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HARMONIC OSCILLATOR IN THE TVM FRAMEWORK - Recognition of the spectrum from the transformation mechanism and the hierarchy of new levels – from the First Principles of the Theory of Time Modulation and the established initial state of the universe

Maričić Zoran

Published Aug 4, 2026• 0 citations
Publisher

Abstract

ABSTRACT In standard quantum mechanics, the spectrum of the harmonic oscillator: is derived by solving the Schrödinger equation for a parabolic potential. This is a mathematical formalism that gives correct numerical results but does not explain why the levels are as they are – why there is a ground state , why the levels are equally spaced, why the difference between them is . In the TVM framework, the spectrum of the harmonic oscillator is recognized through two independent physical processes that are already embedded in the mechanism of the Universe's evolution: Evolution of the first level – how the energy of the original, primary level decreases through successive transformations. Hierarchy of new levels – how, at each transformation, part of the energy separates from the previous level and forms a new, independent level. These two processes are simultaneous and correlated: the first level cannot evolve without creating new levels, and new levels cannot arise without the evolution of the first level. They are two sides of the same process – the impossibility of perfect division upon reaching the transformation limit . In TVM, the ground state of the harmonic oscillator is not a "zero-point energy of the vacuum" – it is a consequence of the first transformation. The formula: comes directly from the TVM mechanism and fractal recursion, where each new level has its own frequency , and the product of amplitude and momentum grows as . Thus, the harmonic oscillator ceases to be a "special" quantum system – it becomes a general pattern that appears wherever a system undergoes a series of transformations, from the initial state of the Universe to local quantum phenomena. Keywords: TVM, harmonic oscillator, spectrum, fractal recursion, transformation, stationarity, energy defect. NOTE: This is the third in a series of supplementary scientific papers that elaborate the application of the TVM formalism to specific physical phenomena. Starting from the foundations laid in the zeroth paper [1], this work derives the harmonic oscillator spectrum from the transformation mechanism and the hierarchy of levels. It is recommended that the reader first study the First supplementary work (de Broglie relation) [2] and the Second supplementary work (Electron shell filling) [3], as well as Chapters 3 and 4 of the main work [1].

DOI
10.5281/zenodo.21790517

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ABSTRACT In standard quantum mechanics, the spectrum of the harmonic oscillator: is derived by solving the Schrödinger equation for a parabolic potential. This is a mathematical formalism that gives correct numerical results but does not explain why the levels are as they are – why there is a ground state , why the levels are equally spaced, why the difference between them is . In the TVM framework, the spectrum of the harmonic oscillator is recognized through two independent physical processes that are already embedded in the mechanism of the Universe's evolution: Evolution of the first level – how the energy of the original, primary level decreases through successive transformations. Hierarchy of new levels – how, at each transformation, part of the energy separates from the previous level and forms a new, independent level. These two processes are simultaneous and correlated: the first level cannot evolve without creating new levels, and new levels cannot arise without the evolution of the first level. They are two sides of the same process – the impossibility of perfect division upon reaching the transformation limit . In TVM, the ground state of the harmonic oscillator is not a "zero-point energy of the vacuum" – it is a consequence of the first transformation. The formula: comes directly from the TVM mechanism and fractal recursion, where each new level has its own frequency , and the product of amplitude and momentum grows as . Thus, the harmonic oscillator ceases to be a "special" quantum system – it becomes a general pattern that appears wherever a system undergoes a series of transformations, from the initial state of the Universe to local quantum phenomena. Keywords: TVM, harmonic oscillator, spectrum, fractal recursion, transformation, stationarity, energy defect. NOTE: This is the third in a series of supplementary scientific papers that elaborate the application of the TVM formalism to specific physical phenomena. Starting from the foundations laid in the zeroth paper [1], this work derives the harmonic oscillator spectrum from the transformation mechanism and the hierarchy of levels. It is recommended that the reader first study the First supplementary work (de Broglie relation) [2] and the Second supplementary work (Electron shell filling) [3], as well as Chapters 3 and 4 of the main work [1].
The study of quantum properties of the harmonic oscillator is important in physics as many real-world systems oscillate harmonically. Motion of systems in a confined space is often modeled as being a quantized harmonic motion or, in first instance, is approximated by a harmonic motion. This chapter helps the reader to solve the equation employing an algebraic operator technique, which is based on the Dirac notation. This approach has several definite advantages and exploits the commutation relations among the operators involved and their properties. The chapter transforms the stationary Schrodinger equation into a second-order differential equation and finds the solution to the equation with the aid of a more advanced mathematical technique that involves special functions. The energy of a harmonic oscillator is quantized, with the sequence of values. The difference in energy between adjacent energy levels is equal to the energy of a single photon, ħω.
Abstract The ground state of the linear harmonic oscillator is the solution to a minimization problem, and this opens up two new ways to completely solve Schrödinger’s time-independent equation. The first exploits systematic patterns that emerge from a formal relationship between two different integral representations of the energy. A remarkable invariant form of the ground-state probability-density equation under specific conditions is the leading principle of the second procedure, where a uniform notation for all mathematical object involved facilitates calculations and generates useful schemes to quickly deduce the main functional relationships of the problem. We believe these procedures may represent useful educational alternatives to standard treatments that address this fundamental problem.
Mechanical motions of atoms near their equilibrium geometry are approximated by a universal mapping on a collection of independent harmonic oscillators. The respective frequencies are characteristic to the material, for example, lattice phonons or molecular vibrations. The quantum description of atomic motions involves the solution of the Schrödinger equation for the harmonic oscillator in the space of proper wave functions. The energy levels obtained are shown to be equally spaced where the level spacing is proportional to the oscillator frequency. The stationary solutions are identified in terms of Hermite polynomials, demonstrating remarkable differences from the classical harmonic oscillator. The importance of the zero-point energy is emphasized in the context of the stability of chemical bonds, where the Harmonic approximation is shown to be reasonably valid for typical interatomic bonds in standard thermal conditions. This explains the relevance of the harmonic approximation for analyzing the absorption spectrum of infrared radiation by molecules.
The time-dependent Schrodinger problem of an oscillator .5kx*x together with the additional potential -x f(t), where f(t) is a function of time (often periodic), has already been solved in the literature (1). The solution involves a variable transformation y= x - b(t), where b(t) is a function of time which will turn out to be related to f(t) and W(x,t)= exp(i y d/dt b(t)) Wosc( y) exp(i Eosc t) . This yields an oscillator type time dependent equation with a term q(t) Wosc(y) which can be removed by using Wosc(y) exp(i c(t)). Wosc( y) is the oscillator wavefunction with the space-time variable y and Eosc the usual oscillator energy levels. Thus, by using the variable transformation together with W(x,t)=Wosc(x)exp(iEosc t) g(x,t) (i.e. a product of factors) one can obtain an “time-independent like” Schrodinger equation in y (after performing one more transformation Wosc(y)exp(i c(t)). In a previous note (2), we argued that if one writes a wavefunction as W(x)=Wo(x)q(x), where Wo(x) is the ground state wavefunction, one obtains a time-independent Schrodinger equation for Wo(x), which contains V(x), and a second eigenvalue equation coupling Wo(x) and q(x) with an eigenvalue equal to E-Eo. In particular, it is the time flux [d/dt W(x,t)]/W(x,t) which becomes linked with q(x) and the spatial flux [d/dx W(x)]/W(x). In (3), we argued that bound state quantum mechanics contains two velocities v(x), the classical velocity and u(x)/m = [d/dx W(x)]/ W(x) and that each behave differently. For example, v(x) behaves according to Newton’s law, while u(x)/m does not (except for the oscillator ground state). In this note, we try to apply these two ideas to the case of the oscillator with the time-dependent extra potential x f(t). We see that writing W(x,t) as a product (after the variable change) leads to two sets equation, one being a time-independent like equation for the oscillator and the other linking u(x)/m to a portion of time flux [d/dt W(x,t) ]/ W(x,t) |x constant. In this case, the time flux is directly related to d/dx [W*(x,t)W(x,t)] i.e. the physical density. Thus, a change in physical density in time is related to a change in physical density in space. Thus, two physical processes, it seems, are occurring. One is the W osc (y) resonance (albeit it a strange one because y=x + b(t)) and the second flux coupling with exp(i b(t) y), part of the oscillator potential and part of the additional potential -x f(t). Given that the problem has been mathematically solved already in the literature (1), our objective is to attempt to see if there is physical meaning in terms of ideas we have presented in previous notes, namely flux, the notion of a product wavefunction leading to a separate resonance type of reaction and double resonances.
Starting with a real abstract algebra which encapsulates the algebraic structures of both classical and quantum mechanics, this paper presents a self–contained real-ization of the latter in phase space. Having both mechanics formulated in the same space opens two new windows into the comparative study of foundations. This stems from the fact that the same physical problem, as defined by a given Hamil-tonian, can be solved in several independent ways. The exact solutions can then be compared. Thus, comparison of the classical and quantum solutions in phase space offers new epistemological insights into Bohr’s correspondence principle, while com-parison of the quantum solutions in the different formalisms of Hilbert space and phase space yields new physical insights. These general ideas are then tested on the harmonic oscillator. The analytic ground work is presented in Part I, the exact solutions will be derived in Part II. 1.
Courses about Quantum Mechanics are generally developed at the level of the mathematical formalism, its syntactic level, seldom treating the interpretation of the theory, its semantic level. In the majority of cases, this first formal approach is not followed by any other course addressing the numerous interpretations of the theory. It has become apparent that this strategy leads to a deficit in the professional formation of the students, exceedingly driving them to technical areas of application of Quantum Mechanics. However, applications such as those on quantum computing and entanglement, as they become more and more encrusted in the deepest foundations of the theory, are asking more and more for the development of this kind of ability. In this paper we develop an approach to deal with second quantization in the realm of Quantum Mechanics. We show that the second quantization Schrodinger equation can be mathematically derived from a classical Hamiltonian written in the phase space (Q,P)(Q,P)(Q,P), obtained by a canonical transformation upon the original phase space (q,p)(q,p)(q,p), using an axiomatic quantization procedure developed elsewhere. We apply this quantization process to a bosonic system, modelled by the harmonic oscillator problem. We then make reverse engineering to show that Schwinger's second quantization approach to fermionic systems furnishes the path to derive a Schrodinger equation explicitly written in terms of the usual momenta and positions operators (q^,p^)(\hat{q},\hat{p})(q^​,p^​) for such systems. Finally, we use these mathematical developments to address the commonly accepted statement that 'the spin has no classical analog'.