The General Solution for a Constant Potential

We have found the general solution of the Schrödinger Equation in a region in which the potential is constant. Assume the potential is equal to $V_0$ and the total energy is equal to $E$. Assume further that we are solving the time independent equation.

$$\bgroup\color{black} {-\hbar^2\over 2m}{d^2u(x)\over dx^2}+V_0u(x)=Eu(x) \egroup$$

$$\bgroup\color{black} {d^2u(x)\over dx^2}=-{2m(E-V_0)\over\hbar^2}u(x) \egroup$$

For $E>V_0$, the general solution is

$$\bgroup\color{black} u(x)=Ae^{+ikx}+Be^{-ikx} \egroup$$

with $k=\sqrt{2m(E-V_0)\over\hbar^2}$ positive and real. We could also use the linear combination of the above two solutions.

$$\bgroup\color{black}u(x)=A\sin(kx)+B\cos(kx)\egroup$$

We should use one set of solutions or the other in a region, not both. There are only two linearly independent solutions.

The solutions are also technically correct for $E<V_0$ but $k$ becomes imaginary. For simplicity, lets write the solutions in terms of $\kappa=\sqrt{2m(V_0-E)\over\hbar^2}$, which again is real and positive. The general solution is

$$\bgroup\color{black} u(x)=Ae^{+\kappa x}+Be^{-\kappa x} .\egroup$$

These are not waves at all, but real exponentials. Note that these are solutions for regions where the particle is not allowed classically, due to energy conservation; the total energy is less than the potential energy. Nevertheless, we will need these solutions in Quantum Mechanics.