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I'm having trouble understanding the legitimacy of solving the Schrödinger equation for a particle confined in an infinite square well. Aren't we supposed to solve it for the whole space and not just some region before we claim it to be the state of the particle?

Also what about the energy eigenvalues obtained this way? If we solve the eigenvalue equation $\hat E\psi=E\psi$ over a subset of whole space and obtain energy eigenvalues, what would guarantee that these eigenvalues are what one would get when one solves the energy eigenvalue equation over whole space?

Another trouble is for $x>L$ or for $x<0$ (the well is $0<x<L$), the energy eigenvalue equation would look something like $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{ dx^2}+V(x)\psi=E\psi\implies0+(\infty)(0)=(E)(0) $$ this equation looks horrendous to me given $E$'s are some finite numbers. What about the well boundaries $x=0$ and $x=L$, how do we know wavefunction would be continuous here?

Frobenius
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Arjun
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    Have you learned about the solutions for the finite square well? – Marius Ladegård Meyer Mar 22 '24 at 13:11
  • You can model the situation by working with the Hilbert space $L^2[0,L]$ and imposing the corresponding boundary conditions on the domain of the Hamiltonian. – Tobias Fünke Mar 22 '24 at 13:20
  • See e.g. this. – Tobias Fünke Mar 22 '24 at 13:22
  • @MariusLadegårdMeyer Nope..I guess you are going to say take the infinite potential well as a limiting case for the finite case..I think I would be fine with such a thing.But the question still remains whether what we do for the infinite case (solving S.E just inside the well saying $\psi$=0 at the boundaries and outside the well) just by itself is a valid method(I.e without taking the limits of the finite thing) – Arjun Mar 22 '24 at 13:43
  • @MariusLadegårdMeyer the infinite case has been taught before the finite case..then if I don't know the finite case solution..is there a way to justify what one does in the infinite case? – Arjun Mar 22 '24 at 13:47
  • @TobiasFünke Thanks for that reference..working in $L^2[0,L]$ makes sense and seems natural .But what about the stuff they do in books just working with $L^2(R)$ but still solving S.E only in a subset of R?Do you think it it wrong? – Arjun Mar 22 '24 at 14:03
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    I don't know what you mean. Can you give a reference? But yes, most intro QM books are a bit "sloppy", and by no means rigorous anyway. – Tobias Fünke Mar 22 '24 at 14:06
  • See https://physics.stackexchange.com/q/744609 – Hyperon Mar 22 '24 at 14:37
  • Indeed the limiting approach is one way to go about it. But to consider the case from a different perspective, consider the wave function of a guitar string tied at both ends. The "cost of vibration" past the two ends is infinite, so we disregard it. Notice this applies even if the string is physically there, having been tied to the ends with knots instead of terminating. – Marius Ladegård Meyer Mar 22 '24 at 18:38
  • @TobiasFünke see griffiths qm section 2.2 "The infinite square well" He says $\psi$=0 outside the well,but inside the well V(x)=0, hence inside the well $\psi=Asinkx+Bcoskx$ ..then he says $\psi$ is zero at the boundaries due to continuity and obtains values for k – Arjun Mar 22 '24 at 19:46

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