  Due to the exponential growth of the state space of coupled quantum systems
it is not possible, in general, to numerically store the state of a very large
number of quantum systems within a classical computer. We demonstrate a method
for modelling the dynamical behaviour of measurable quantities for very large
numbers of interacting quantum systems. Our approach makes use of a symbolic
non-commutative algebra engine that we have recently developed in conjunction
with the well-known Ehrenfest theorem. Here we show the possibility of
determining the dynamics of experimentally observable quantities, without
approximation, for very large numbers of interacting harmonic oscillators. Our
analysis removes a large number of significant constraints present in previous
analysis of this example system (such as having no entanglement in the initial
state). This method will be of value in simulating the operation of large
quantum machines, emergent behaviour in quantum systems, open quantum systems
and quantum chemistry to name but a few.
