Three Notes on the Quantum Theory of Aperiodic Effects

In Section 1 it is shown that the normalization of the characteristic functions corresponding to a continuous spectrum, which has been introduced by Hellinger and Weyl, satisfies the requirements of the $\ensuremath{\delta}$-normalization of the Dirac-Jordan transformation theory. It is further shown that this normalization makes the flux to and from infinity of systems for which an integral of motion $\ensuremath{\beta}$ lies in the little range $\ensuremath{\Delta}{\ensuremath{\beta}}^{\ensuremath{'}}$ equal to $(\frac{\ensuremath{\partial}E}{h\ensuremath{\partial}{\ensuremath{\beta}}^{\ensuremath{'}}})\ensuremath{\Delta}{\ensuremath{\beta}}^{\ensuremath{'}}.$In Section 2 the condition for the validity of classical mechanics in the form grad $\ensuremath{\lambda}\ensuremath{\ll}1$, where $\ensuremath{\lambda}$ is the instantaneous wave length $\ensuremath{\lambda}=(\frac{h}{2\ensuremath{\pi}}){[2M(E\ensuremath{-}U)]}^{\ensuremath{-}\frac{1}{2}}$, is applied to establish Rutherford's formula for the scattering of $\ensuremath{\alpha}$-particles.In Section 3 a method is developed for computing the transition probabilities between states of the same energy, and which are represented by almost orthogonal eigenfunctions. The theory is applied to the ionization of hydrogen atoms in a constant electric field. The period of ionization in a field of 1 volt per cm is ${10}^{{10}^{10}}$ sec. The bearing of such transitions on the problem of metallic conduction is discussed.