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Most proofs I see that all bound states of a quantum system are non-degenerate use the wave-function representation. How can we prove it using only Dirac notation, without resorting to the wave picture?

Phineas Nicolson
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    Do you mean non-degenerate in your question body? Also, $\psi(x)=\langle x|\psi\rangle$, so couldn't you say the arguments are the same, or maybe I am missing something. Perhaps you could expand on both what you have seen and what you want to be shown. – BioPhysicist Nov 26 '18 at 04:48
  • Yes, my bad. I've already fixed it. – Phineas Nicolson Nov 26 '18 at 04:50
  • Perhaps you might translate as much of this related question to observe that your are very much considering the space dependence of the potential and the wave function involved and wf normalizability. The question is why on earth would you like to abstract away from space features? – Cosmas Zachos Nov 26 '18 at 17:20
  • I believe the question would be why not? – Phineas Nicolson Nov 26 '18 at 17:39
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    Why not hike the Alps in flipflops? – Cosmas Zachos Nov 27 '18 at 01:06
  • I mean I want to learn how to. – Phineas Nicolson Nov 27 '18 at 02:56
  • Seriously, my point is that for TISE x and p are very asymetric, and it is all about x properties of the potentials and their functional analysis. Abstracting further to include possible velocity-dependent potentials, which would be the only excuse for the bra-ket generalization would muddy the waters instead of illuminating it. The Messiah QM text overcomplete treatment reminds you why. – Cosmas Zachos Nov 27 '18 at 15:02
  • I don't want to include velocity-dependent potentials, I just want to know how to prove it in bra-ket notation. – Phineas Nicolson Nov 27 '18 at 15:31

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