How External Boundaries Affect the Number of Bound States in a Quantum Well

This study examines the number of bound (negative energy) states in one-dimensional quantum wells of width w confined within an infinite square well (ISW) of width W. Six different quantum wells were explored: a finite square well and one to five delta wells inside the ISW. These systems admit zero-energy states with piecewise linear eigenfunctions for certain combinations of the ratio r = W/w and the strength (α) of the delta wells or depth (V₀) of the finite square well. Using these zero-energy conditions, we can determine how the number of negative-energy states changes when the ratio r is changed. We determine the zero-energy conditions for each case analytically. The zero-energy curves divide the parameter space (α/V₀ vs. r) into regions with different numbers of negative energy states. Passing over a zero-energy curve changes the number of negative energy states by one. For the systems with the finite square well, the single delta well, and the two delta wells inside the ISW, the number of bound states changes by at most one when changing r alone. For the systems containing three, four, or five delta wells inside the ISW, the number of bound states changes by at most two when changing r alone.