Interpretations of Quantum Mechanics

Abstract

Quantum mechanics was initially formulated in what appeared to be two fundamentally distinct ways, as Matrix mechanics with operator matrices satisfying $$[Q_i,P_j]=i\hbar\delta_{ij}$$ , and Wave mechanics with the function $$\psi$$ satisfying $$i\hbar(\partial\psi(\vec{q})/\partial t)=H(\vec{q},\vec{p})\psi(\vec{q})$$ , the former developed by Heisenberg, Born and Jordan [73, 74],, and the latter by Schrödinger [393]. Dirac, Jordan, Pauli, and Schrödinger subsequently provided arguments for the equivalence of these two approaches. However, the Dirac–Jordan equivalence proof made use of the Dirac $$\delta$$ ‘function,’ which is not well defined as a function because it takes an infinite value at a single point although it can be given a proper definition as distribution (or “improper function”). Von Neumann finally rigorously proved the equivalence and derived the hydrogen atom energy eigenvalue spectrum by making use of Hilbert space, a separable complete vector space with an inner product and a countable, potentially infinite basis (cf. [282] and [281], Appendix 4), capturing the theory’s mathematical essence [473 – 477]. Much later, exploring some ideas of Dirac involving the Lagrangian and action [139], Feynman also produced a third, mathematically equivalent formulation of the theory [168].