Oscillator in an Electromagnetic Field

Abstract

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

Appendices

Appendix I

Here we obtain the relation (1.53) between the spectral density of energy of a random stationary field and the autocorrelation function of the electric field strength. By definition, the spatial density of the energy E of the uniform electromagnetic field occupying the volume V is

$$ u = \frac{\rm E}{V} = \left\langle {\frac{{{\bf{E}}^{2} +{\bf{H}}^{2} }}{8\,\pi }} \right\rangle = \frac{{\left\langle {{\bf{E}}^{2} \left( t \right)} \right\rangle }}{4\,\pi }. $$

(A.1)

The angle brackets denote averaging over the electromagnetic field state, that is, over amplitudes, phases, and polarizations of its monochromatic components. In the third equality of (A.1), we use the fact that, for an electromagnetic wave in vacuum, the strengths of the electric and magnetic fields coincide. By definition, the stationary electric field strength correlator is

$$ KE_{ik} \left( \tau \right) = \left\langle {E_{i} \left( t \right)\,E_{k} \left( {t + \tau } \right)} \right\rangle = \int\limits_{ - \infty }^{\infty } {KE_{ik} \left( \omega \right)} \,\exp \left( { - i\,\omega \,\tau } \right)\,\frac{d\omega }{2\,\pi }. $$

(A.2)

The Fourier transform of the correlator $KE_{ik} \left( \omega \right)$ is called the field spectral density tensor.

Using (A.2), (A.1) can be rewritten as

$$ u = \frac{{\left\langle {{\bf{E}}^{2} \left( t \right)} \right\rangle }}{4\,\pi } = \frac{{KE_{ii} \left( {\tau = 0} \right)}}{4\,\pi } = \frac{1}{{\left( {2\,\pi } \right)^{2} }}\int\limits_{0}^{\infty } {KE_{ii} \left( \omega \right)} \,d\omega . $$

(A.3)

Here the Einstein summation convention is implied. In going to integration over positive frequencies alone, we use the fact that the function $KE_{ii} \left( \omega \right)$ is even, which is easily shown from its definition and from the reality of the electric field strength $E_{i} \left( t \right).$

We now compare (A.3) with the determination of the spectral density of the field

$$ u = \int\limits_{0}^{\infty } {\rho \left( \omega \right)\,d\omega } , $$

(A.4)

from which it follows that

$$ \rho \left( \omega \right) = \frac{1}{{\left( {2\,\pi } \right)^{2} }}KE_{ii} \left( \omega \right), $$

(A.5)

$$ KE_{ii} \left( \omega \right) = \int\limits_{ - \infty }^{\infty } {KE_{ii} \left( \tau \right)} \,e^{i\,\omega \,\tau } \,d\tau = \int\limits_{ - \infty }^{\infty } {e^{i\,\omega \,\tau } \,\left\langle {{\bf{E}}\left( t \right)\,{\bf{E}}\left( {t + \tau } \right)} \right\rangle \,d\tau } . $$

(A.6)

Equations (A.5)–(A.6) imply (1.53).

Appendix II

Here we derive the formula (1.77) for the power of dipole radiation of the charged Morse oscillator averaged over the oscillation period after cessation of an exciting electromagnetic field pulse $(t \gg \Updelta t),$ when the oscillations can be considered to be free. We proceed from the expression for the instantaneous dipole radiation power of one-dimensional oscillations of the oscillator with energy $\varepsilon:$

$$ Q\left( {t,\,\varepsilon } \right) = \frac{{2\,\left| {\ddot{d}\left( {t,\varepsilon } \right)} \right|^{2} }}{{3\,c^{3} }}, $$

(A.7)

where $d\left( {t,\,\varepsilon } \right) = q\,x\left( {t,\,\varepsilon } \right)$ is the dipole moment of the oscillator, $k = \omega_{0} \,\sqrt {{m \mathord{\left/ {\vphantom {m {2\,D}}} \right. \kern-0pt} {2\,D}}}$ is the Morse potential parameter, D is the binding energy, and m is the oscillator mass. Let us rewrite (A.1) in dimensionless variables $\rho = k\,x$ and $\tau = \omega_{0} \,t$ and average over the dimensionless oscillation period

$$ \tilde{T}^{{\left( {Morse} \right)}} = T^{{\left( {Morse} \right)}} \omega_{0} = \frac{2\,\pi }{{\,\sqrt {1 - \tilde{\varepsilon }} }}, $$

(A.8)

where $\tilde{\varepsilon } = {\varepsilon \mathord{\left/ {\vphantom {\varepsilon D}} \right. \kern-0pt} D}$ is the normalized energy. The result is

$$ \left\langle {Q\left( {t,\,\varepsilon } \right)} \right\rangle_{T} \,=\, \frac{{4\,q^{2} \,\omega_{0}^{2} \,D}}{{\,3\,m\,c^{3} }}\,\left\langle {\ddot{\rho }_{{\tau^{2} }}^{2} } \right\rangle_{{\tilde{T}}} . $$

(A.9)

We must therefore calculate the average of the squared dimensionless second derivative $\left\langle {\ddot{\rho }_{{\tau^{2} }}^{2} } \right\rangle_{{\tilde{T}}} .$ This average can be written

$$ \begin{aligned} \left\langle {\ddot{\rho }_{{\tau^{2} }}^{2} } \right\rangle_{{\tilde{T}}} \,=\, & \,\frac{2}{{\tilde{T}}}\,\int\limits_{{\rho_{1} \left( {\tilde{\varepsilon }} \right)}}^{{\rho_{2} \left( {\tilde{\varepsilon }} \right)}} {\frac{{\ddot{\rho }^{2} }}{{\dot{\rho }}}\,d\rho \,=\, } 2\,\frac{{\sqrt {1 - \tilde{\varepsilon }} }}{2\,\pi }I_{M} \left( {\tilde{\varepsilon }} \right) \\ \,=\, & \,2\,\frac{{\sqrt {1 - \tilde{\varepsilon }} }}{2\,\pi }\,\int\limits_{{\rho_{1} \left( {\tilde{\varepsilon }} \right)}}^{{\rho_{2} \left( {\tilde{\varepsilon }} \right)}} {\frac{{\left( {\exp \left( { - 2\,\rho } \right) - \exp \left( { - \rho } \right)} \right)^{2} }}{{\sqrt {\tilde{\varepsilon } - 1 + 2\exp \left( { - \rho } \right) - \exp \left( { - 2\rho } \right)} }}\,d\rho ,} \\ \end{aligned} $$

(A.10)

where $\rho_{1} = - \ln \left( {1 + \sqrt {\tilde{\varepsilon }} } \right)$ and $\rho_{2} = - \ln \left( {1 - \sqrt {\tilde{\varepsilon }} } \right)$ are the turning points at which the oscillator velocity is zero. They correspond to zeros of the radicand in the denominator of the right-hand side of (A.10). In writing (A.10), we used the fact that the electromagnetic pulse had ceased and that the dynamics of the Morse oscillator is defined solely by its potential energy, validating the expression for acceleration of the free oscillator (1.75) and its velocity following from (1.67).

It is not difficult to calculate the integral $I_{M} \left( {\tilde{\varepsilon }} \right)$ on the right-hand side of (A.10) if we make the change of variable $\rho = - \ln \left( {1 + \sqrt {\tilde{\varepsilon }} y} \right).$ Then after elementary transformations we find

$$ I_{M} \left( {\tilde{\varepsilon }} \right) = \tilde{\varepsilon }\,\int\limits_{ - 1}^{1} {\frac{{y^{2} }}{{\sqrt {1 - y^{2} } }}dy = \frac{\pi }{2}\,\tilde{\varepsilon }} . $$

(A.11)

Substituting the right-hand side of (A.6) into (A.5) and the resulting expression into (A.4), we arrive at the formula (1.77) $\tilde{\varepsilon } = {\varepsilon \mathord{\left/ {\vphantom {\varepsilon D}} \right. \kern-0pt} D}$:

$$ \left\langle {Q\left( {t,\,\varepsilon } \right)} \right\rangle_{T} = \frac{{2\,q^{2} \,\omega_{0}^{2} \,D}}{{3\,m\,c^{3} }}\,\tilde{\varepsilon }\,\sqrt {1 - \tilde{\varepsilon }} . $$

(A.12)