In this short note, the the key to unlocking the secrets of quantum physics will be elucidated by exploring the fundamentals of Schrodinger's wave mechanics approach to describing quantum phenomenon. We will show that de Broglie's wave-particle duality hypothesis which lies at the heart of Schrodinger's wave-function \psi produces a complex wave equation whose mathematical structure can be described by Euler's famous equation e^{i\theta}=cos(\theta)+isin(\theta) which basically describes a helical wave in 3D space. By comparing and contrasting the electromagnetic wave with that of a helical wave which Euler's equation represents, we may have discovered the geometric basis for spin and helicity and antimatter with negative energies that Dirac uncovered in his relativistic reformulation of Schrodinger's equation.
In this short note, the the key to unlocking the secrets of quantum physics will be elucidated by exploring the fundamentals of Schrodinger's wave mechanics approach to describing quantum phenomenon. We will show that de Broglie's wave-particle duality hypothesis which lies at the heart of Schrodinger's wave-function \psi produces a complex wave equation whose mathematical structure can be described by Euler's famous equation e^{i\theta}=cos(\theta)+isin(\theta) which basically describes a helical wave in 3D space. By comparing and contrasting the electromagnetic wave with that of a helical wave which Euler's equation represents, we may have discovered the geometric basis for spin and helicity and antimatter with negative energies that Dirac uncovered in his relativistic reformulation of Schrodinger's equation.
Copenhagen physicists said; The electronics that we use today-radio, televisions, computers, lasers, MRI machines GPS-communications- depend on calculations using Schrodinger’s equation plus a lot of other Mathematics and Measurements, our idea of “shut up and calculate” has worked very well for us technologically. My comment’s these technological achievements do not justify that the equation itself as (I will try show) has any true scientific basis. In 1925, Erwin Schrodinger wrote excitingly to Einstein. “I have been intensely concerned these days with Louis de Broglie’s theory [1]. It is extraordinary, exciting, but still has some very grave difficulties”. So regardless of you are consider Schrodinger’s equation is a fundamental equation (has no derivation from simples forms) or derived one, you cannot deny first that Schrodinger’s equation is a wave equation and second it is shows how (de Broglie hypothesis of) matter wave change with time. i.e. Schrodinger equation describe the evaluation of the physical system in which the quantum effects, the “Wave-particle” duality are significant.
Schrodinger's equation is the most important and famous equation in physics, which rivals Maxwell equations, it lies in the heart of quantum mechanics and it forms the basis of our understanding of all our universe.Before we discuss the origin of Schrodinger's equation, let us take a quick look at how physicists use such equations.First, they determine the wave function associated with a particle of mass m influenced by forces from Schrodinger's equation as the following.a) They specify the potential energy function V(r, t) via the forces acted on the particle.b) They solve this equation to find the wave function Ψ(r, t).
Finding that in the formula of expansion of a function into a series of wave-like functions the coefficients are its Fourier transforms, if existed, we deduce mathematically all the principles and hypothesis that illustrated physicists utilized to build quantum mechanics a century ago, beginning with the duality particle-wave principle of Planck and including the Schrödinger equations. By the way, we find a simple Fourier transform relation between Dirac momentum and position bras and a useful permutation relation between operators in phase and Hilbert spaces. Moreover, from the found particle-wave duality formula we prove and obtain again essentially by mathematical analysis all the laws of wave optics concerning reflections, refractions, polarizations, diffractions by one or many identical 3D objects with various forms and dimensions.
Credible reasons are presented to reveal that many of the lingering century old enigmas, surrounding the behavior of at least an individual quantum particle, can be comprehended in terms of an objectively real specific wave function. This wave function is gleaned from the single particle energy-momentum eigenstate offered by the theory of space filling universal quantum fields that is an inevitable outcome of Dirac's pioneering masterpiece. Examples of these well-known enigmas are wave particle duality, the de Broglie hypothesis, the uncertainty principle, wave function collapse, and predictions of measurement outcomes in terms of probability instead of certainty. Paul Dirac successfully incorporated special theory of relativity into quantum mechanics for the first time. This was accomplished through his ingenious use of matrices that allowed the equations of motion to maintain the necessary first order time derivative feature necessary for positive probability density. The ensuing Dirac equation for the electron led to the recognition of the mystifying quantized spin and magnetic moment as intrinsic properties in contrast to earlier ad hoc assumptions. The solution of his relativistic equation for the hydrogen atom produced results in perfect agreement with experimental data available at the time. The most far reaching prediction of the celebrated Dirac equation was the totally unexpected existence of anti-particles, culminating in the eventual development of the quantum field theory of the Standard Model that reveals the deepest secrets of the universe known to date. Quanta 2019; 8: 88–100.
We present the complete, ontically grounded wave mechanics derived from the de Broglie equation λp = h, interpreted literally as a real, physical wave. By treating ψ = 1/λ as a spatial compaction field and anchoring it on the Euler ComplexPlane (ECP)—a two-faced, rotating, compact topological surface—we derive the nonlinear wave equation, the soliton identity 1 = 1, the logarithmic spiral geodesic, the dual velocities and frequencies, the eigen-energy spectrum, and the photon asthe uncompacted reference state. We show that the Maxwell and d’Alembert equations are linear, uncompacted limits of the Prince nonlinear wave equation. The framework resolves the wave-particle duality, quantizes time and energy, and vindicatesde Broglie’s law of phase harmony as the foundational principle of quantum mechanics.
Abstract This paper proposes a geometric theory of wave-particle duality based on the algebra of three-dimensional numbers (3D Numbers). In the 3D number space T = σ + δe + ti (with e²=1, i²=-1, ei=0), the motion of a photon naturally takes the form of a helical trajectory T(t) = e^{δe}·e^{iωt}. The projection of this helix onto the (σ,i) plane yields the standard quantum mechanical wave function ψ(t) = cosh(δ)·e^{±iωt}, while the hyperbolic stretch along the e-direction corresponds to particle nature. Through five numerical experiments, we verify: (1) the helical structure is exact (radius deviation < 2×10⁻¹⁶); (2) projection to the complex plane gives the standard wave function; (3) all quantum mechanical quantities (E=ℏω, p=ℏk, λ=2π/k) are derived directly; (4) δ-parameter scan reveals a continuous wave-particle spectrum; (5) 3D numbers are the unique number system supporting both wave and particle nature. The central thesis: wave-particle duality is not a superposition of two attributes, but the projection decomposition of a single helical motion onto two orthogonal directions. Keywords: 3D numbers, helical motion, wave-particle duality, mid integration, foundations of quantum mechanics, non-associative algebra 摘要 本文提出一种基于三维数(3D Number)代数结构的光子波粒二象性几何理论。在三维数空间 T = σ + δe + ti(其中 e²=1, i²=-1, ei=0)中,光子的运动自然地表现为螺旋运动 T(t) = e^{δe}·e^{iωt}。该螺旋运动在 (σ,i) 平面上的投影给出标准量子力学波函数 ψ(t) = cosh(δ)·e^{±iωt},在 e 方向上的双曲伸缩对应粒子性。通过五组数值实验,我们验证了:(1) 螺旋结构严格成立(半径偏差 < 2×10⁻¹⁶);(2) 投影到复平面给出标准波函数;(3) 从螺旋运动可直接推导能量 E=ℏω、动量 p=ℏk、de Broglie 波长 λ=2π/k 等全部量子力学量;(4) δ 参数扫描揭示了波动-粒子连续谱;(5) 三维数是唯一同时支持波动性和粒子性的数系。本工作表明:波粒二象性的本质不是两种属性的叠加,而是同一螺旋运动在两个正交方向上的投影分解。 关键词:三维数、螺旋运动、波粒二象性、中道积分、量子力学基础、非结合代数