We outline a procedure for using matrix mechanics to compute energy eigenvalues and eigenstates for two and three interacting particles in a confining trap, in one dimension. Such calculations can bridge a gap in the undergraduate physics curriculum between single-particle and many-particle quantum systems, and can also provide a pathway from standard quantum mechanics course material to understanding current research on cold-atom systems. In particular, we illustrate the notion of “fermionization” and how it occurs not only for the ground state in the presence of strong repulsive interactions, but also for excited states, in both the strongly attractive and strongly repulsive regimes.

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See Figs. 4 and 5 in Ref. 12, as the dimensionless range parameter b/a approaches zero.

16.
See supplementary material at http://dx.doi.org/10.1119/1.4985063 for a collection of MATLAB routines used to perform many of the calculations described in this paper.
17.
Beware, however, that different conventions are followed by physicists and mathematicians. We use the physicists' convention, where Hn(z) has a coefficient of zn equal to 2n, i.e., H0(z) = 1, H1(z) = 2z, H2(z) = 4z2–2, etc.
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Supplementary Material

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