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I was in my undergraduate QM lecture and we just finished with the Dirac barrier. My question is as follows: We know that the Dirac well’s complete set of solutions requires one bound state and an infinite set of scattering states (plane waves). The solutions to the Dirac barrier are a only a set of scattering states (plane waves again). How can both sets of solutions be complete? I asked this question in lecture and was told that no one-to-one correspondence exists between the two sets of plane waves, so the comparison is improper, but I’ve been unable to find anything else on the subject.

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  • $\begingroup$ Can you expand on your reasoning as to why you think they shouldn't be complete? $\endgroup$ – BioPhysicist Nov 14 '18 at 2:34
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I'm not quite sure what OP asks, but the potential profile $V$ clearly matters. E.g. if $\lim_{|x|\to \infty}V(x)=\infty$, such as, the harmonic oscillator, or the infinite well, then there are no scattering states at all!

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