From Griffiths, if we have some potential $V(x)$ and energy $E$ such that $E=V(0)$ where $V(x)<E$ for all $x<0$ and $V(x)>E$ for all $x>0$. In the patching region, Griffiths uses only one of the asymptotic Airy functions: $$\psi_{patch} = a\mathrm{Ai}(\alpha x)$$ instead of the general linear combination of both Airy functions. What is the justification behind this? I tried deriving the connection formulas without ignoring the second airy function, and I end up with two independent equations with 3 unknowns so I'm not sure why Griffiths is allowed to simply 'choose' his patching function.

  • $\begingroup$ Related: physics.stackexchange.com/q/281816/2451 and links therein. $\endgroup$ – Qmechanic May 11 '18 at 4:23
  • $\begingroup$ Make sure you understand where [8.39] and [8.40] are coming from. Vanishing of the coefficient in front of the second Airy function follows from matching these two expressions. $\endgroup$ – mavzolej May 11 '18 at 5:52
  • $\begingroup$ To make the question more concise (so that potential answerers don't have to repeat the whole argument in Griffiths), consider to add equation number, page, edition. $\endgroup$ – Qmechanic May 11 '18 at 11:52

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