In view of the remarkable progress in microrheology to monitor the random motion of Brownian particles with size as small as few nanometers, in association that de Broglie matter waves have been experimentally observed for large molecules of comparable nanometer size; we examine whether Brownian particles can manifest a particle-wave duality without employing a priori arguments from quantum decoherence. First, we examine the case where Brownian particles are immersed in a memoryless viscous fluid with a time-independent diffusion coefficient; and the requirement for the Brownian particles to manifest a particle-wave duality leads to the untenable result that the diffusion coefficient has to be proportional to the inverse time; therefore, diverging at early times. This finding agrees with past conclusions by Grabert et al., Phys. Rev. A 119 (1979), that quantum mechanics is not equivalent to a Markovian diffusion process. Next, we examine the case where the Brownian particle is trapped in a harmonic potential well with and without dissipation. Both solutions of the Fokker-Plank equation for the case with dissipation, and of the Schrödinger equation for the case without dissipation lead to the same physically acceptable result—that for the Brownian particle to manifest a particle-wave duality, its mean kinetic energy K_BT/2 needs to be ½ the ground-state energy, of the quantum harmonic oscillator. Our one-dimensional calculations show that for this to happen, the trapping needs to be very strong so that a Brownian particle needs to be embedded in an extremely stiff solid.
In view of the remarkable progress in microrheology to monitor the random motion of Brownian particles with size as small as few nanometers, in association that de Broglie matter waves have been experimentally observed for large molecules of comparable nanometer size; we examine whether Brownian particles can manifest a particle-wave duality without employing a priori arguments from quantum decoherence. First, we examine the case where Brownian particles are immersed in a memoryless viscous fluid with a time-independent diffusion coefficient; and the requirement for the Brownian particles to manifest a particle-wave duality leads to the untenable result that the diffusion coefficient has to be proportional to the inverse time; therefore, diverging at early times. This finding agrees with past conclusions by Grabert et al., Phys. Rev. A 119 (1979), that quantum mechanics is not equivalent to a Markovian diffusion process. Next, we examine the case where the Brownian particle is trapped in a harmonic potential well with and without dissipation. Both solutions of the Fokker-Plank equation for the case with dissipation, and of the Schrödinger equation for the case without dissipation lead to the same physically acceptable result—that for the Brownian particle to manifest a particle-wave duality, its mean kinetic energy K_BT/2 needs to be ½ the ground-state energy, of the quantum harmonic oscillator. Our one-dimensional calculations show that for this to happen, the trapping needs to be very strong so that a Brownian particle needs to be embedded in an extremely stiff solid.
In view of the remarkable progress in microrheology to monitor the random motion of Brownian particles with size as small as few nanometers, in association that de Broglie matter waves have been experimentally observed for large molecules of comparable nanometer size; we examine whether Brownian particles can manifest a particle-wave duality without employing a priori arguments from quantum decoherence. First, we examine the case where Brownian particles are immersed in a memoryless viscous fluid with a time-independent diffusion coefficient; and the requirement for the Brownian particles to manifest a particle-wave duality leads to the untenable result that the diffusion coefficient has to be proportional to the inverse time; therefore, diverging at early times. This finding agrees with past conclusions by Grabert et al., Phys. Rev. A 119 (1979), that quantum mechanics is not equivalent to a Markovian diffusion process. Next, we examine the case where the Brownian particle is trapped in a harmonic potential well with and without dissipation. Both solutions of the Fokker-Plank equation for the case with dissipation, and of the Schrödinger equation for the case without dissipation lead to the same physically acceptable result—that for the Brownian particle to manifest a particle-wave duality, its mean kinetic energy K_BT/2 needs to be ½ the ground-state energy, of the quantum harmonic oscillator. Our one-dimensional calculations show that for this to happen, the trapping needs to be very strong so that a Brownian particle needs to be embedded in an extremely stiff solid.
In view of the remarkable progress in microrheology to monitor the random motion of Brownian particles with a size as small as a few nanometers, and given that de Broglie matter waves have been experimentally observed for large molecules of comparable nanometer size, we examine whether Brownian particles can manifest a particle-wave duality without employing a priori arguments from quantum decoherence. First, we examine the case where Brownian particles are immersed in a memoryless viscous fluid with a time-independent diffusion coefficient, and the requirement for the Brownian particles to manifest a particle-wave duality leads to the untenable result that the diffusion coefficient has to be proportional to the inverse time, therefore, diverging at early times. This finding agrees with past conclusions published in the literature, that quantum mechanics is not equivalent to a Markovian diffusion process. Next, we examine the case where the Brownian particle is trapped in a harmonic potential well with and without dissipation. Both solutions of the Fokker–Planck equation for the case with dissipation, and of the Schrödinger equation for the case without dissipation, lead to the same physically acceptable result—that for the Brownian particle to manifest a particle-wave duality, its mean kinetic energy kBT/2 needs to be ½ the ground-state energy, E0=12ℏω of the quantum harmonic oscillator. Our one-dimensional calculations show that for this to happen, the trapping needs to be very strong so that a Brownian particle with mass m and radius R needs to be embedded in an extremely stiff solid with shear modulus, G proportional to m/RkBT/ℏ2 .
In view of the remarkable progress in microrheology to monitor the random motion of Brownian particles with a size as small as a few nanometers, and given that de Broglie matter waves have been experimentally observed for large molecules of comparable nanometer size, we examine whether Brownian particles can manifest a particle-wave duality without employing a priori arguments from quantum decoherence. First, we examine the case where Brownian particles are immersed in a memoryless viscous fluid with a time-independent diffusion coefficient, and the requirement for the Brownian particles to manifest a particle-wave duality leads to the untenable result that the diffusion coefficient has to be proportional to the inverse time, therefore, diverging at early times. This finding agrees with past conclusions published in the literature, that quantum mechanics is not equivalent to a Markovian diffusion process. Next, we examine the case where the Brownian particle is trapped in a harmonic potential well with and without dissipation. Both solutions of the Fokker–Planck equation for the case with dissipation, and of the Schrödinger equation for the case without dissipation, lead to the same physically acceptable result—that for the Brownian particle to manifest a particle-wave duality, its mean kinetic energy kBT/2 needs to be ½ the ground-state energy, E0=12ℏω of the quantum harmonic oscillator. Our one-dimensional calculations show that for this to happen, the trapping needs to be very strong so that a Brownian particle with mass m and radius R needs to be embedded in an extremely stiff solid with shear modulus, G proportional to m/RkBT/ℏ2.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.
We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear function of the position of the particle. Physically, this corresponds to a configuration in which damping and diffusion are spatially inhomogeneous. We derive systematically the quantum master equation for the Brownian particle in the Born-Markov approximation and we discuss the appearance of additional terms, for various polynomials forms of the coupling. We discuss the cases of linear and quadratic coupling in great detail and we derive, using Wigner function techniques, the stationary solutions of the master equation for a Brownian particle in a harmonic trapping potential. We predict quite generally Gaussian stationary states, and we compute the aspect ratio and the spread of the distributions. In particular, we find that these solutions may be squeezed (superlocalized) with respect to the position of the Brownian particle. We analyze various restrictions to the validity of our theory posed by non-Markovian effects and by the Heisenberg principle. We further study the dynamical stability of the system, by applying a Gaussian approximation to the time-dependent Wigner function, and we compute the decoherence rates of coherent quantum superpositions in position space. Finally, we propose a possible experimental realization of the physics discussed here, by considering an impurity particle embedded in a degenerate quantum gas.
Wave-particle duality is widely regarded as one of the conceptual pillars of modern quantum theory, providing the framework through which the behavior of microscopic matter can be consistently interpreted. According to this principle, microscopic entities such as electrons, photons, atoms and even complex molecules exhibit both wave-like and particle-like properties depending on the experimental configuration used to observe them. This duality is quantitatively formalized through the De Broglie relation, which associates a wavelength with any particle possessing momentum. The present work provides a comprehensive theoretical analysis of the physical and mathematical foundations of wave-particle duality, focusing particularly on the De Broglie wavelength and the kinetic energy of the electron in both non-relativistic and relativistic regimes. The connection between the De Broglie hypothesis and Schrödinger’s equation is discussed in detail, highlighting the role of the kinetic energy operator and the interpretation of matter waves in terms of quantum wave packets. Experimental evidence supporting the wave nature of matter, including electron diffraction experiments and interferometry techniques, is also reviewed. Beyond their conceptual importance, these principles underpin several modern technologies, including electron microscopy, nanoscale semiconductor devices and quantum metrology. The analysis demonstrates that the De Broglie relation provides a unifying framework connecting momentum, wavelength and kinetic energy, thereby forming a fundamental bridge between classical mechanics, quantum theory and modern technological applications.