Abstract Recent researches in experimental and theoretical Physics strongly indicate that both matter and radiation possess a dual aspect exhibiting, on the one hand, the properties of undulatory phenomena extensible throughout the space-time continuum, and on the other, those of “parcel” or particle phenomena concentrated in small regions, the so-called “world lines” of particles. The wave mechanics of de Broglie and Schrödinger is an attempt to bridge the gulf, so to speak, between matter and radiation; and the question naturally arises as to whether these two apparently distinct entities may not represent the same, or at least similar, world conditions. Let (Ox, Oy,Oz) be a rectangular system of axes at rest relative to a material observer, and consider the motion of an electron relative to the observer. First, let the electron be conceived as a particle, as a region of the observer’s space bounded by a movable surface. It will be necessary for the following analysis to assign to this surface a definite form, and I shall choose the form of a Heaviside ellipsoid associated with the Lorentz electron, since this is the only form that is known to satisfy the geometrical conditions of the space-time continuum.
Abstract Recent researches in experimental and theoretical Physics strongly indicate that both matter and radiation possess a dual aspect exhibiting, on the one hand, the properties of undulatory phenomena extensible throughout the space-time continuum, and on the other, those of “parcel” or particle phenomena concentrated in small regions, the so-called “world lines” of particles. The wave mechanics of de Broglie and Schrödinger is an attempt to bridge the gulf, so to speak, between matter and radiation; and the question naturally arises as to whether these two apparently distinct entities may not represent the same, or at least similar, world conditions. Let (Ox, Oy,Oz) be a rectangular system of axes at rest relative to a material observer, and consider the motion of an electron relative to the observer. First, let the electron be conceived as a particle, as a region of the observer’s space bounded by a movable surface. It will be necessary for the following analysis to assign to this surface a definite form, and I shall choose the form of a Heaviside ellipsoid associated with the Lorentz electron, since this is the only form that is known to satisfy the geometrical conditions of the space-time continuum.
Wave-particle duality is widely regarded as one of the conceptual pillars of modern quantum theory, providing the framework through which the behavior of microscopic matter can be consistently interpreted. According to this principle, microscopic entities such as electrons, photons, atoms and even complex molecules exhibit both wave-like and particle-like properties depending on the experimental configuration used to observe them. This duality is quantitatively formalized through the De Broglie relation, which associates a wavelength with any particle possessing momentum. The present work provides a comprehensive theoretical analysis of the physical and mathematical foundations of wave-particle duality, focusing particularly on the De Broglie wavelength and the kinetic energy of the electron in both non-relativistic and relativistic regimes. The connection between the De Broglie hypothesis and Schrödinger’s equation is discussed in detail, highlighting the role of the kinetic energy operator and the interpretation of matter waves in terms of quantum wave packets. Experimental evidence supporting the wave nature of matter, including electron diffraction experiments and interferometry techniques, is also reviewed. Beyond their conceptual importance, these principles underpin several modern technologies, including electron microscopy, nanoscale semiconductor devices and quantum metrology. The analysis demonstrates that the De Broglie relation provides a unifying framework connecting momentum, wavelength and kinetic energy, thereby forming a fundamental bridge between classical mechanics, quantum theory and modern technological applications.
Abstract This paper is based on the idea that wave mechanics is primarily suitable for giving an indirect description of nature. The world to which the equations of the, theory here proposed are directly applicable is an unnatural one, containing sources and sinks of matter which give to its radiational laws a symmetry lacking in nature. The theory may be regarded as consisting of two postulates. The first (which may, if desired, be regarded as a definition) specifies the character and properties of an unnatural assembly of systems, each consisting of an electron in a static potential field U. It states that, of the energy radiated in an elementary solid angle of standard magnitude, the part which will pass through an analyzer which transmits the component of electric field in a given direction (perpendicular to the line of observation) is determined by applying the usual Schrödinger interpretations (of density, current and material energy) to a Schrödinger equation perturbed by a term which is a certain constant multiplied by the electric field, in that direction, acting on the electron, due to the potential field U. (For comparison with classical radiation theory, it will be noted that the perturbation term vanishes in any region in which, from the classical standpoint, the electron is not accelerated, in that direction, by the field U). In the process of specifying the radiation from the assembly, the Schrödinger interpretations will have specified also the nature and magnitude of the sources and sinks of electrons (each in a field U). Their presence is due, mathematically speaking, to the multiplying constant mentioned above being a complex quantity. From these results the properties of a natural assembly are specified by means of the second postulate, which states that when the sources and sinks are removed, any process which in their absence is meaningless or physically inconceivable will cease, and all other processes will remain unaffected. The calculation is readily extended to include the effect of an external field of radiation. The theory requires no representation of radiation other than the Maxwellian wave, and will be free from the difficulties encountered in other theories.
This chapter discusses the conceptual foundations of quantum mechanics in the context of matter. Classical physics identified matter with particles and radiation with "waves." Although this description accounted for a large number of very important facts, it became apparent around the turn of the century that other phenomena, such as blackbody radiation and photoelectric effect, are satisfactorily explained only if radiation is described in terms of particles. Sometime later the hypothesis that, conversely, matter has wave properties was proposed by Louis de Broglie and was confirmed by the Davisson-Germer experiment of electron diffraction. In this experiment and similar ones, an electron beam was passed through a regular lattice (crystal), and subsequently showed the well-known diffraction patterns that are characteristic of waves. These results confirmed the theoretical predictions that a monoenergetic electron beam should be described by a plane wave and that the wave vector k of the wave should be related to the momentum p of an individual electron.
Abstract It will probably be agreed that among all the recent developments of the quantum theory, one of the least satisfactory is the theory of radiation. The present paper is intended as a preliminary to a new line of attack on the subject. It was begun some time ago, but owing to lack of success in carrying it to a conclusion, its publication has been much delayed. In the meantime other papers have appeared, which in some respects follow the same train of thought. The authors of these works have carried their methods further in some directions than I have attempted, but there is still perhaps room for the discussion of a number of questions from the rather different point of view adopted here. 1. The main principle of the present work is the idea that, since matter and light both possess the dual characters of particle and wave, a similar mathematical treatment ought to be applied to both, and that this has not yet bee done as fully as should be possible. Whereas we have a fairly complete calculus for dealing with the behaviour of any number of electrons or atoms, for photons the existing processes are much less satisfactory. The central difficulty, which makes it hard to apply the ordinary methods of wave mechanics to light, is the fact that (at least according to our present ideas) photons can be created and annihilated, and to represent this in a wave system we have to be able to think of a medium suddenly coming into existence and then going out again, when the light that it was carrying is absorbed. Such behaviour is a grave difficulty in the way of allowing us to think of the photon as a wave, and tends to make us think with more favour of its particle aspect, until we recall that after all it is quite unlike any known particle to come into existence and later to disappear without trace. The theories at present current, such as that of Heisenberg and Pauli, avoid these difficulties because they are mainly formal generalisations of the classical theory; this frees them from the above difficulties, but they pay for it in being highly abstract, and, as it has turned out, rather unsuccessful.
The simplest but deepest questions of the philosophy of quantum mechanics are these: Are quantum systems waves, or are they particles? Are they both or are they neither? When light bounces an electron out of the surface of an alkali metal it behaves like a particle. Light carries momentum but also a punch. Just as there is no such thing as half a punch, so there is no such thing as half a quantum of light. On the other hand, the shadow formed on a screen by an object has soft edges, edges which get softer as the object is moved away from the screen. Light exhibits the phenomena of diffraction and interference and so is wavelike. This dual behaviour of light, and indeed of matter itself, is called wave-particle duality. The wave nature of a quantum system is reflected in wave mechanics which assigns a wave-function to the system. But the dual nature of light was noticed before Schrodinger developed the wave-mechanical formalism in 1926. In fact, it goes back at least as far as Einsteins explanation of the photoelectric effect in 1905. Wave-particle duality is not something that arises within the quantum mechanical formalism. It is something which is expressed within it, and is something which is best discussed in terms of the paradoxes that quantum experiments force upon physics. In fact, the simplest and most immediately disturbing of the quantummechanical paradoxes arise out of wave-particle duality. We take the two-slit experiment and the delayed-choice experiment as examples.
The objective of the work is to reveal the Association of special theory of relativity with De-Broglie wavelength of a particle. It shows that wave nature of a particle is always associated with it’s relativistic properties. The phenomena of interference, diffraction and polarization of light can only be explained on the basis of wave theory of light. These phenomena show that light possesses wave nature. However, certain phenomena like photoelectric effect, Compton effect and discrete emission and absorption of radiation can only be explained on the basis of quantum theory. According to this theory , light propagates in a small packets or quanta and behave like corpuscles or particle. This shows that light possesses particle nature. thus, we can say that light possesses dual nature. The matter also possesses the same dual nature. In order to have wave nature, the particle must satisfy the equation Where m denotes the mass of particle when it is moving with certain velocity. h=6.62×10-34 is the plank’s constant. C is the velocity of light in free space. This shows that wavelength of a material Particle depends on mass. When mathematical operation is done on above equation, it give complete quantitative properties about the wave nature of a particle.