Piecewise constant potential

We can analytically find the energy eigenstates for a piecewise constant potential. Consider one constant interval, with potential $V_1$ . From the time independent Schrodinger equation, we have

\begin{align*} E \psi_1(x) &= -\frac{\hbar^2}{2m} \frac{\partial^2 \psi_1}{\partial x^2} + V_1 \psi_1(x) \\ \frac{\partial^2\psi}{\partial x^2} &= -\frac{2m(E-V_1)}{\hbar^2} \psi_1(x). \end{align*}

We see that if $E > V_1$ , the second derivative of $\psi_1$ is a negative constant times $\psi_1$ . This gives us a complex exponential solution $\psi_1 = Ae^{ikx} + Be^{-ikx}$ where $k^2 = \frac{2m(E-V_1)}{\hbar^2}$ .

If $E<V_1$ , then the second derivative is a positive constant times $\psi_1$ , giving us a real exponential solution $\psi_1 = Ae^{\kappa x} + Be^{-\kappa x}$ .

At steps in the potential, we have the condition that both $\psi$ and $\psi'$ be continuous.