  In the `Many Interacting Worlds' (MIW) discrete Hamiltonian system
approximation of Schr\"odinger's wave equation, introduced in \cite{hall_2014},
convergence of ground states to the Normal ground state of the quantum harmonic
oscillator, via Stein's method, in Wasserstein-$1$ distance with rate
$\mathcal{O}(\sqrt{\log N}/N)$ has been shown in McKeague-Levin (2016),
Chen-Thanh (2023), McKeague-Swan (2023). In this context, we construct
approximate higher energy states of the MIW system, and show their convergence
with the same rate in Wasserstein-$1$ distance to higher energy states of the
quantum harmonic oscillator. In terms of techniques, we apply the `differential
equation' approach to Stein's method, which allows to handle behavior near
zeros of the higher energy states.
