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The Lorentz Oscillator Model

The Lorentz Oscillator model offers the simplest picture of atom--field interactions. It is purely classical; however, this model is an elegant tool for visualizing atom--field interactions. The Lorentz Oscillator model also bears a number of basic insights into this problem. For a basic discussion of this model see [1].

Lorentz was a late nineteenth century physicist, and quantum mechanics had not yet been discovered. However, he did understand the results of classical mechanics and electromagnetic theory. Therefore, it is not surprising that he described the problem of atom--field interactions in these terms. Lorentz thought of an atom as a mass ( the nucleus ) connected to another smaller mass ( the electron ) by a spring, Figure gif. The spring would be set into motion by an electric field interacting with the charge of the electron. The field would either repel or attract the electron which would result in either compressing or stretching the spring.

  
Figure: Lorentz Oscillator Model

Lorentz was not positing the existence of a physical spring connecting the electron to an atom; however, he did postulate that the force binding the two could be described by Hooke's Law, i.e., , where x is the displacement from equilibrium. Any student of quantum mechanics should recall that this assumption is the dipole approximation of electron--atom interaction. Therefore, this perspective is quite valid.

If Lorentz's system comes into contact with an electric field, then the electron will simply be displaced from equilibrium. The oscillating electric field of the electromagnetic wave will set the electron into harmonic motion. The effect of the magnetic field can be omitted because it is miniscule compared to the electric field. Lorentz also considered the possibility of damping in his model.

The result of Lorentz's work was the Abraham--Lorentz Equation:

 

This equation has one of the most well--known forms of mathematical physics. It has analogs in the LRC circuit, the damped Spring--Mass system, and countless other systems. Its solution is quite well--known, and Lorentz was aware of it. Lorentz understood the origin of all the terms in eq:LE except for the damping term. He really could not quantify . But this result is not surprising because the full interpretation of requires at least a semiclassical treatment of this problem. The main sources of are atomic collisions, the Doppler effect, and spontaneous emission.

The specific form of eq:LE which was most enlightening to Lorentz had an electric field, , of the form

where is a real number. This represents an electromagnetic wave with frequency and amplitude . The characteristic frequency of the electron oscillation, , is defined by the value of the spring constant k and the mass of the electron. and are constants set by the atomic system; only , the frequency of the electric field, can be varied by the experimenter.

The solution of eq:LE will take the form of a sum of a characteristic solution and a particular solution for this , . These solutions are:

where . The characteristic solution of this differential equation is exponentially damped in time; therefore, is a transient solution, i.e., it only exists as the atom first begins to interact with the field. The length of time that this transient exists depends on the magnitude of the damping coefficient. The greater the damping coefficient, the shorter the amount of time the characteristic solution will exist. The particular solution on the other hand has no decaying exponential and will be a steady--state solution. The interesting physical effects are in this later solution.

oscillates sinusoidally in time with some amplitude and phase shift relative to the electric field. This amplitude is a function of the frequency of the driving electric field. The frequency that will maximize the amplitude of this for a given set of and values is the resonant frequency, . It corresponds to the point at which the field is driving the atom's electron the most constructively---the most energy is being transferred from the field to the system at resonance.

Finding the resonant frequency is as easy as taking derivatives of the amplitude function, setting the result equal to zero, and finding the root of the resulting polynomial:

The resonant frequency is . If there is no damping, then the resonant frequency would simply be the characteristic frequency, . To make sure that this frequency does indeed maximize the amplitude, one can apply the second derivative test, and verify that

This resonant frequency should not be a surprise for it is the same for any damped harmonic oscillator with a sinusoidal driving term. The effect of damping in this type of system is usually quantified in terms of the quality factor, . A large quality factor implies that the amount of damping is small and vice versa.

  
Figure: Sample plots of amplitude curves

  
Figure: Sample plots of phase curves

Some plots of the amplitude function at various quality factors have been included in Figure gif to illustrate how the damping coefficient, , affects the resonance condition. Notice that at high damping (low quality factor) the resonance peak is barely noticeable and the resonant frequency moves off of as is predicted. The phase shift, , in Figure gif varies in a more linear fashion as the damping is increased.

  
Figure: Particular Solution of Abraham--Lorentz Equation for Q = 100

When the effects of the phase shift, amplitude, and driving frequency are factored into the particular solution, a number of interesting results emerge. At high values of Q as moves off of the resonant frequency by a small amount, the amplitude of the particular solution's oscillation drops significantly as can be observed in Figure gif. There is also a discernible phase shift in the oscillations as one moves off of resonance. If one examines a solution for the electric field at the resonant frequency and one at a higher frequency, then one will discover that the frequency of the oscillation has increased. If the driving frequency is below resonance, then the oscillation frequency is lower.

These observations are all based on a purely classical model. Yet, it shall become clear that these effects occur in quantum mechanical models as well. However, this model does not suffice to explain a large portion of the observed effects in atom--field interactions, especially when a weak field is interacting with a small number of atoms. For this situation and others, at least a partially quantum mechanical treatment is required.



next up previous contents index
Next: The Two--State Atom Up: An Examination of Atom--Field Previous: Preface



Andy Antonelli
Wed May 17 14:34:24 EDT 1995