I.1.1 The Variables of a Moving Particle

The position and velocity vectors of a particle are illustrated in Fig. I.1. The position vector r extends from the origin to the particle, while the velocity vector v points in the direction of the particle’s motion. Other variables, which are appropriate for describing a moving particle, can be defined in terms of these elementary variables.

The momentum p of the particle is equal to the product of the mass and velocity v of the particle

We shall find that the momentum is useful for describing the motion of electrons in an extended system such as a crystal.

The motion of a particle moving about a center of force can be described using the angular momentum, which is defined to be the cross product of the position and momentum vectors

The cross product of two vectors is a vector having a magnitude equal to the product of the magnitudes of the two vectors times the sine of the angle between them. Denoting the angle between the momentum and position vectors by θ as in Fig. I.1, the magnitude of the angular momentum vector can be written

This expression for the angular momentum may be written more simply in terms of the distance between the line of motion of the particle and the origin, which is denoted by r0 in Fig. I.1. We have

The angular momentum is thus equal to the distance between the line of motion of the particle and the origin times the momentum of the particle. The direction of the angular momentum vector generally is taken to be normal to the plane of the particle’s motion. For a classical particle moving under the influence of a central force, the angular momentum is conserved. The angular momentum will be used in later chapters to describe the motion of electrons about the nucleus of an atom.

The kinetic energy of a particle with mass m and velocity v is defined by the equation

where v is the magnitude of the velocity or the speed of the particle. The concept of potential energy is useful for describing the motion of particles under the influence of conservative forces. In order to define the potential energy of a particle, we choose a point of reference denoted by R. The potential energy of a particle at a point P is defined as the negative of the work carried out on the particle by the force field as the particle moves from R to P. For a one-dimensional problem described by a variable x, the definition of the potential energy can be written

(I.1)

As a first example of how the potential energy is defined we consider the harmonic oscillator illustrated in Fig. I.2(a). The harmonic oscillator consists of a body of mass m moving under the influence of a linear restoring force

(I.2)

where x denotes the distance of the body from its equilibrium position. The constant k, which occurs in eq. (I.2), is called the force constant. The restoring force is proportional to the displacement of the body and points in the direction opposite to the displacement. If the body is displaced to the right, for instance, the restoring force points to the left. It is natural to take the reference position R in the definition of the potential energy of the oscillator to be the equilibrium position for which x = 0. The definition of the potential energy (I.1) then becomes

(I.3)

Here x’ is used within the integration in place of x to distinguish the variable of integration from the limit of integration.

If we were to pull the mass shown in Fig. I.2(a) from its equilibrium position and release it, the mass would oscillate with a frequency independent of the initial displacement. The angular frequency of the oscillator is related to the force constant of the oscillator and the mass of the particle by the equation

or

Substituting this expression for k into eq. (I.3), we obtain the following expression for the potential energy of the oscillator:

(1.4)

The oscillator potential is illustrated in Fig. I.2(b). The harmonic oscillator provides a useful model for a number of important problems in physics. It may be used, for instance, to describe the vibration of the atoms in a crystal about their equilibrium positions.

As a further example of potential energy, we consider the potential energy of a particle with electric charge q moving under the influence of a charge Q. According to Coulomb’s law the electromagnetic force between the two charges is equal to

where r is the distance between the two charges and ε0 is the permittivity of free space. The reference point for the potential energy for this problem can be conveniently chosen to be at infinity where r = ∞ and the force is equal to zero. Using eq. (I.1), the potential energy of the particle with charge q at a distance r from the charge Q can be written

Evaluating this integral, we find that the potential energy of the particle is

An application of this last formula will arise when we consider the motion of electrons in an atom. For an electron with charge −e moving in the field of an atomic nucleus having Z protons and hence a nuclear charge of eZ, the formula for the potential energy becomes

(I.5)

The energy of a body is defined to be the sum of its kinetic and potential energies

For an object moving under the influence of a conservative force, the energy is a constant of the motion.

I.1.2 Elementary Properties of Waves

We consider now some of the elementary properties of waves. Various kinds of waves arise in classical physics, and we shall encounter other examples of wave motion when we apply the new quantum theory to microscopic systems.