I'm working with the Schrodinger equation
$$\frac{-\hbar ^2}{2m}\frac{\partial^2 \psi}{\partial x^2}=E \psi$$
and trying to use the Euler Method to approximate a wavefunction. The SEQ can be rewritten as
$$\psi '' = -\frac{2mE}{\hbar ^2}\psi$$
My question is 2-fold: How do I rewrite the SEQ as a set of first order differential equations? Am I using Euler Method correctly?
My attempt:
I think it is this, but I'm not sure:
Let $w=\psi '$ and $w' = \psi ''$. Now we have
$$w' = \frac{-2mE}{\hbar ^2}\psi$$
and to use Euler method it is
$$w_{i+1} = w_i + hw' \to \psi_{i+1}'=\psi_i'+h\psi_i''$$
$$\psi_{i+1}=\psi_i + hw \to \psi_{i+1} = \psi_i + h\psi_{i+1}'$$
$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ It's convenient to rewrite the original equation as two first order ones. Namely, $$ \varphi'\pars{x} = -k^{2}\psi\pars{x}\,,\quad \psi'\pars{x} = \varphi\pars{x}\,;\qquad k^{2} = {2mE \over h^{2}} $$ \begin{align} \totald{}{x}{\psi\pars{x} \choose \varphi\pars{x}} = \underbrace{\pars{\begin{array}{cc} \ds{0} & \ds{1} \\ \ds{-k^{2}} & \ds{0} \end{array}}}_{\ds{\mathsf{A}}}\ \underbrace{% {\psi\pars{x} \choose \varphi\pars{x}}}_{\ds{\Psi\pars{x}}} \implies \Psi\,'\pars{x} = \mathsf{A}\Psi\pars{x} \end{align}
$$ \bbx{\ds{\Psi_{n + 1} = \Psi_{n} + h\,\mathsf{A}\Psi_{n}\,,\qquad Euler}} $$