Brownian Particles and Matter Waves

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.