Editing Bound state (section)

===Position-bound states===
Consider the one-particle Schrödinger equation. If a state has energy <math display="inline"> E < \max{\left(\lim_{x\to\infty}{V(x)}, \lim_{x\to-\infty}{V(x)}\right)}</math>, then the wavefunction {{mvar|ψ}} satisfies, for some <math>X > 0</math>

:<math>\frac{\psi^{\prime\prime}}{\psi}=\frac{2m}{\hbar^2}(V(x)-E) > 0\text{ for }x > X</math>

so that {{mvar|ψ}} is exponentially suppressed at large {{mvar|x}}. This behaviour is well-studied for smoothly varying potentials in the [[WKB approximation]] for wavefunction, where an oscillatory behaviour is observed if the right hand side of the equation is negative and growing/decaying behaviour if it is positive.<ref>{{Cite book |last=Hall |first=Brian C. |title=Quantum theory for mathematicians |date=2013 |publisher=Springer |isbn=978-1-4614-7115-8 |series=Graduate texts in mathematics |location=New York Heidelberg$fDordrecht London |page=316-320}}</ref> Hence, negative energy-states are bound if <math>V(x)</math> vanishes at infinity.