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Quantum field theory in curved spacetime (QFTCS) is a field of study that focuses on problems that arise when considering a quantum field on a fixed, curved spacetime. It allows the study of quantum effects in strong gravitational fields, and has led to many interesting conclusions, such as the Unruh effect and the Hawking effect.
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How does the quantum wave function behave within a black hole?
There is no answer yet to your question. This is the problem of unification of gravity and quantum mechanics. See, e.g., string theory.
P.S.: no one knows what happens at the singularity. See, infor …