# Quantum inequalities in quantum mechanics

December 17, 2003

We study a phenomenon occuring in various areas of quantum physics, in which
an observable density (such as an energy density) which is classically
pointwise nonnegative may assume arbitrarily negative expectation values after
quantisation, even though the spatially integrated density remains nonnegative.
Two prominent examples which have previously been studied are the energy
density (in quantum field theory) and the probability flux of rightwards-moving
particles (in quantum mechanics). However, in the quantum field context, it has
been shown that the magnitude and space-time extension of negative energy
densities are not arbitrary, but restricted by relations which have come to be
known as `quantum inequalities'. In the present work, we explore the extent to
which such quantum inequalities hold for typical quantum mechanical systems. We
derive quantum inequalities of two types. The first are `kinematical' quantum
inequalities where spatially averaged densities are shown to be bounded below.
Such quantum inequalities are directly related to Garding inequalities. The
second type are `dynamical' quantum inequalities where one obtains bounds from
below on temporally averaged densities. We derive such quantum inequalities in
the case of the energy density in general quantum mechanical systems having
suitable decay properties on the negative spectral axis of the total energy.
Furthermore, we obtain explicit numerical values for the quantum inequalities
on the one-dimensional current density, using various spatial averaging weight
functions. We also improve the numerical value of the related `backflow
constant' previously investigated by Bracken and Melloy. In many cases our
numerical results are controlled by rigorous error estimates.

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