The physical significance of probability current density (PCD) is usually given as a local conservation of probability. The time rate of reduction of probability at a point is accompanied by a local outward PCD.
In a momentum eigen state, say $\phi(x)=Ae^{ik_0x}$, the probability density $P=|\phi(x)|^2=|A|^2$, a constant. So $\frac{\partial}{\partial t}P=0.$ Applying the formula for PCD, we get $$j_x=\frac{\hbar}{2im}\left[\psi^*\frac{\partial\psi}{\partial x}-\psi\frac{\partial\psi^*}{\partial x}\right]=\frac{\hbar k_0}{m}|A|^2$$
In a way, the existence of non-zero $j_x$ makes sense as $\phi(x)$ is a non-zero momentum state. In the other way again, it does not make sense because if probability flows from one point to another, how come it remain constant at all point? I mean, if probability flows from a point A to another B, that intuitively suggests that $P$ is reduced at A and increased by the same amount at B, which is not going with P being constant everywhere. Where is the catch?
Is there something that I am missing?
Looking for help
