Geonium lineshape

The most precise determination of the magnetic moments of the electron and positron come from the University of Washington Geonium experiments. Present improvements in the accuracy of these experiments now require that the detailed shapes of the lines which they detect must be know to extract the moments with a corresponding precision. The linewidths arise because the trapped particle oscillates in a magnetic field which is made slightly inhomogeneous so that it varies quadratically with the oscillator displacement. The oscillaator undergoes a random, thermal, Brownian motion. Thus the lineshapes are related to a nonlinear noise problem. This problem is solved by writing the stochastic average of the oscillator motion as a functional integral which transcribes it into an easily solved one-dimensional, quantum-mechanical barrier penetration problem. Although this paper concentrates on a particular experiment, the techniques developed here should be useful in a wide variety of problems