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Are there any techniques in guessing the ground state wave function in any given potential?

For example, for a given potential like $$ \frac{1}{1-x^2}$$ or $$ \frac{1}{1-x^3}~?$$ I know wave function is sine or cosine inside the well and outside would be like exponential growth or decay depends on which side. Could I find the turning point by equating kinetic energy, $$ \frac{1}{2} m x'^2 ~?$$

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    $\begingroup$ Have a look at the WKB approximation: en.wikipedia.org/wiki/… $\endgroup$ – Gert Dec 4 '15 at 0:38
  • $\begingroup$ Technically speaking, the WKB approximation is for highly exited states/continuum limit (rather than the ground state), cf. e.g. this Phys.SE post. $\endgroup$ – Qmechanic Dec 4 '15 at 11:41

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