Schrödinger Equation

Figure 8 illustrates the final application we shall consider in this set of notes, finding the probability ${\mathcal P}(x)$ for the energy states of a particle of mass $m$ and energy $E$ moving along the $x$-axis under the influence of a force $F(x)$ which we describe by the usual potential energy $U(x) \equiv - \int F\,dx$. Unlike the particle in a box, this problem cannot be solved by simple analogy to classical waves. It requires our general approach to new problems of (1) identifying the degrees of freedom, (2) finding the equation of motion, (3) solving the equation of motion. The next set of notes explores how to solve the equation of motion. Here, we shall complete the first two of these phases.

Figure 8: Realization of particle of mass $m$ with energy $E$ moving in potential $U(x)$.