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Say the potential is defined to be 0 outside of the well, and -V0 inside it.

Analyzing the case when -V0 < E < 0, one finds that the probability of finding the particle outside the well is not zero. This result means that a particle might have a negative kinetic energy - a situation which is impossible from a classical point of view.

So if such a situation is acceptable, why won't QM allow a situation where E < -V0 ?

Why a particle with such energy cannot exist, if we permit negative kinetic energy?