To my understanding if I have a finite barrier with potential $V(x)>E$, then to the left of the barrier, the wavefunction can be represented as two exponentials:
$$\psi= e^{(ik_{left} x)} + e^{-(ik_{left} x)}$$
Where the negative exponent represents a wave travelling in the opposite direction. Firstly is this correct, or me making things up?
So in the barrier, as $V>E$, $k$ is imaginary so we get a solution of the form:
$$\psi = e^{-k x}$$
i.e. exponential decay.
However I can't understand why it shouldn't be reflected upon leaving the finite barrier. If this is so, then it would produce a similar wave but of negative exponent to the incident wave, i.e. a wave which is exponentially growing from left to right? Is this correct? Is this allowed?