A Note on the Particle-Wave Correspondence

Bhargava Kotur

Ronald Geballe

Попов Игорь Павлович

It is noted that for a monochromatic de Broglie wavelength corresponding to the free inertial particle wave group consists of a single wave, so the group velocity is identically equal to the phase. Justified by the presence in the expression for the wave kinetic energy factor ½, which is the same as in classical case which is due to the inertia of the particle. The representation of particle as a set of sub-particles, of which N is infinite, and therefore receive a confirmation of the equality group and phase velocities is described. It is shown that with the ratio for wave kinetic energy, expression for the total energy of the wave, the wave function and the Schrodinger equation for a free particle are inextricably linked

Paul Peter Urone, Roger A. Hinrichs

Noriaki Kihara

Observational note (thought-experiment style; not a proof or claim paper). A sequel to the preceding note, which defined a single particle-like state as a phase-bearing local mode on the subjective geometry S^4(R_U) with spread W_a = 2 R_a and deferred the multi-particle case. Here we treat the minimal case: two particle-like states with the same spread phase, on a closed one-dimensional relational setting (period-1 phase line, a single great circle of S^4). Representing each particle as a rectangular standing wave (odd harmonics only), the n-th harmonic amplitude of the superposition is A_n = (2/n) cos(n pi f), where f is the relative position in fundamental wavelengths, and the total squared norm (energy-like quantity) is the triangle wave E(f) = (pi^2/4)(1-2f) for 0 <= f <= 1/2 (the autocorrelation of a square wave). Within the continuous f, discrete feature points appear: fusion (f=0), orthogonality / independent coexistence (f=1/4), and complete cancellation (f=1/2). Depending on whether the superposition is symmetrized or antisymmetrized, complete overlap (f=0) is allowed (boson-like) or makes the field vanish, i.e. exclusive (fermion-like). For unequal frequencies, interference arises only from shared odd harmonics (the irreducible ratio r/s with both r,s odd, i.e. matching 2-adic valuation); without sharing, interference vanishes and placement is free. All results are presented as observations of structural correspondence (with standing-wave quantization and the exchange symmetry of quantum statistics); no derivation of quantum statistics or the spin-statistics theorem is claimed. Notably the existence of the three feature points is universal to any half-wave-antisymmetric (odd-harmonic) profile, while only the linear (triangle-wave) envelope is specific to the square-wave (1/n) choice. This record contains Japanese and English versions (Markdown, LaTeX, PDF) and one figure.

Hal Tasaki

This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general $N$ body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)