The New Kinematics and its Exploration

Abstract

The avowed aim of Heisenberg’s paper was “to try and establish a theoretical quantum mechanics, analogous to classical mechanics, but in which only relations between observable quantities occur.”¹ The attempt was a confused one, if only on account of the variety of conflicting notations,² but the fundamental idea was clear: to take over the classical equation of motion,

$$\ddot q + f\left(q \right) = 0$$

((1))

but to replace the classical acceleration and potential f(q) by quantum- theoretical representations derived from their series or integral Fourier expressions. In the case of the position function q(t) of the periodic motion of an electron in an atom, the position itself was not observable: as Heisenberg and Pauli had long recognised, the kinematic conception of an orbit was in this case operationally meaningless.³ But the terms of the Fourier expansion of the position could be directly related to observables. Classically the position vector of an oscillating electron could be expanded as a Fourier series,

$$\left({n,t} \right) = \sum\limits_{\alpha = - \infty}^\infty {{q_\alpha}\left(n \right){e^{i\omega \left(n \right)\alpha t}}}$$

((2))

and the radiation corresponding to each harmonic was proportional to the real part of the Fourier component, IR (qα(n) e^iω(n)αt). In quantum theory the Bohr frequency condition,

$$v\left({n,n - \alpha} \right) = {1 \over h}\left\{{W\left(n \right) - W\left({n - \alpha} \right)} \right\}$$

((3))

with W(n) the energy of the nth state, led to the requirement that the harmonic components take the form of expressions q(n, n — α) e^{iv(n, n — α)t,}corresponding to pairs of states or transitions.