Smearing effect due to the spread of a probe particle on the Brownian motion near a perfectly reflecting boundary

Quantum fluctuations of an electromagnetic vacuum are investigated in a half-space bounded by a perfectly reflecting plate by introducing a probe described by a charged wave-packet distribution in time direction. The wave-packet distribution of the probe enables one to investigate the smearing effect upon the measured vacuum fluctuations caused by the quantum nature of the probe particle. It is shown that the wave-packet spread of the probe particle significantly influences the measured velocity dispersion of the probe. In particular, the asymptotic late-time behavior of its z component ⟨Δv_z²⟩ for the wave-packet case is quite different from the test point-particle case (z is the coordinate normal to the plate). The result for the wave packet is ⟨Δv_z²⟩∼1/τ² at late time (τ is the measuring time), instead of the reported late-time behavior ⟨Δv_z²⟩∼1/z² for a point-particle probe. This result can be quite significant for further investigations on the measurement of vacuum fluctuations.