How to solve this differential equation for $\psi_n$?:
$$\frac{1}{2}\frac{\partial^2}{\partial x^2}\psi_n=\lambda_n\psi_n$$
apparently this is a heat equation but I cannot find information on this. Any help is much appreciated. thanks.
EDIT
The boundary conditions are for initial and terminal, respectively, $\psi_n(x_0)=0$, and $\psi_n(x_T)=0$.
No, this is not "a heat equation", but solving it (with appropriate boundary conditions) is typically one of the steps in solving a heat equation boundary value problem using separation of variables. The general solution of your equation is $\psi_n = c_1 \exp(\sqrt{2\lambda_n} x) + c_2 \exp(-\sqrt{2\lambda_n} x)$.