Quantum Theory of a Damped Electrical Oscillator and Noise. II. The Radiation Resistance

The results of a previous paper are extended to include damping and noise due to a radiation resistance. The average electromagnetic field energy of an oscillator of natural frequency $\omega$, with inductance $L$, coupled to a radiation resistance $R$, as a function of time $t$ is given by $U=\left[\frac{1}{2}\hslash \omega +\frac{\hslash \omega }{\exp \left(\frac{\hslash \omega }{\mathrm{kT}}\right)-1\mathrm{}}\right][1-e^{-{\left(\frac{R}{L}\right)}^t}]+U_0{e}^{-{\left(\frac{R}{L}\right)}^t}.$ Using first-order perturbation theory and quantum statistics, an expression is derived for the radiation resistance of a circuit. This agrees exactly with the classical value.