Hall, Deckert and Wiseman (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical “worlds.” A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured by these authors and proven by McKeague and Levin (2015, arXiv:1412.1563) using Stein’s method. In this talk we discuss new connections between Stein’s method and Many Interacting Worlds theory. In particular, we show that quantum position probability densities for higher energy levels beyond the ground state arise as distributional fixed points in a new generalization of Stein’s method. These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution. The talk is based on joint work with Erol Pekoz and Yvik Swan.