Why there is no Gibb's phenomenon in QM? Why we don't see any Gibb's phenomenon in quantum mechanics?
EDIT
At sharp edges (discontinuities), we usually find ringing. This can be observed in many physical phenomenon (eg. shock waves). Naturally, whenever there is a shrap discontinuity in wave functions, I'd expect a ringing in the probability of finding a particle around that edge.
 A: First of all, the Gibbs phenomenon is a mathematical effect – it is the appearance of narrow but intense oscillations around the right value whenever a function with a discontinuity is approximated by its Fourier expansion that is truncated.
This phenomenon doesn't occur when the relevant function is a wave function $\psi(x,y,z)$ in quantum mechanics for a simple reason: the wave functions don't have such discontinuities. If a wave function had a jump of this sort, then its derivative $\psi'$ or $\nabla x$ would have a $\delta$-function at the point of the jump, and its square would integrate to infinity ($\delta^2$ is infinitely times greater than just $\delta$). 
But this integral is proportional to a formula for the expectation value of the kinetic energy 
$$\int_{-\infty}^\infty |\psi'|^2 dx = -\int_{\infty}^\infty \psi^* \psi'' dx$$
by integration by parts so whenever it diverges, it means that the energy is infinite which is physically impossible. That's why real-world, finite-energy wave functions can't have jump-like discontinuities as a function of spatial coordinates although their first derivatives are already allowed to have such jumps.
A: Remember that
$$p\Psi=-i\hbar\nabla\Psi $$  
Also,          
$$E\Psi=i\hbar\partial_t\Psi $$    
If there is discontinuity in the wavefunction, the 4-momentum would be infinite.
