Abstract In the early presentation of the wave theory of matter de Broglie considered that every material particle is the seat of a "periodic phenomenon” of a frequency dependent upon the mass of the particle. A consideration of how this periodicity would manifest itself to observers, with respect to whom the particle is in motion, led to the development of some of the basic principles of the wave theory and, in particular, the expression of the mechanical quantities of energy and of momentum in terms of the theory led to the equation v0 = m0c2/h. (1) v0 denotes the frequency of the periodic phenomenon as it would be measured by an observer at rest with respect to the particle and m0 is the rest mass. We have shown in these Proceedings (Flint 1928, pp. 636, 638) that the interval of time τ0 = 1/v0 can be regarded as representing the smallest interval of time which has any significance in the physical world in relation to a material particle. This result was first developed from the Wilson-Sommerfeld quantum conditions and later from Planck’s principle of the quantum of action.
Abstract In the early presentation of the wave theory of matter de Broglie considered that every material particle is the seat of a "periodic phenomenon” of a frequency dependent upon the mass of the particle. A consideration of how this periodicity would manifest itself to observers, with respect to whom the particle is in motion, led to the development of some of the basic principles of the wave theory and, in particular, the expression of the mechanical quantities of energy and of momentum in terms of the theory led to the equation v0 = m0c2/h. (1) v0 denotes the frequency of the periodic phenomenon as it would be measured by an observer at rest with respect to the particle and m0 is the rest mass. We have shown in these Proceedings (Flint 1928, pp. 636, 638) that the interval of time τ0 = 1/v0 can be regarded as representing the smallest interval of time which has any significance in the physical world in relation to a material particle. This result was first developed from the Wilson-Sommerfeld quantum conditions and later from Planck’s principle of the quantum of action.
Every orbit of a material particle is composed of a finite number of parts of space lengths and time durations which depend on the particle’s momentum and energy. These are the “point” and the “moment” in space and time respectively. The uncertainty principles are attributed to the space-time through the physicisation of the geometrical concept of “point”. This new interpretation of uncertainty principles derives new conclusions in wave theory. The results of the discontinuity of space time, gives a new meaning in the foundations of quantum mechanics. These results are the new interpretation of photon, of matter waves and complementarity, Bohr’s postulates, as mathematical conclusions in the bases of Calculus.
Since the interest in the quantum phenomena started, there have been two main approaches to the quantum theory, the first approach investigates the motion of a wave-particle object by assuming the wave has the characteristics of a moving particle, momentum and energy, this approach was developed by physicists Max Planck, Albert Einstein, and others. The second approach investigates the motion of a particle-wave object by assuming the particle has the characteristics of a moving wave, wavelength, wave vector, and frequency, this approach was developed by physicists De Broglie, Schrodinger and others which led to the presence of quantum mechanics. This article focuses on the second approach and merging it with classical mechanics. An early mathematical formulation for the quantum theory was proposed by De Broglie’s relations and later on, another formulation was given by Schrodinger’s equation. A successful attempt to bridge the gap between classical mechanics with quantum theory implemented via a mathematical derivation for De Broglie’s relations and Schrodinger formulation, The derivation process overcame a main obstacle was the convergence of the deterministic nature of classical mechanics with the probabilistic nature of the quantum theory by adopting an innovative method based upon replacing constant values in a deterministic relation with probabilistic variables in the same relation, and this led to switch the relation from deterministic representation to probabilistic representation. Both classical and quantum explanation are given for the deduced formulas. An energy approach is instrumented. Since classical mechanics formulas are not sufficient to explain the quantum phenomena, a new concept of complementary energy introduced in order to connect the parameters of a moving particle with the parameters of an associated wave, the value of the complementary energy for each particle adjusted the total energy equation of N moving particles without changing the magnitude of the total energy of a system of N particles. In special cases, the value of the complementary energy for some particles equals the value of the total energy for the particles. The derivation process indicates that complementary energy corresponds to the potential energy on the surface of the material which prevents the particles to escape from the boundary of the matter to the outer space and maintains its coherence, complementary energy concept also manifest the impact of the moving particles on a single particle due to the energy conveyed during direct contact in case of collision or by the remote influence through an existing field in the space, so it represents an energy of dual nature for both matter and radiation wave combined. The complementary energy is not introduced as mathematical assumption in order to manipulate the total energy equation, it’s actually exists in the total energy equation and its validity has not been applied before. A general format for any potential energy is obtained regardless the nature of field it represents (a universal format for potential energy) that is by using mathematical manipulation for the equation of total energy of N moving particles. The derivation process led to a particle-wave formula. The substitution into the particle-wave formula with the speed of light, De Broglie’s relations, revealed. Some additional applications for the particle-wave formula are suggested, also as a well known energy quantization formula obtained by applying all of De Broglie’s relation, a general energy, and speed formulas resulting during the derivation process. Matching time independent equation called Schrodinger’s amplitude equation with classical mechanics formulation demonstrated by deducing both the wave function and the magnitude of the wave vector from the classical formulation after converting it to probabilistic formulation. A dedicated section for wave function solutions for Schrodinger’s time dependent equation was given in terms of the wave functions and the magnitude of the wave vectors obtained from the matching process of Schrodinger’s amplitude equation with classical mechanics.
Several fundamental physical relations share a mathematical structure in which the product of two conjugate quantities equals an invariant constant. Representative examples include the energy-period relation ET = h (where T = 1/f is the oscillation period and h is Planck’s constant), the momentum-wavelength relation pλ = h (the de Broglie relation), and the frequency-wavelength dispersion relation fλ = c. Although these expressions originate from quantum mechanics, wave mechanics, and electromagnetism respectively, each belongs to the generalized class XY = K, where K is an invariant constant. Their shared algebraic form raises the question of whether a common organizational principle underlies physical representation. This paper proposes the Foundational Principle of Closure Representation as an interpretational framework. Within this framework, action is adopted as the foundational quantity, and closure is introduced as a representation-independent measure of cyclic completion, defined by the condition Δθ = 2π. Planck’s constant h is interpreted as the action associated with one completed closure—consistent with the Bohr–Sommerfeld quantization condition ∮p dq = nh. The Planck relation E = hf is accordingly understood as the product of action per closure and closure rate per unit time. A representation-transfer relation (RX)(Y/R) = XY is introduced, where R is a dimensionless representation coordinate. A measurable realization is given by R = λC/λdB = p/(mc), coinciding with the standard relativistic quantity γβ. This relation is a mathematical identity; its physical significance lies in the proposed interpretation of R as a representation coordinate characterizing the distribution of closure action between conjugate observables. The proposed framework does not introduce new physical laws, modify established equations, or claim to supersede existing theories. It provides a unified interpretational architecture within which several established reciprocal relations—connecting action, closure, wavelength, frequency, momentum, and propagation—may be understood as instances of a common closure-action structure.
In classical physics, velocity follows from exact measurements of space and time intervals (i.e perfect resolution of both). Inertial mass follows from a resistance of an object to a force (push or pull). Momentum, however, which nonrelativistically is mv and should be associated with ideas of inertial mass, and space time resolution may be instead linked to the physical idea of an impulse. An impulse seems to be like a hit and does not appear to be linked directly to space time physically, although its consequences are i.e. a particle receiving the impulse moves in space and time. In classical mechanics, x(t) and so one considers either equally spaced dt intervals and corresponding dx(t) or vice versa (constant dx) if an integral only involves space dependent values. In quantum mechanics, one has two uncertainty principles, one related to E and t and the other to p and x. We argue that p is linked to the notion of an impulse which does not on the surface involve space or time. Given that p, even in nonrelativistic quantum mechanics, is mv there should at least be some built in notion of dx and dt. From the uncertainty principles one sees that dx (i.e. the wavelength) is linked to hbar/p and dt to hbar/E. Thus neither dt nor dx are constants, but depend on the motion. In the high energy quantum limit dt —> W(x)W(x) dx →1/v(x) dx where v(x) is classical velocity and W(x) the bound state wavefunction. The bound state wavefunction, however, applies to the envelope of WW (not the actual crests and troughs) and so the dx-dt scheme which emerges is a smoothed over one and not the actual dx -dp of quantum mechanics. We try to examine some of these ideas in this note.
Time – what it is and its nature – has from the beginning held mysteries for layman, philosopher, and physicist alike. Periodic phenomena from motions of heavenly bodies and pendulums to quantal atomic clocks provide the means to measure time intervals, and Newton’s absolute time is a background flow against which classical and non-relativistic quantum physics is played out. While internally consistent, these two fields of physics are incompatible with the requirement of treating space and time on an equal footing. Quantum field theories provide the answer, as both treated as parameters, physics residing instead in the states and operators. But even in classical physics and quantum mechanics, time-independent descriptions provide an alternative to an explicitly time-dependent one. These, including a general method of approach called invariant imbedding, are discussed as alternative representations of an underlying reality that is the concern of physics.