Potential Scattering #

In this section, we will address scattering problems. Although primarily concerned with quantum scattering, we will also deal with classical scattering problems in some cases.

1D-TISE

Numerical Method for One-dimensional Time-independent Schrödinger Equation # This section describes the numerical solution methods for the one-dimensional time-independent Schrödinger equation. Finite Difference Finite Difference Method # The finite difference method approximates differential equations with difference equations for numerical solutions. A web tool implementing this method is available at the link below: Solving the 1D Time-Independent Schrödinger Equation Using the Finite Difference Method 1. Schrödinger Equation # Consider the Schrödinger equation for a quantum system in atomic units ($m=1, \hbar=1$): $$ \begin{align}\label{eq1} \left[-\frac{1}{2}\frac{d^2}{dx^2}+V(x)\right]\psi(x)=E\psi(x) \end{align} $$ ここで、$V(x)$はポテンシャル、$E$はエネルギー固有値を表します。また、$x$は区間$[x_a, x_b]$で定義され、境界条件として固定端条件を考えます。つまり、 where ( V(x) ) is the potential, ( E ) is the energy eigenvalue, and ( x ) is defined in the interval ([x_a, x_b]) with fixed boundary conditions: