Vacuum Fluctuation Noise and Dissipation

An electron beam interacting with the fields in an enclosed region is considered. The induced noise caused by the vacuum fluctuations and the thermal fluctuations is calculated in terms of the dissipation function R(ω) and the electron transit time τ. The observable mean squared electromotive force is given by ${〈{V}²〉}Av={2}{{π}}{∫}{0}^{{{ω}}ₘₐₓ}R({ω})[{{}{ω}}{2}+{{}{ω}}{exp({{}{ω}}{kT}){-}1}]{[{sin1/2{ω}{τ}}{1/2{ω}{τ}}]}²d{ω}+{1}{{π}}{∫}{{{ω}}ₘₐₓ}^{{∞}}[{R({ω}){}{ω}}{exp({{}{ω}}{kT}){-}1}]{[{sin1/2{ω}{τ}}{1/2{ω}{τ}}]}²d{ω}.$ ${{ω}}ₘₐₓ$ is defined by ${}{{ω}}ₘₐₓ=U$, where $U$ is the energy of the incident electrons. The effect of the fluctuations of the vacuum is given by the temperature-independent part of the first integral.The above expression is employed to calculate the fluctuations induced in an electron beam which interacts with a damped electrical oscillator. The results are valid even if the oscillator is heavily damped. As the damping becomes very large the fluctuations are shown to approach zero. For a weakly damped oscillator the fluctuations reduce to those already given in an earlier paper.The differential form of the fluctuation dissipation theorem is shown to be very useful for the evaluation of matrix elements.