Interpretations of Quantum Mechanics

  • Gregg JaegerEmail author
Part of the The Frontiers Collection book series (FRONTCOLL)

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].


Quantum Mechanic Quantum State Quantum Theory Ontological Commitment Quantum Probability 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.College of General Studies Quantum Imaging LabBoston UniversityBostonUSA

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