Expressing complementarity and the x-p commutation relation through further quantum inequalities

Abstract

Complementarity and the commutation relation of position (x) and momentum (p) imply much more than the fundamental x-p uncertainty inequality. Here, we display some further consequences of the former that could have certain pedagogical interest and, so, contribute to the teaching of quantum mechanics. Inspired by an elementary derivation of the x-p uncertainty inequality, based upon a positive quadratic polynomial, we explore one possible extension, via quartic polynomials and simple algebra and integrations. Our analysis, aimed at providing some further pedagogic expression of genuine quantum behaviours, yields other quantum inequalities for expectation values, expressed through suitable discriminants associated with quartic algebraic equations, which differ from (and are not a strict consequence of) the x-p uncertainty inequality. Those quantum inequalities are confirmed, and genuine non-classical behaviours are exhibited, for simple cases: a harmonic oscillator, a hydrogenic atom and free Gaussian wave packets. The physical interest of the expectation values involved in the quantum inequalities and of the latter is discussed, in the framework of quantum optics and squeezing phenomena.