One-Body Problems without Spin

One-dimensional problems, though in a sense oversimplifications, may be used with advantage in order to understand the essential features of quantum mechanics. They may be derived from the three-dimensional wave equation, (A.1)$$ - {\mkern 1mu} \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2}\psi = + V(x,t)\psi - \frac{\hbar }{i}\frac{{\partial \psi }}{{\partial t}},$$if the potential depends upon only one rectangular coordinate x, by factorization: (A.2)$$\psi = {e^{i({k_2}y + {k_3}z)}}{\mkern 1mu} \varphi (x,t).$$