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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