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I don't see that as truly essential; that depends on an additional
assumption, namely energy proportional to k^2. For electromagnetism
including optics, and for sound, and for other massless excitations,
the energy only goes like |k| to the first power.
*) I get almost 8000 hits from
https://www.google.com/search?q=%22quantum+dot%22+%22square+well%22
I think the data in figure 2 of this paper does a great job of showing the
decrease in spacing in energy as the quantum number increases, as well as the
change in energy of a particular state with well thickness:
"Quantum well states and short period oscillations of the density of states at
the Fermi level in Cu films grown on fcc Co(100) " P. Segovia, E.G. Michel,
J.E. Ortega Physical Review Letters 77 (16) (1996)
3455-3458<tel:(1996)%203455-3458>.