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By properly ordering functions of noncommuting operators, a one-to-one transformation between operator functions and corresponding functions of commuting algebraic variables can be made. With this transformation, boson operator equations such as the Schrödinger equation can be converted to differential equations for the transformed functions, the resulting equations containing solely commuting variables. Once the solution to the transformed equation is obtained, the inverse transformation may be applied to yield the solution to the original operator equation. The method is extended to include angular momentum operators.

The variable parameter (or parametric) principle of amplification is characterized by a typical arrangement in which a variable energy storage element, such as an inductor or a capacitor, is suitably coupled to two resonant circuits. If the value of the energy storage element is made to vary in the proper way, energy is fed from the source which drives the element (that is, the pump) to the fields of both the resonant circuits. This paper describes the behavior relative to gain, band width, and noise of this type of amplifier. Specifically, it is shown that to increase gain, the Q of one of the resonant circuits, the one commonly called the idling circuit, must be increased or the variation in the variable reactance must be increased. The band width is inversely proportional to this Q and to the voltage gain. Hence, for high gain, the amplifier is normally a narrow band device. One of the most important sources of noise is the thermal noise originating in the idling circuit. However, in principle this source can be reduced indefinitely by making the idling frequency approach the pumping frequency or by artificially cooling the idling circuit. In this fashion very low noise figures should be possible. The parametric principle can also be applied to producing frequency conversion with large conversion gain. The appendix presents the expressions for gain, band width, and noise figure for this application. The behavior of the converter relative to gain, band width, and noise is quite similar to that of the amplifier.