Analytical Mechanics and Electromagnetism

Springer Science+Business Media New York 2015

Massimo Rudan Physics of Semiconductor Devices 10.1007/978-1-4939-1151-6_1

1. Analytical Mechanics

Massimo Rudan 1

(1)

University of Bologna, Bologna, Italy

Massimo Rudan

Email:

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1.1 Introduction

The differential equations that govern the dynamical problems can be derived from variational principles, upon which the theories of Euler and Lagrange, and those of Hamilton and Jacobi, are based. Apart from being the source of the greatest intellectual enjoyment, these theories have the definite advantage of generality. Their concepts can in fact be extended to cases where the Newtonian equation of dynamics does not apply. Among such cases there are the equations governing the electromagnetic field and those related to the quantum description of the particles motion.

The invariance property of the Lagrange equations with respect to a change of coordinates gives origin to the concept of generalized coordinates and conjugate momenta; in turn, the introduction of the Hamilton equations provides a picture in which the intrinsic symmetry of the roles of coordinates and momenta becomes apparent. Basing on the concept of conjugate coordinates, the time evolution of a particle or of a system of particles is described in the phase space instead of the coordinate space. This chapter and the next one illustrate the basic principles of Analytical Mechanics. Their purpose is to introduce a number of concepts that are not only useful per se , but also constitute a basis for the concepts of Quantum Mechanics that are introduced in later chapters. A third chapter devoted to Analytical Mechanics describes a number of important examples, that will be applied to later developments illustrated in the book.

As the velocity of particles within a semiconductor device is small with respect to that of light, the non-relativistic form of the mechanical laws is sufficient for the purposes of this book. The relativistic form is used only in a few paragraphs belonging to the chapter devoted to examples, to the purpose of describing a specific type of collision between particles. This chapter starts with the description of the Lagrangian function and the Lagrange equations, that are derived as a consequence of the variational calculus, followed by the derivation of the Hamiltonian function and Hamilton equations. Next, the HamiltonJacobi equation is derived after discussing the timeenergy conjugacy. The chapter continues with the definition of the Poisson brackets and the derivation of some properties of theirs, and concludes with the description of the phase space and state space.

1.2 Variational Calculus

Consider a real function defined in the interval and differentiable in its interior at least twice. The first two derivatives will be indicated with e . Now, define the integral

(1.1)

where the form of the function is prescribed. If () establishes a correspondence between a set of functions and a set of numbers. Such a correspondence is called functional .

It is interesting to extend to the case of functionals some concepts and procedures that apply to functions proper; among these it is important the concept of extremum. In fact, one defines the extremum function of a functional by a method similar to that used for defining the extremum point of a function: some w is an extremum of G if a variation in () produces a variation that is infinitesimal of an order higher than that of . The procedure by which the extremum functions are calculated is called variational calculus .

To proceed it is necessary to define the variation . For this one lets , with an arbitrary function defined in and differentiable in its interior, and a real parameter. The function thus defined is the finite variation of w . The sum tends to w in the limit . As a consequence, such a limit provides the infinitesimal variation . For simplicity it is convenient to restrict the choice of to the case , so that coincides with w at the integration boundaries for any value of .

Now, replacing w with in () makes G a function of , whose derivative is

(1.2)

According to the definition given above, if w is an extremum function of G then it must be ; in this case, in fact, the first-order term in the power expansion of G with respect to vanishes, and the variation of G becomes second order in or higher. In conclusion, one proceeds by imposing that the right hand side of () vanishes for

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