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[tex]\psi = \exp ( - a{x^2})[/tex] (neglecting nomalization)

And we have a detector far in space monitoring the local probability of finding a particle. Now if we suddenly turn off the harmonic potential, the wavefunction will evolve as free particle, and

[tex]\Psi (x,t) = \frac{{\exp (\frac{{ - a{x^2}}}{{1 + 2iat/m}})}}{{\sqrt {1 + 2iat/m} }}[/tex].

We see no matter how far the detector is, the local probability will start to change immediately after we turn off the potential. So will the detector record a change in number of particles detected? If so, it seems there's a superluminal signal transmitted since the detector could be very far from the origin .