  The Leggett-Garg (LG) inequalities were introduced to test for the possible
presence of macroscopic quantum coherence. Since such effects may be found in
various types of macroscopic oscillators, we consider the application of the LG
approach to the one-dimensional quantum harmonic oscillator and also to more
general bound systems, using a single dichotomic variable $Q$ given by the sign
of the oscillator position. We present a simple method to calculate the
temporal correlation functions appearing in the LG inequalities for any bound
system for which the eigenspectrum is (exactly or numerically) known. We apply
this result to the quantum harmonic oscillator for a variety of experimentally
accessible states, namely energy eigenstates, and superpositions thereof. For
the subspace of states spanned by only the ground state and first excited
state, we readily find substantial regions of parameter space in which the LG
inequalities at two, three and four times can each be independently violated or
satisfied. We also find that the violations persist, although are reduced, when
the sign function defining $Q$ is smeared to reflect experimental imprecision.
For higher energy eigenstates, we find that LG violations diminish, showing the
expected classicalization. With a $Q$ defined using a more general type of
position coarse graining, we find two-time LG violations even in the ground
state. We also show that two-time LG violations in a gaussian state are readily
found if the dichotomic variable at one of the times is taken to be the parity
operator. To demonstrate the versatility of the approach, we repeat much of the
LG analysis for the Morse potential, finding qualitatively similar physical
results.
