[Paper Review] von Neumann-Landau equation for wave functions, wave-particle duality and collapses of wave functions

Zeqian Chen|ArXiv.org|Mar 22, 2007

Geophysics and Sensor Technology4 references3 citations

TL;DR

This paper introduces the von Neumann-Landau equation for wave functions (vNLW) as an extension of Schrödinger's equation, using 'bipartite' wave functions to mathematically formalize wave-particle duality. It shows that von Neumann’s entropy quantitatively measures complementarity between wave-like and particle-like behavior, and reinterprets wave function collapse as a simultaneous transition from multiple quantum levels to a single one.

ABSTRACT

It is shown that von Neumann-Landau equation for wave functions can present a mathematical formalism of motion of quantum mechanics. The wave functions of von Neumann-Landau equation for a single particle are `bipartite', in which the associated Schrödinger's wave functions correspond to those `bipartite' wave functions of product forms. This formalism establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. Furthermore, this extension of Schrödinger's form suggests that collapses of Schrödinger's wave functions can be regarded as the simultaneous transition of the particle from many levels to one.

Motivation & Objective

• To establish a mathematical formalism for wave-particle duality in quantum mechanics using an extended wave equation.
• To show that von Neumann’s entropy serves as a quantitative measure of complementarity between wave-like and particle-like behaviors.
• To reinterpret the collapse of Schrödinger’s wave functions as a simultaneous transition of a particle from multiple energy levels to a single level.
• To provide a measurement-free expression of wave-particle duality, avoiding reliance on the uncertainty principle.

Proposed method

• Derives the von Neumann-Landau equation for wave functions (vNLW) as a differential equation for 'bipartite' wave functions Ψ(x,y;t), governed by iℏ ∂Ψ/∂t = (Ĥ(x) − Ĥ(y))Ψ.
• Defines 'bipartite' wave functions as elements in L² space over two spatial variables, generalizing single-particle Schrödinger wave functions.
• Expresses general solutions of vNLW as superpositions of product states ψₙ(x)ψₘ*(y), with coefficients cₙₘ representing transition amplitudes between energy levels.
• Introduces von Neumann entropy S(Ψ) = −Tr[ρ log ρ] for the density matrix ρ derived from Ψ, to quantify wave-particle duality.
• Analyzes two limiting cases: Ψ_W (coherent superposition, wave-like) and Ψ_P (incoherent mixture, particle-like), showing S(Ψ) ranges from 0 to (1/2)ln2.
• Uses the formalism to reinterpret wave function collapse as a probabilistic transition from multiple levels to a single level, with probabilities pₘ = Σₙ|cₙₘ|².

Experimental results

Research questions

• RQ1 Can the von Neumann-Landau equation for wave functions provide a mathematical formalism for wave-particle duality?
• RQ2 How does von Neumann’s entropy quantify the complementarity between wave-like and particle-like behaviors in a single quantum system?
• RQ3 Can the collapse of a Schrödinger wave function be interpreted as a simultaneous transition from multiple quantum levels to a single level?
• RQ4 Does the vNLW formalism allow a measurement-free description of wave-particle duality, independent of the uncertainty principle?
• RQ5 What is the role of 'bipartite' wave functions in unifying the description of superposition and measurement outcomes?

Key findings

• The von Neumann-Landau equation for wave functions (vNLW) is a valid extension of Schrödinger’s equation, governing 'bipartite' wave functions Ψ(x,y;t) via iℏ ∂Ψ/∂t = (Ĥ(x) − Ĥ(y))Ψ.
• The von Neumann entropy S(Ψ) of the density matrix derived from Ψ quantitatively measures the complementarity between wave-like and particle-like behaviors, with 0 ≤ S(Ψ) ≤ (1/2)ln2.
• For a single electron in a double-slit setup, Ψ_W corresponds to wave-like interference (S(Ψ_W) > 0), while Ψ_P corresponds to particle-like localization (S(Ψ_P) = (1/2)ln2).
• The collapse of a superposition state ψ = Σₙ aₙψₙ is interpreted as the simultaneous transition of the particle from all levels ψₙ to a single level ψₘ with probability |aₘ|².
• The energy change upon collapse is △Eₘ = Σₙ |cₙₘ|² (Eₙ − Eₘ), with cₙₘ = aₙaₘ* in the standard superposition case.
• The formalism provides a measurement-free expression of wave-particle duality, where the degree of complementarity is directly encoded in the entropy S(Ψ).