Why wavefunction becomes exponentially smaller during quantum tunneling? I am interested in quantum tunneling and I am wondering why the wavefunction of a particle would becomes smaller so that there is a slight possibility of finding it at the other side of a big energy barrier? Is there any interaction otherwise how can the wavefunction knows there is a barrier? 
 A: Wavefunctions are solutions of quantum mechanical  differential equations, with given boundary conditions for the problem at hand, i.e. tunneling:


Is there any interaction otherwise how can the wavefunction knows there is a barrier?

The boundary condition of a barrier defines the wavefunction by construction. It has been found that experiments validate this model and its predictions.
A: The classic step barrier is the easiest to solve but the general behavior holds for other functions.  The comment addresses the physics of your question, i.e. "the presence of a potential barrier is what defines an interaction".  This is the same for classical mechanics.  One could ask, "How does the earth know to move in an ellipse when gravity is there, is there an interaction"?
Specific details come from the math.  Schrodinger's equation is set up in each interval of the x-axis (assuming a 1-dim problem) with the appropriate value of V(x) in each region.  If the total energy is greater than the barrier there will be a modification of the wavelength in the region of the barrier.  If the total energy of the particle is less than the barrier then the wave number will become imaginary, cancelling the imaginary factor, i, in the exponential leading to a decaying solution.  If the barrier continues to exist for all space after it is encountered the decay will eventually cause the wave function to vanish, this is the phenomenon of penetration.  If the barrier is finite in extent then the particle will make it out the other side, tunneling.  It is fairly easy to derive the exact solution for this case since the potential is constant, they are exp(+ikx) and exp(-ikx), with appropriate boundary conditions.   
