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When solving the TISE $$E\psi_E (x)=V(x)\psi_E (x)-\frac{\hbar^2}{2m}\psi_E ''(x)$$ in the position basis, the goal is to find all the values of $E$ for which the solution the TISE admits solutions that are either square-integrable (bound states) or non square-integrable but nonetheless a part of the "rigged Hilbert space" (scattering states).

There is a theorem (the proofs I've seen are not completely rigorous but the arguments do seem like something that can be made rigorous with sufficient mathematical analysis) that shows that for any normalizable solution/bound state, we must have $E \in [V_{min}, \min V(\pm \infty)]$. Moreover, clearly, given a solution $\phi_E(x)$, there is a clear prescription to determine whether it corresponds to a bound state or not: we simply check if $\int_{\mathbb{R}} |\phi_E|^2 dx \in \mathbb{R}^+$.

However, I still don't understand how exactly does one distinguish b/w non-normalizable solutions that are a part of the "rigged Hilbert space" and thus constitute 'generalized eigenfunctions' and non-normalizable solutions that must be rejected:

  • Do all non-normalizable solutions for $E \in \mathbb{R} - [V_{min}, \min V(\pm \infty)]$ constitute scattering states?
  • Is the set of energies $E$ for which there exists a scattering state always a continuous interval?
  • Can the non-normalizable solutions for $E \in [V_{min}, \min V(\pm \infty)]$ constitute scattering states?
  • Is there a general prescription to tell whether a particular function corresponds to a scattering state (What I've implicitly been doing is to rejected any unbounded solutions to TISE and call everything else a scattering state).

The main issue I'm having in resolving these issues is that I'm not too familiar with the mathematics that puts these notions of "Rigged Hilbert spaces" and "generalized eigenfunctions" on rigorous footing.

Vulgar Mechanick
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1 Answers1

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Solving TISE is finding the energies for which the equation can be solved, then using those energies we find the appropriate wave functions. This is exactly the same as diagonalizing the Hamiltonian operator. If you find a solution for an energy that is higher than the maximum of the potential this is called an unbound state. For an answer why is this spectrum continuous, check this out.

To answer your questions specifically:

  • Yes $\langle \psi| \psi \rangle $ diverging is required if the particle is not localized/unbound
  • For a scattering state, you don't have a constriction for any energy besides the bounds, so it must be continous
  • Like I said, non normalizable solutions of the TISE are scattering states
  • Finally, there is, all the solutions which are physical and can't be normalized/are in the continous spectrum, are the scattering states
Filip Pavicic
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