Question

Quantum Mechanics (but the question is about calculus): I was studying the proof of the conservation of ∫|ψ|²d³r (ψ is the wave function) but I don't understand the following step, in which the divergence theorem is used.

Answer 1

The following integral is performed over the whole space and ψ* is the complex conjugate of ψ:

\[\int\limits -i \hbar {\partial{| \psi|^2} \over \partial t} d^3r = - {\hbar^2 \over 2m} \int\limits \nabla \cdot [ \psi^* \nabla \psi- \psi \nabla \psi^*] d^3r
\]

Now, I don't understand this step:
\[
\int\limits \nabla \cdot [ \psi^* \nabla \psi- \psi \nabla \psi^*] d^3r =0
\]

QUESTION: How has the divergence theorem been used?