Monday, February 25, 2019

quantum mechanics - Normalizable wavefunction that does not vanish at infinity


I was recently reading Griffiths' Introduction to Quantum Mechanics, and I stuck upon a following sentence:



but $\Psi$ must go to zero as $x$ goes to $\pm\infty$ - otherwise the wave function would not be normalizable.



The author also added a footer: "A good mathematician can supply you with pathological counterexamples, but they do not arise in physics (...)".


Can anybody give such a counterexample?




Answer



Take a gaussian (or any function that decays sufficiently quickly), chop it up every unit, and turn all the pieces sideways.


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