PX156 - C1 - the schrodinger equation

in $1962$ , erwin schrodinger postulated that free particle of mass, $m$ , in 1D case, given by the schrodinger equation:

- \frac{ℏ^{2}}{2 m} \frac{\partial^{2} ψ}{\partial x^{2}} + V ψ = i ℏ \frac{\partial ψ}{\partial t}

where, $ψ =$ wave function, $ψ (x, t)$

this is the time dependent schrodinger equation (TDSE), as it uses $\frac{\partial ψ}{\partial t}$ , $\frac{\partial ψ}{\partial x}$
let $ψ_{1} = \exp (i k x - i ω t) :$

\begin{aligned} - \frac{ℏ}{2 m} (i k)^{2} \exp (i k x - i ω t) & = i ℏ (- i ω) \exp (i k x - i ω t) \\ (\frac{ℏ^{2} k^{2}}{2 m} - ℏ ω) ψ & = 0 \end{aligned}

ℏ ω (k) = \frac{ℏ^{2} k^{2}}{2 m}