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DEGENERACY I N QUANTUM MECHANICS 64iwhere D' is defined as in Eq. ( 19) and A4 and 84 are arbitrary constants. This solution represents a state of the rotating space charge in which "spokes" of increased density maintain a fixed position relative to a coordinate system rotating with angular velocity Qp. Fig. 2 is intended to represent this state of affairs. The a.c. at any point is then due only to changes in charge density and not at all to changes in electron velocity. DIscUssIoNWe have assumed throughout that the boundaries are perfectly conducting and that electrons travel in circles around the axis so that no current Rows from the inner to the outer conductor. Thus no d.c. power is being fed into the system, and we cannot expect to make any estimates of a.c. power output. This is also evident from the fact that currents and voltages calculated from our solutions are always ~/2 out of phase. For the same reason it is impossible to estimate the resistance which a tube will present to an a.c. signal. This situation could be remedied by introducing a small d.c. radial velocity, but this increases the complexity of the a.c. equations so that they are quite unmanageable by our present technique.It is a pleasure to acknowledge our indebtedness to Mr. W. C. Hahn for many stimulating discussions and suggestions.

A new kind of quantum mechanics using inner products, matrix elements, and coefficients assuming values that are quaternionic (and thus noncommutative) instead of complex is developed. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The role played by the new imaginaries is studied. The principal conceptual difficulty concerns the theory of composite systems where the ordinary tensor product fails due to noncommutativity. It is shown that the natural resolution of this difficulty introduces new degrees of freedom similar to isospin and hypercharge. The problem of the Schrodinger equation, "which i should appear?" is studied and a generalization of Stone's theorem is used to resolve this problem.1. WHY QUATERNION QUANTUM MECHANICS? ' )pQq= (lfl*cO+b ' *c ' )+i2(lflc l -b 1 cO). We separated the quaternion with respect to i2 and identified i3 with the complex i. But, of course, we could have used any pair of anticommuting units as well.