The Hamiltonian Formalism of Damping in a Tuned Circuit
K W H Stevens
The idea behind this work is the supposition that it should be possible to introduce damping into the quantum mechanical discussion of a harmonic oscillator without a detailed description of the damping mechanism. A study is therefore made of an L, C circuit coupled to an artificial (or transmission) line, using a Hamiltonian formalism. The whole system is conservative but, for the transmission line case, the L, C circuit behaves as if it is damped. The most probable form for the Hamiltonian does not exhibit the damping in an obvious way. It is shown that by a suitable change of variables the Hamiltonian can be split into two time-dependent commuting parts, one of which has the form e -2α t ( P 2 /2 C ) + e 2α t ( Q 2 /2 L ) which has been previously used in a study of 2-level masers, and which does display the damping. The new operators Q and P are studied and it is shown that they are related to the charges and currents in the L, C circuit, but with the noise components due to the resistive loss (the transmission line) omitted. They are thus perhaps more natural variables to use than the actual charges and currents which do contain noise components. The problem of coupling a spin system to the current in the inductance is studied and the transition is made to the analogous problem of a spin system interacting with a damped mode of a cavity resonator. The analysis is compared with one given previously by Stevens and Josephson in 1959 and it is shown that the results previously obtained in a heuristic way are valid provided that noise fluctuations are negligible.
Proceedings of the Physical Society•Feb 1, 1961•19 citations