As we all know, the probability current density in quantum mechanics is defined as: $$\textbf{J}=\dfrac{\hbar}{2mi}(\Psi^* \nabla \Psi-\Psi \nabla \Psi^*)$$ For simplicity let us work in one dimension and let us suppose a wave function $\Psi= A\ \text{cos}\ {kx}$. Applying the above definition an...

Suppose I have a fan blowing air. By Bernoulli's equation, the air pressure in the stream is lower than the air in the surrounding. Here is my confusion: the moment I turn on the fan, the fan starts pumping extra air into the space in front of it and the air pressure there should actually be high...