In a Tunneling problem, if the $E_o<V$, we can show that the scattering wavefunction inside a rectangular barrier is a decaying exponential. The solution being real implies that the probability current $J$ is zero at every point inside the barrier.
From equation of continuity,
$\frac{\partial|\psi|^2}{\partial t}=0$
Does this mean that the probability density at each point inside the barrier is fixed? (Clearly it can't be so). How exactly is this to be interpreted?
Does this mean that the probability density at each point inside the barrier is fixed?
Yes.
(Clearly it can't be so). How exactly is this to be interpreted?
The solution you mentioned (time-independent decaying exponential) is a solution of the time-independent Schroedinger equation $$ \hat{H}\psi = E\psi $$ Since the solutions of this equation depend on time only via factor $e^{\frac{Et}{i\hbar}}$, the resulting probability density $|\psi|^2$ is time-independent.
To obtain time-dependent probability density and non-zero probability current, you can attempt to solve time-dependent Schroedigner equation $$ \partial_t\psi = \frac{1}{i\hbar}\hat{H}\psi $$ for initial condition $\psi(x,t_0)$ that is no equal to above $\psi$ function, for example some localized wave packet. But this is quite difficult and may require numeric computer calculation, so it rarely occurs in textbooks.
