Quantum Effects in the Interaction between Electrons and High-Frequency Fields. I

The passage of an electron through a cavity resonator is considered for the purpose of studying the appearance of quantum mechanical effects in an essentially classical experiment as the frequency of the electromagnetic field becomes high. Both the electron and the field must be represented, initially, by wave packets—in space and momentum coordinates for the former, and in electric field and vector potential coordinates for the latter—in order to correspond to classically meaningful systems. The velocity of the electron at the output of the cavity is investigated. It is found that the expectation value of velocity has a quantum-mechanical correction term. A more significant effect, however, is the nonvanishing of the expectation value of the square of the deviation from the expectation value (mean square deviation from the mean) of the velocity. This deviation is a random phenomenon which will produce noise in an electron beam. An expression for the mean square deviation, in which the results of electron and field quantization are separately apparent, is derived, and its significance is discussed. An order-of-magnitude calculation is made for the ratio of minimum possible mean square deviation of velocity to velocity increment due to the field, and the frequency for which this ratio becomes unity is calculated for a particular set of conditions to be of the order of 3×${10}^{12}$/sec.