PX262 - B2 - time-independent schrödinger equation

for systems in which the potential is independent of time, ie: $V (x, t) = V (x)$ , using separation of variables:

\begin{matrix} ψ (x, t) = ϕ (x) T (t) \\ i ℏ \frac{\partial}{\partial t} [ϕ (x) T (t)] = \frac{- ℏ^{2}}{2 M} \frac{\partial^{2}}{\partial x^{2}} [ϕ (x) T (t)] + V (x) ϕ (x) T (t) \\ ϕ i ℏ \frac{\partial T}{\partial t} = - \frac{ℏ^{2}}{2 M} T \frac{\partial^{2} ϕ}{\partial x^{2}} + V ϕ T \\ \frac{i ℏ}{T} \frac{\partial T}{\partial t} = - \frac{ℏ^{2}}{2 M} \frac{1}{ϕ} \frac{\partial^{2} ϕ}{\partial x^{2}} + V \end{matrix}

\begin{aligned} i ℏ \frac{d T}{d t} & = E T \\ ⟹ T (t) & = \exp (- \frac{i E t}{ℏ}) \end{aligned}

- universal time dependence in systems where potential is time independent

$R H S :$
$- \frac{ℏ^{2}}{2 M} \frac{d^{2} ϕ}{d x^{2}} + V ϕ = E ϕ$
- this is the time-independent schrödinger equation
for a free particle, $V (x) = 0 :$
$ϕ (x) = A \exp (i k x)$
where, $k = \sqrt{\frac{2 M E}{ℏ^{2}}}$
full wave function:
$ψ (x, t) = A \exp [- (k x - ω t)]$