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## Forced Oscillations and Resonance

Access provided by: anon Sign Out. Accurate calculation of high harmonics generated by interactions between very intense laser fields and electron plasmas Abstract: Summary form only given. In a recent paper we proved that the analytical expression of the intensity of the relativistic Thomson scattered field for a system composed of an electron interacting with a plane electromagnetic field can be written as a periodic function of only one variable, that is the phase of the incident field. Arguably any interpretation of the correspondence principle faces the following four challenges: First, determining which or which combination of the various analogies or relations between classical and quantum mechanics Bohr intended to designate by the correspondence principle; second, determining the scope of the correspondence principle i.

One of the most influential discussions of Bohr's correspondence principle appears in Max Jammer's book The Conceptual Development of Quantum Mechanics. Jammer takes the correspondence principle to be a relation between the kinematics of the electron and the properties of the emitted radiation. Like many interpreters of the correspondence principle, he focuses primarily on the frequency relation. He notes that Bohr also found an asymptotic correspondence between the intensities of the spectral lines and the amplitudes in the classical harmonic components, as well as a correspondence between the polarization of the emitted radiation and the character of the classical motion.

Although Jammer notes these other correspondences, he seems to interpret the frequency correspondence as the primary correspondence principle. Regarding the status of this correspondence as a principle, Jammer writes,. Jammer is rather dismissive of Bohr's claim that the correspondence principle should be thought of as a law of quantum theory.

On Jammer's view, Bohr's claim that the correspondence principle is a law is simply an attempt to cover up the inconsistent foundations of the old quantum theory. It is perhaps for this reason that Jagdish Mehra and Helmut Rechenberg , in their comprehensive history of the development of quantum theory, make no concrete commitment as to which of the several correspondence relations discussed by Bohr should be designated as the correspondence principle.

Like Jammer they struggle to understand why Bohr thought the correspondence principle applied to small quantum numbers and why it should be considered a law of quantum theory. Although they offer one of the more technically detailed discussions of the correspondence principle, they do not discuss either Bohr's claim that it should be considered a law nor Bohr's claim that the correspondence principle is formalized into the new quantum mechanics. An alternative interpretation of Bohr's correspondence principle has been defended by Olivier Darrigol in his excellent and highly readable history of quantum theory From c -Numbers to q -Numbers.

Strictly speaking this intensity correspondence is exact only in the limit of large quantum numbers, and cannot be extended to small quantum numbers. The one exception to this is when the classical amplitude is zero, the quantum transition probability will also be zero a special case of what was earlier called Bohr's selection rule , which does hold for all quantum numbers. Unlike some interpreters of the correspondence principle Darrigol does believe that there is a coherent and rational account of Bohr's correspondence principle in terms of the intensity correspondence.

Despite this continuity, Darrigol argues that there was a fundamental, though implicit, shift in Bohr's understanding of the correspondence principle:. Hans Radder has also argued that Bohr's understanding of the correspondence principle evolved over his career. Radder identifies three different phases. Radder's chief concern is relating the various incarnations of Bohr's correspondence principle to the more general correspondence and heuristics arguments in the philosophy of science literature see Section 7 below.

## Solid-state harmonics beyond the atomic limit

Batterman rightly notes that the correspondence principle is not the claim that quantum mechanics must contain classical mechanics as a limiting case. He cogently argues that the asymptotic agreement of classical and quantum frequencies the frequency interpretation is not the correspondence principle, but rather something that is justified and explained by the correspondence principle.

Similarly, Scott Tanona rejects the view that Bohr's correspondence principle is about an asymptotic agreement of quantum and classical theories. He argues instead that Bohr's correspondence principle should be understood primarily as a connection between spectra radiation and orbital motion. More recently, Alisa Bokulich has argued that Bohr's correspondence principle should be understood in terms of what she calls Bohr's selection rule. Thus, the correspondence principle, on this reading of Bohr, is defined as the statement that each allowed quantum transition between stationary states corresponds to one harmonic component of the classical motion.

She argues that the asymptotic agreements of frequencies and intensities emphasized by Jammer and Darrigol respectively are applications or consequences of the correspondence principle, but not the correspondence principle itself. Moreover, she argues that the selection rule interpretation of Bohr's correspondence principle best makes sense of Bohr's claims that the correspondence principle applies to small n , that it is a law of quantum theory, and that it is formalized and preserved in matrix mechanics.

One typically finds a very different understanding of Bohr's correspondence principle in the current physics literature.

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Even as early as Max Born's classic text on the new quantum theory, first published in , one finds the correspondence principle defined as follows:. Liboff goes on to note that even this restricted form of the correspondence principle is violated for systems such as a particle in a box and a rigid rotor. Instead, Bohr considered that for large quantum numbers the quantum result was very nearly the classical result. The correspondence principle on this reading is understood broadly as the attempt to get a mathematically precise formulation of the classical limit.

Much of the contemporary physics community's interest in Bohr's correspondence principle variously understood is in prima facie violations of this principle by particular classes of physical systems. Is It Complete? Their concern is whether quantum mechanics can account for the behavior of chaotic classical systems.

While the contemporary physics literature's understanding of the correspondence principle is largely in line with Max Born's construal of this principle given above , it was not what Bohr himself meant by the correspondence principle. Although Bohr agreed that quantum mechanics ought to be able to recover the empirically confirmed predictions of classical mechanics, he explicitly rejected this reading of the correspondence principle.

The requirement that the quantum theory should go over to the classical description for low modes of frequency, is not at all a principle.

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The generalized correspondence principle here is seen both as a constraint on the development of new theories and as an account of how successor theories are related to their predecessors. Regarding the specific case of the quantum-classical relation, Post argues that the general correspondence principle fails:. Although Post's generalized correspondence principle bears some similarity to the physicist's conception of Bohr's correspondence principle, it is quite different from Bohr's own understanding.

Indeed, according to Post, the generalized correspondence principle does not even hold for the specific theory pair that was Bohr's subject. In contrast to Post, Radder has claimed that the generalized correspondence principle does apply to the case of the quantum-classical relation p. Background and Scientific Context 2. The Correspondence Principle Defined 3. Bohr's Writings on the Correspondence Principle — 4.

Early Responses 4. Interpretations in the History and Philosophy Literature 6. Interpretations in the Current Physics Literature 7. Background and Scientific Context Niels Bohr was a Danish physicist who lived from until ; he was born and died in Copenhagen. On one hand such a conception seems to offer the only simple possibility of accounting for the phenomena of photoelectric action, if we adhere to an unrestricted application of the notions of conservation of energy and momentum.

### 1. Introduction

On the other hand, it does not appear reconcilable with the phenomena of interference of light, which constitute our only means of analysing radiation in its harmonic constituents and determining the frequency and state of polarisation of each of these constituents. Bohr a unpublished; BCW 3, pp. The Correspondence Principle Defined Among current Bohr scholars there is a consensus that Bohr did not intend his correspondence principle to designate some sort of general requirement that quantum mechanics recover the predictions of classical mechanics in the classical limit, despite the prevalence of this interpretation in the physics literature see Section 6.

According to the selection rule interpretation, the correspondence principle is Bohr's insight that each allowed transition between stationary states corresponds to one harmonic component of the classical motion. Based on Fig. He writes, In Q. L [Bohr ] this designation has not yet been used, but the substance of the principle is referred to there as a formal analogy between the quantum theory and the classical theory.

Such expressions might cause misunderstanding, since in fact—as we shall see later on—this Correspondence Principle must be regarded purely as a law of the quantum theory, which can in no way diminish the contrast between the postulates and electrodynamic theory. Bohr [] , fn. This correspondence is of such a nature, that the present theory of spectra is in a certain sense to be regarded as a rational generalization of the ordinary theory of radiation. Bohr , pp. This is equivalent to the statement that, when the quantum numbers are large, the relative probability of a particular transition is connected in a simple manner with the amplitude of the corresponding harmonic component in the motion.

This peculiar relation suggests a general law for the occurrence of transitions between stationary states.

Thus we shall assume that even when the quantum numbers are small the possibility of transition between two stationary states is connected with the presence of a certain harmonic component in the motion of the system. He writes, The above view, which may be termed the correspondence principle … has offered an immediate interpretation of the apparent capriciousness, involved in the application of the principle of combination of spectral lines, which consists in the circumstance, that only a small part of the spectral lines, which might be anticipated from an unrestricted application of this [Rydberg-Ritz combination] principle, are actually observed in the experiments.

Bohr b unpublished; BCW 4, p. To understand why it is a law of quantum theory as opposed to a law of classical electrodynamics it is helpful to consider Bohr's following remarks: [T]he occurrence of radiative transitions is conditioned by the presence of the corresponding vibrations in the motion of the atom. As to our right to regard the asymptotic relation obtained as the intimation of a general law of the quantum theory for the occurrence of radiation, as it is assumed to be in the Correspondence Principle mentioned above, let it be once more recalled that in the limiting region of large quantum numbers there is no wise a question of a gradual diminution of the difference between the description by the quantum theory of the phenomena of radiation and the ideas of classical electrodynamics, but only an asymptotic agreement of the statistical results.

Bohr [] , p. Bohr unpublished lecture; BCW4, p. Not only do the frequencies of the corresponding harmonic components agree asymptotically with the values obtained from the frequency condition in the limit where the energies of the stationary states converge, but also the amplitudes of the mechanical oscillatory components give in this limit an asymptotic measure for the probabilities of the transition processes on which the intensities of the observable spectral lines depend.

Bohr then turns to a discussion of Heisenberg's matrix mechanics paper, arguing that It [the new matrix mechanics] operates with manifolds of quantities, which replace the harmonic oscillating components of the motion and symbolise the possibilities of transitions between stationary states in conformity with the correspondence principle… In brief, the whole apparatus of the quantum mechanics can be regarded as a precise formulation of the tendencies embodied in the correspondence principle.

Bohr , p.

In his Como lecture, for example, he writes, The aim of regarding the quantum theory as a rational generalisation of the classical theories led to the formulation of the so-called correspondence principle. The utilisation of this principle for the interpretation of spectroscopic results was based on a symbolical application of classical electrodynamics, in which the individual transition processes were each associated with a harmonic in the motion of the atomic particles to be expected according to ordinary mechanics. Early Responses Early responders to Bohr's correspondence principle can be roughly divided into three categories: those who misunderstood the principle e.

Sommereld [] , p. In a letter written to Bohr in November of Sommerfeld writes, In the appendices of my book, you can see that I have taken some pains to show the value of your correspondence principle better than in the 1.

We see this, for example, in Sommerfeld's article on Bohr's atomic models, where he writes, The magic of the correspondence principle has proved itself generally through the selection rules of the quantum numbers, in the series and band spectra… Nonetheless I cannot view it as ultimately satisfying on account of its mixing of quantum-theoretical and classical viewpoints.

Sommerfeld , p. One was an effort to bring abstract order to the new ideas by looking for a key to translate classical mechanics and electrodynamics into quantum language which would form a logical generalization of these. Do you still cling to your … application of the Correspondence Principle in this case? Pauli to Bohr, December 12th, ; quoted in Serwer , p.

It would be much more satisfying if we could understand directly on the grounds of a more general quantum mechanics one that deviates from classical mechanics.

## Forced Oscillations and Resonance – College Physics

Pauli to Bohr December 31st, ; quoted in Heilbron , p. This is especially clear in a letter Heisenberg wrote to Pauli on September 30 th , Together with Bohr I have again examined the problem carefully, and we arrived at the conclusion that it is not—as Sommerfeld says—that the sum-rules cannot be understood with the help of the correspondence principle; on the contrary they are a necessary consequence of the correspondence principle… We are very happy about this interpretation for now the attacks against the correspondence principle are completely refuted… [S]ince recently the correspondence principle has been blamed so much, it would be good to publish it [your results confirming the correspondence principle] ad majorem correspondentiae principii gloriam [for the greater glory of the correspondence principle].

Heisenberg to Pauli, September 30 th , ; quoted in Mehra and Rechenberg pp. This visit was crucial to Heisenberg. Under pressure from Pauli, he began to change his views on many issues…[including] the Correspondence Principle. Serwer , p. In general, the use of peak amplitude is simple and unambiguous only for symmetric periodic waves, like a sine wave , a square wave , or a triangle wave. For an asymmetric wave periodic pulses in one direction, for example , the peak amplitude becomes ambiguous.

This is because the value is different depending on whether the maximum positive signal is measured relative to the mean, the maximum negative signal is measured relative to the mean, or the maximum positive signal is measured relative to the maximum negative signal the peak-to-peak amplitude and then divided by two.

Strictly speaking, this is no longer amplitude since there is the possibility that a constant DC component is included in the measurement. In telecommunication , pulse amplitude is the magnitude of a pulse parameter, such as the voltage level, current level, field intensity , or power level. Pulse amplitude is measured with respect to a specified reference and therefore should be modified by qualifiers, such as average , instantaneous , peak , or root-mean-square.

Pulse amplitude also applies to the amplitude of frequency - and phase -modulated waveform envelopes. The units of the amplitude depend on the type of wave, but are always in the same units as the oscillating variable. A more general representation of the wave equation is more complex, but the role of amplitude remains analogous to this simple case. For waves on a string , or in medium such as water , the amplitude is a displacement. The amplitude of sound waves and audio signals which relates to the volume conventionally refers to the amplitude of the air pressure in the wave, but sometimes the amplitude of the displacement movements of the air or the diaphragm of a speaker is described.

Loudness is related to amplitude and intensity and is one of the most salient qualities of a sound, although in general sounds can be recognized independently of amplitude. The square of the amplitude is proportional to the intensity of the wave. For electromagnetic radiation , the amplitude of a photon corresponds to the changes in the electric field of the wave.

However, radio signals may be carried by electromagnetic radiation; the intensity of the radiation amplitude modulation or the frequency of the radiation frequency modulation is oscillated and then the individual oscillations are varied modulated to produce the signal. A steady state amplitude remains constant during time, thus is represented by a scalar. Otherwise, the amplitude is transient and must be represented as either a continuous function or a discrete vector. For audio, transient amplitude envelopes model signals better because many common sounds have a transient loudness attack, decay, sustain, and release.

With waveforms containing many overtones, complex transient timbres can be achieved by assigning each overtone to its own distinct transient amplitude envelope. Unfortunately, this has the effect of modulating the loudness of the sound as well.