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I'm doing a project with lead (II) sulfide quantum dots because they have optical transitions in the near infrared. I understand the simple "particle in a box" approach where confinement strengthens exponentially when the box shrinks.

What I don't get is how this confinement results in discrete energy levels? Can someone tell me how this shrinking of molecular size and confinement of the electron hole pair (exciton Bohr radius) results in discrete energy levels? Is it the electron shells being changed?

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As a rule of dumb, you can have a continuous spectrum if electrons are delocalized. This happens (i) for extended systems (like solids) or (ii) for unbound electrons. In both cases the electrons move in an infinite space, and due to position momentum commutator rules and the fact that momentum is related to energy, one can have a continuum spectrum.

Another way of looking into it is that in a confined space you can use a discrete basis set to describe your wave-functions. Since the eigenstate of the hamiltonian are a possible basis set, then the eigen-states are discrete.

Quantum dots are an example of a way to localize electrons and thus to get discrete energy levels for the bound part of the spectrum. Same happens in a box (where all electrons are bound by a potential with an infinite barrier).

See also the discussion in this post.

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