Answer
$k$ will be positive and real if $U_0$ is greater than $E$ in the region $x > L$. Mathematically this is an acceptable solution. But physically this is unrealistic. The integral of the probability density should be unity. But it increases exponentially with x and this is impossible.
Work Step by Step
We have already proved that $\psi= De^{2kx}$ is a solution of Schrödinger’s equation in its one dimensional form, where D is a constant and k is positive provided $k=\frac{\pi}{h}\sqrt{2m(U_0-E)}$.
$k$ will be positive and real if $U_0$ is greater than $E$ in the region $x > L$. Mathematically this is an acceptable solution. But physically this is unrealistic. The integral of the probability density should be unity. But it increases exponentially with x and this is impossible.
