= Schrödinger equation for a one dimensional particle
{c}
We select for the general <equation Schrodinger equation>:
* $\vv{x} = x$, the linear <cartesian coordinate> in the x direction
* $\hat{H} = -\frac{\hbar^2}{2m}\pdv{^2}{x^2} + V(x, t)$, which analogous to the sum of <kinetic energy>[kinetic] and <potential energy> in <classical mechanics>
giving the full explicit <partial differential equation>:
$$
i\hbar\pdv{\psi(x, t)}{t} = \left[ -\frac{\hbar^2}{2m}\pdv{^2}{x^2} + V(x, t) \right]\psi(x, t)
$$
{title=<Schrödinger equation> for a one dimensional particle}
The corresponding <time-independent Schrödinger equation> for this equation is:
$$
\left[-\frac{\hbar^2}{2m} \pdv{^2}{x^2} + V(x)\right]\psi(x) = E \psi(x)
$$
{title=<time-independent Schrödinger equation> for a one dimensional particle}