Valeriy Astapenko - Interaction of Ultrashort Electromagnetic Pulses with Matter

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Valeriy Astapenko SpringerBriefs in Physics Interaction of Ultrashort Electromagnetic Pulses with Matter 2013 10.1007/978-3-642-35969-9_1 The Author(s) 2013

1. Oscillator in an Electromagnetic Field

Valeriy Astapenko 1

(1)

Moscow Institute of Physics and Technology, Institutskii per. 9, Moscow, 141700, Russia

Valeriy Astapenko

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Abstract

The model of a harmonic oscillator is widely used in the most diverse fields of physics.

The model of a harmonic oscillator is widely used in the most diverse fields of physics. This is connected first of all with the fact that, for small deviations from the equilibrium position the potential energy of the system is approximately described by a quadratic x -coordinate dependence, and a quadratic dependence of the potential energy is a characteristic feature of a harmonic oscillator. In actual fact, in the equilibrium position the force f acting on a particle is equal to zero. Since the force is defined by the first derivative of the potential energy with respect to the coordinate the linear term in the potential energy expansion in terms of the deviation from the equilibrium position is also equal to zero, and the quadratic dependence remains:

(1.1)

Given that the choice of potential energy origin is arbitrary, the potential energy of the system at the equilibrium point may be assumed to be zero, so that . Now, the total energy of a harmonic oscillator is equal to the sum of the kinetic and potential energies (for simplicity we consider a one-dimensional case):

(1.2)

where is the velocity, m is the mass, and is the eigenfrequency of the oscillator. Comparing the potential energy of the harmonic oscillator [the second summand on the right-hand side of (), we find that

(1.3)

where the double prime denotes the second derivative of the potential energy with respect to the coordinate. It is assumed that .

Thus for small deviations from the equilibrium position, any physical system can be treated approximately as a harmonic oscillator, the eigenfrequency of which is determined in the one-dimensional case by ().

The harmonic oscillator model is widely used to study the physics of oscillation processes. It can also be applied to the description of quantum transitions of electrons between stationary states under the action of an electromagnetic field on various physical systems, using the spectroscopic conformity principle.

1.1 Harmonic Oscillator in a Monochromatic Field

We begin by considering an elementary case of interaction between a charged harmonic oscillator and the electric field of a monochromatic electromagnetic wave. Hereafter we will assume that the wavelength is much longer than the oscillator dimension a :

(1.4)

The inequality () is true for a wide wavelength range if one considers the interaction of radiation with atomic particles, the characteristic size of which is defined by the Bohr radius: (it will be recalled that 1 = 108 cm). In the classical picture, the Bohr radius defines the size of the electron orbit nearest to the nucleus in a hydrogen atom.

When () is satisfied, the coordinate dependence of the electromagnetic field can be neglected, and for a monochromatic electric field strength, one can set

(1.5)

where are the amplitude, frequency, and initial phase of the electric field, respectively. A monochromatic field is characterized by the fact that its amplitude and initial phase are constants. The electric field amplitude determines the radiation intensity in vacuum averaged over the period according to the equation

(1.6)

where c is the velocity of light in free space. It will be recalled that the intensity is the quantity of radiant energy passing through a unit area per unit time, usually measured in watts per square centimeter.

For the quantitative characterization of an electric field in atomic physics, the usual reference quantity is the atomic strength equal to the strength of the atomic field at the first Bohr orbit of a hydrogen atom:

(1.7)

where is the electron mass. According to () is

(1.8)

The harmonic oscillator approximation can be used to describe the interaction between radiation and atomic particles for small enough electric field amplitudes and radiation intensities:

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