Harmonic Oscillators, Heisenberg's Uncertainty Principle and Simultaneous Measurement Precision for Position and Momentum

Donald J. Kouri, Cameron L. Williams, Bernhard G. Bodmann

Published 2014-09-08Version 1

There is no question as to the validity of Heisenberg's uncertainty principle, which follows from an abstract analysis of the tenets of quantum mechanics. Herein, however, we reconsider the implications of Heisenberg's Uncertainty Principle for the simultaneous measurement of position and momentum. We show that one can, with a suitable modification of the Fourier transform (which reflects the specifics of the system mass and force constant), obtain a data analysis kernel that enables one to improve significantly the simultaneous measurement precision for position and momentum for the particular harmonic oscillator under study. Our results show that 1) the simultaneous precision for measuring the position and the corresponding momentum depends on the physical parameters of the harmonic oscillator under study 2) one can simultaneously squeeze coherent states by the same amount in both x and wave number, k. The results also suggest that each physical system may, in fact, determine its own optimum transform between representations of non-commuting observables so as to decrease their simultaneous measurement uncertainty limit.