Abstract
Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position–momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.
Highlights
Following numerous attempts at the beginning of the 20th century, a satisfactory solution to the wave–particle duality problem was found by Heisenberg, Born, and Jordan [1,2,3]in 1925 with matrix mechanics and, almost simultaneously, by Schrödinger [4,5,6] with wave mechanics
Complex numbers became a conceptual element of the very foundations of physics: the fundamental equations of matrix mechanics and wave mechanics: Published: 23 December 2021 pq − qp Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
Considering, Schrödinger’s wave mechanics, the corresponding wave equation is linear, so one could assume that changing from a real to a complex wave function would not cause much difference; it would just produce two identical equations for real and imaginary parts of the complex wave function
Summary
Following numerous attempts at the beginning of the 20th century, a satisfactory solution to the wave–particle duality problem was found by Heisenberg, Born, and Jordan [1,2,3]. Considering, Schrödinger’s wave mechanics, the corresponding wave equation is linear, so one could assume that changing from a real to a complex wave function would not cause much difference; it would just produce two (almost, apart from the real potential V) identical equations for real and imaginary parts of the complex wave function This is not the case, because i = −1 occurs explicitly, i.e., in position space. Bonilla and Schuch clarified how Madelung’s hydrodynamic picture can be extended consistently to include both position and momentum representations In this context, the connection of the position and momentum uncertainties, and the corresponding uncertainty product with the hydrodynamic quantities in the Madelung picture, need further examination, regarding what the contributions of phase and amplitude to these quantities are.
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