vol III chap 9 sect 3
Previous: 9.2. Correspondence and uncertainty principles in Quantum Physics.
9.3. Communication forms for describing scientific texts.¶
Information contained in scientific texts to be shared can be expressed in terms of different communication forms. In this section we deal with four communication forms related to a couple of communication strategies: a strategy for the organization of contents to be communicated in terms of diagrams or designs, and a strategy for the application of procedures to be communicated in terms of dialogues or discourses. See Table 9.2 for a description of these communication forms.
Next, we analyze how the four previous communication forms can be used to describe different aspects of the Physics Nobel Lectures concerning matter waves and electron diffraction as well as the correspondence and uncertainty principles. Appropriate references of those Nobel Lectures are given in MLA format.
(1). Flux DIAGRAM for obtaining de Broglie equation \(𝑝= ℎ/𝜆\) (as described in Section 9.1 and explained in his Nobel Lecture).¶
(2). Experimental DESIGNS explained by Davisson and by Thomson in their Nobel Lectures.¶
Diffraction of electrons by crystals was independently observed by Davisson and Germer and by Thomson using quite different experimental settings. Both experiments demonstrated the physical existence of electron waves and therefore that electrons satisfy de Broglie’s equation \(p = h/λ\). They obtained equivalent experimental values: Davisson and Germer obtained \(λ = 1.66 Å\) for the electron wavelength, and Thomson obtained \(λ = 1.65 Å\).
In Davisson-Germer experiment a gun produced an electron beam that was directed perpendicular onto the surface of a crystal that can be rotated about the beam axis. Afterwards, scattered waves were registered in a collector galvanometer (Figure 9.3a). Strong reflections were observed at an angle \(θ = 50°\) when the accelerated voltage was 54 volts (Figure 9.3b). At this voltage the observed peak in the pattern of dispersed electrons was a qualitative demonstration of the validity of de Broglie´s hypothesis \(p = h/λ\).
Source: based on Figures 1 and 2 in Davisson’s Nobel Lecture. MLA style: Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Mon. 20 Feb 2023. https://www.nobelprize.org/prizes/physics/1937/davisson/lecture/
Figure 9.3. Davisson- Germer experiment: (a) experimental arrangement, (b) scattered intensity as a function of the angle \(θ\) for two different values of the voltage V.
Thomson experiments were different in two aspects:
• The accelerated energy for the electrons was much higher (in between 10,000 - 40,000 eV instead of in between 30-600 eV) making possible that the electrons could pass through thin foils of solid materials and be observed by transmission rather than by reflection.
• The diffracting material was not a single crystal, but a microcrystalline aggregate composed by randomly oriented individual crystals. This fact resulted in diffraction patterns of concentric circles that were received in a photographic plate.
It is interesting to note that in 1897 Joseph John Thomson (1856-1940), the father, discovered the electron and received the 1906 Physics Nobel Prize for showing that the electron is a particle, and that in 1927 George Paget Thomson (1892-1975), the son, discovered the diffraction of the electron and received the 1937 Physics Nobel Prize for showing that the electron is a wave.
WORK: “The idea that electricity is transmitted by a tiny particle related to the atom was first forwarded in the 1830s. In the 1890s, J.J. Thomson managed to estimate its magnitude by performing experiments with charged particles in gases. In 1897 he showed that cathode rays (radiation emitted when a voltage is applied between two metal plates inside a glass tube filled with low-pressure gas) consist of particles— electrons—that conduct electricity. Thomson also concluded that electrons are part of atoms.”
MLA style: J.J. Thomson – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Wed. 5 Jun 2024. https://www.nobelprize.org/prizes/physics/1906/thomson/facts/
NOBEL LECTURE: Carriers of Negative Electricity by Thomson.
- Introductory
- Electric deflection of the rays
- Determination of e/m
- Corpuscles very widely distributed
- Magnitude of the electric charge carried by the corpuscle
(3). DIALOGUES between Bohr and Heisenberg about correspondence and uncertainty principles.¶
This is a hypothetical dialogue between the two laureates that has been prepared by selecting and then quoting some paragraphs from Bohr´s Lecture presented in 1922 and from Heisenberg´s Lecture presented in 1932 (Figure 9.4)
Figure 9.4. Werner Heisenberg and Niels Bohr in 1934.
- Bohr´s Lecture The Structure of the Atom contains the section called The correspondence principle.
…..
“According to the correspondence principle, it is assumed that every transition process between two stationary states can be co-ordinated with a corresponding harmonic vibration component in such a way that the probability of the occurrence of the transition is dependent on the amplitude of the vibration.”
…….
“ the case of the Stark effect, where, on the other hand, the classical theory was completely at a loss, the quantum theory explanation could be so extended with the help of the correspondence principle as to account for the polarization of the different components into which the lines are split, and also for the characteristic intensity distribution exhibited by the components.”
……
….. “Indeed, it was possible, with the aid of the correspondence principle, to account completely for the characteristic rules which govern the seemingly capricious occurrence of combination lines, and it is not too much to say that the quantum theory has not only provided a simple interpretation of the combination principle, but has further contributed materially to the clearing up of the mystery that has long rested over the application of this principle.”
“The same viewpoints have also proved fruitful in the investigation of the so-called band spectra. These do not originate, as do series spectra, from individual atoms, but from molecules; and the fact that these spectra are so rich in lines is due to the complexity of the motion entailed by the vibrations of the atomic nuclei relative to each other and the rotations of the molecule as a whole.”
……
“It is true we no longer think that the rotation is reflected in the spectra in the way claimed by classical electrodynamics, but rather that the line components are due to transitions between stationary states which differ as regards rotational motion. That the phenomenon retains its essential feature, however, is a typical consequence of the correspondence principle.”
- Heisenber´s Lecture The Development of Quantum Mechanics contains comments on the correspondence principle as well as explanations concerning the uncertainty principle.
“Quantum mechanics, on which I am to speak here, arose, in its formal content, from the endeavour to expand Bohr’s principle of correspondence to a complete mathematical scheme by refining his assertions. The physically new viewpoints that distinguish quantum mechanics from classical physics were prepared by the researches of various investigators engaged in analyzing the difficulties posed in Bohr’s theory of atomic structure and in the radiation theory of light.”
…..
…. “Classical physics seemed the limiting case of visualization of a fundamentally unvisualizable microphysics, the more accurately realizable the more Planck’s constant vanishes relative to the parameters of the system. This view of classical mechanics as a limiting case of quantum mechanics also gave rise to Bohr’s principle of correspondence which, at least in qualitative terms, transferred a number of conclusions formulated in classical mechanics to quantum mechanics. In connection with the principle of correspondence there was also discussion whether the quantum-mechanical laws could in principle be of a statistical nature; the possibility became particularly apparent in Einstein’s derivation of Planck’s law of radiation.”
…..
“To each stationary state of an atom corresponds a whole complex of parameters which specify the probability of transition from this state to another. There is no direct relation between the radiation classically emitted by an orbiting electron and those parameters defining the probability of emission; nevertheless Bohr’s principle of correspondence enables a specific term of the Fourier expansion of the classical path to be assigned to each transition of the atom, and the probability for the particular transition follows qualitatively similar laws as the intensity of those Fourier components.”
…..
“It thus seemed consistent simply to adopt in quantum mechanics the equations of motion of classical physics, regarding them as a relation between the matrices representing the classical variables. The Bohr-Sommerfeld quantum conditions could also be re-interpreted in a relation between the matrices, and together with the equations of motion they were sufficient to define all matrices and hence the experimentally observable properties of the atom.”
……
….. “Closer examination of the formalism shows that between the accuracy with which the location of a particle can be ascertained and the accuracy with which its momentum can simultaneously be known, there is a relation according to which the product of the probable errors in the measurement of the location and momentum is invariably at least as large as Planck’s constant divided by 4π. In a very general form, therefore, we should have
where p and q are canonically conjugated variables. These uncertainty relations for the results of the measurement of classical variables form the necessary conditions for enabling the result of a measurement to be expressed in the formalism of the quantum theory. Bohr has shown in a series of examples how the perturbation necessarily associated with each observation indeed ensures that one cannot go below the limit set by the uncertainty relations. He contends that in the final analysis an uncertainty introduced by the concept of measurement itself is responsible for part of that perturbation remaining fundamentally unknown.”
……
….. “The behaviour of the observer as well as his measuring apparatus must therefore be discussed according to the laws of classical physics, as otherwise there is no further physical problem whatsoever. Within the measuring apparatus, as emphasized by Bohr, all events in the sense of the classical theory will therefore be regarded as determined, this also being a necessary condition before one can, from a result of measurements, unequivocally conclude what has happened.”
….
….. “A visual description for the atomic events is possible only within certain limits of accuracy - but within these limits the laws of classical physics also still apply. Owing to these limits of accuracy as defined by the uncertainty relations, moreover, a visual picture of the atom free from ambiguity has not been determined. On the contrary the corpuscular and the wave concepts are equally serviceable as a basis for visual interpretation.”
…..
“For the clearest analysis of the conceptual principles of quantum mechanics we are indebted to Bohr who, in particular, applied the concept of complementarity to interpret the validity of the quantum-mechanical laws. The uncertainty relations alone afford an instance of how in quantum mechanics the exact knowledge of one variable can exclude the exact knowledge of another. This complementary relationship between different aspects of one and the same physical process is indeed characteristic for the whole structure of quantum mechanics. I had just mentioned that, for example, the determination of energetic relations excludes the detailed description of spacetime processes.”
(4). DISCOURSE presented by de Broglie when he received the 1929 Physics Nobel Prize (excerpts from his Nobel Lecture The Wave Nature of the Electron).¶
…..
“For a long time physicists had been wondering whether light was composed of small, rapidly moving corpuscles. This idea was put forward by the philosophers of antiquity and upheld by Newton in the 18th century. After Thomas Young’s discovery of interference phenomena and following the admirable work of Augustin Fresnel, the hypothesis of a granular structure of light was entirely abandoned and the wave theory unanimously adopted.”…... “Experiment also yielded decisive proof in favour of an atomic constitution of electricity; the concept of the electricity corpuscle owes its appearance to Sir J. J. Thomson and you will all be familiar with H. A. Lorentz’s use of it in his theory of electrons.”
“Some thirty years ago, physics was hence divided into two: firstly the physics of matter based on the concept of corpuscles and atoms which were supposed to obey Newton’s classical laws of mechanics, and secondly radiation physics based on the concept of wave propagation in a hypothetical continuous medium, i.e. the light ether or electromagnetic ether.” … “By an intuition of his genius Planck realized the way of avoiding it: instead of assuming, in common with the classical wave theory, that a light source emits its radiation continuously, it had to be assumed on the contrary that it emits equal and finite quantities, quanta. The energy of each quantum has, moreover, a value proportional to the frequency v of the radiation. It is equal to \(hv\), \(h\) being a universal constant since referred to as Planck’s constant.”
“The success of Planck’s ideas entailed serious consequences. If light is emitted as quanta, ought it not, once emitted, to have a granular structure? The existence of radiation quanta thus implies the corpuscular concept of light. “…. “It must therefore be assumed that traditional dynamics, even as modified by Einstein’s theory of relativity, is incapable of accounting for motion on a very small scale.”
“The existence of a granular structure of light and of other radiations was confirmed by the discovery of the photoelectric effect.” …. “Nevertheless, it was still necessary to adopt the wave theory to account for interference and diffraction phenomena and no way whatsoever of reconciling the wave theory with the existence of light corpuscles could be visualized.”
“As stated, Planck’s investigations cast doubts on the validity of very small scale mechanics.” …. “On the other hand Planck was led to assume that only certain preferred motions, quantized motions, are possible or at least stable, since energy can only assume values forming a discontinuous sequence.” ….
…..
“The necessity of assuming for light two contradictory theories-that of waves and that of corpuscles - and the inability to understand why, among the infinity of motions which an electron ought to be able to have in the atom according to classical concepts, only certain ones were possible: such were the enigmas confronting physicists at the time I resumed my studies of theoretical physics.”
“When I started to ponder these difficulties two things struck me in the main. Firstly the light-quantum theory cannot be regarded as satisfactory since it defines the energy of a light corpuscle by the relation \(W = hv\) which contains a frequency \(v\). Now a purely corpuscular theory does not contain any element permitting the definition of a frequency. This reason alone renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity.”
“On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations. That suggested the idea to me that electrons themselves could not be represented as simple corpuscles either, but that a periodicity had also to be assigned to them too.”
“I thus arrived at the following overall concept which guided my studies: for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. In other words the existence of corpuscles accompanied by waves has to be assumed in all cases. However, since corpuscles and waves cannot be independent because, according to Bohr’s expression, they constitute two complementary forces of reality, it must be possible to establish a certain parallelism between the motion of a corpuscle and the propagation of the associated wave. The first objective to achieve had, therefore, to be to establish this correspondence.”
“With that in view I started by considering the simplest case: that of an isolated corpuscle, i.e. a corpuscle free from all outside influence. We wish to associate a wave with it. Let us consider first of all a reference system (\(x_0y_0z_0\)) in which the corpuscle is immobile: this is the "intrinsic" system of the corpuscle in the sense of the relativity theory. In this system the wave will be stationary since the corpuscle is immobile: its phase will be the same at every point; it will be represented by an expression of the form \(sin[2πν_0(t_0 - τ_0)]\); \(t_0\) being the intrinsic time of the corpuscle and \(τ_0\) a constant.”
“In accordance with the principle of inertia in every Galilean system, the corpuscle will have a rectilinear and uniform motion. Let us consider such a Galilean system and let \(v = βc\) be the velocity of the corpuscle in this system; we shall not restrict generality by taking the direction of the motion as the x-axis. In compliance with Lorentz’ transformation, the time t used by an observer of this new system will be associated with the intrinsic time \(t_0\) by the relation:”
\(t_0=\frac{t-\frac{βx}{c}}{√(1-β^2 )}\)
“and hence for this observer the phase of the wave will be given by”
\(sin[(\frac{ν_0}{√(1-β^2 )})(t-\frac{βx}{c}- τ_0)]\)
“For him the wave will thus have a frequency:”
\(ν = \frac{ν_0}{√(1-β^2)}\)
“and will propagate in the direction of the x-axis at the phase velocity:”
\(V=\frac{c}{β}=\frac{c^2}{v}\)
“By the elimination of β between the two preceding formulae the following relation can readily be derived which defines the refractive index of the vacuum n for the waves considered:”
\(n= \frac{V}{v}=\frac{1}{1-(\frac{ν_0}{ν})^2}\)
“A «group velocity» corresponds to this «law of dispersion». You will be aware that the group velocity is the velocity of the resultant amplitude of a group of waves of very close frequencies. Lord Rayleigh showed that this velocity \(U\) satisfies equation:”
\(\frac{1}{U}= \frac{δ(nν)}{δν}\)
“Here \(U = v\), that is to say that the group velocity of the waves in the system xyzt is equal to the velocity of the corpuscle in this system.” ..…
“The corpuscle is thus defined in the system \(xyzt\) by the frequency \(ν\) and the phase velocity \(V\) of its associated wave. To establish the parallelism of which we have spoken, we must seek to link these parameters to the mechanical parameters, energy and quantity of motion. Since the proportionality between energy and frequency is one of the most characteristic relations of the quantum theory, and since, moreover, the frequency and the energy transform in the same way when the Galilean reference system is changed, we may simply write”
\(energy = h \times frequency\), or \(W = hν\)
“where \(h\) is Planck’s constant. This relation must apply in all Galilean systems and in the intrinsic system of the corpuscle where the energy of the corpuscle, according to Einstein, reduces to its internal energy \(m_0c^2\) (\(m_0\) being the rest mass) we have”
\(hν_0= m_0 c^2\)
“This relation defines the frequency \(ν_0\) as a function of the rest mass \(m_0\), or inversely.”
“The quantity of movement is a vector \(p\) equal to”
\(\frac{m_0v}{√(1-β^2)}\)
and we have
\((p)=\frac{m_0v}{√(1-β^2)}= \frac{Wv}{c^2} = \frac{hν}{V}= \frac{h}{λ}\)
"The quantity \(λ\) is the distance between two consecutive peaks of the wave, i.e. the "wavelength". Hence:”
\(λ= \frac{h}{p}\)
“The whole of the foregoing relates to the very simple case where there is no field of force at all acting on the corpuscles. I shall show you very briefly how to generalize the theory in the case of a corpuscle moving in a constant field of force deriving from a potential function \(F(xyz)\).“……
…..
“Here again it is demonstrated that the group velocity of the waves is equal to the velocity of the corpuscle. The parallelism thus established between the corpuscle and its wave enables us to identify Fermat’s principle for the waves and the principle of least action for the corpuscles (constant fields).” …..
……
“These concepts lead to an interpretation of the conditions of stability introduced by the quantum theory. Actually, if we consider a closed trajectory C in a constant field, it is very natural to assume that the phase of the associated wave must be a uniform function along this trajectory. Hence we may write:”
\(∫_C \frac{dl}{λ} = ∫_C \frac{p}{h}dl=integer\)
“This is precisely Planck’s condition of stability for periodic atomic motions. The conditions of quantum stability thus emerge as analogous to resonance phenomena and the appearance of integers becomes as natural here as in the theory of vibrating cords and plates.”
“The general formulae which establish the parallelism between waves and corpuscles may be applied to corpuscles of light on the assumption that here the rest mass \(m_0\) is infinitely small. Actually, if for a given value of the energy \(W\), \(m_0\) is made to tend towards zero, \(v\) and \(V\) are both found to tend towards \(c\) and at the limit the two fundamental formulae are obtained on which Einstein had based his light-quantum theory”
\(W=hν\) \(p=\frac{hν}{c}\)
“Such are the main ideas which I developed in my initial studies. They showed clearly that it was possible to establish a correspondence between waves and corpuscles such that the laws of mechanics correspond to the laws of geometrical optics. In the wave theory, however, as you will know, geometrical optics is only an approximation: this approximation has its limits of validity and particularly when interference and diffraction phenomena are involved, it is quite inadequate. This prompted the thought that classical mechanics is also only an approximation relative to a vaster wave mechanics. I stated as much almost at the outset of my studies, i.e. "A new mechanics must be developed which is to classical mechanics what wave optics is to geometrical optics". This new mechanics has since been developed, thanks mainly to the fine work done by Schrödinger. It is based on wave propagation equations and strictly defines the evolution in time of the wave associated with a corpuscle.” ….
…..
….. “The two mechanics, wave and quantum, are equivalent from the mathematical point of view. “
“We shall content ourselves here by considering the general significance of the results obtained. To sum up the meaning of wave mechanics it can be stated that: "A wave must be associated with each corpuscle and only the study of the wave’s propagation will yield information to us on the successive positions of the corpuscle in space".” …. “I must restrict myself to the assertion that when an observation is carried out enabling the localization of the corpuscle, the observer is invariably induced to assign to the corpuscle a position in the interior of the wave and the probability of it being at a particular point M of the wave is proportional to the square of the amplitude, that is to say the intensity at M.”
“This may be expressed in the following manner. If we consider a cloud of corpuscles associated with the same wave, the intensity of the wave at each point is proportional to the cloud density at that point (i.e. to the number of corpuscles per unit volume around that point). This hypothesis is necessary to explain how, in the case of light interferences, the light energy is concentrated at the points where the wave intensity is maximum: if in fact it is assumed that the light energy is carried by light corpuscles, photons, then the photon density in the wave must be proportional to the intensity.”
“This rule in itself will enable us to understand how it was possible to verify the wave theory of the electron by experiment.” …..
“Since the wavelength of the electron waves is of the order of that of X-rays, it must be expected that crystals can cause diffraction of these waves completely analogous to the Laue phenomenon.” …..
……
“For X-rays the phenomenon of diffraction by crystals was a natural consequence of the idea that X-rays are waves analogous to light and differ from it only by having a smaller wavelength. For electrons nothing similar could be foreseen as long as the electron was regarded as a simple small corpuscle. However, if the electron is assumed to be associated with a wave and the density of an electron cloud is measured by the intensity of the associated wave, then a phenomenon analogous to the Laue phenomenon ought to be expected for electrons. …….”Since, according to our general principle, the intensity of the diffracted wave is a measure of the density of the cloud of diffracted electrons, we must expect to find a great many diffracted electrons in the directions of the maxima. If the phenomenon actually exists it should thus provide decisive experimental proof in favour of the existence of a wave associated with the electron with wavelength \(h/mv\), and so the fundamental idea of wave mechanics will rest on firm experimental foundations. “
“Now, experiment which is the final judge of theories, has shown that the phenomenon of electron diffraction by crystals actually exists and that it obeys exactly and quantitatively the laws of wave mechanics.” …..
…..
“Thus to describe the properties of matter as well as those of light, waves and corpuscles have to be referred to at one and the same time. The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted. It has thus been possible to predict a whole group of phenomena without their actually having been discovered. And it is on this concept of the duality of waves and corpuscles in Nature, expressed in a more or less abstract form, that the whole recent development of theoretical physics has been founded and that all future development of this science will apparently have to be founded.”
REFERENCES¶
FRENCH, A.P. TAYLOR, E.F. Introduction to Quantum Physics. The MIT Introductory Physics Series. New York, W.W. Norton, (1978)
10.1. The Michelson interferometer and the Michelson-Morley experiment.