PX262 - N1 - klein-gordon equation

looking at the schrodinger equation for a free particle:

i ℏ \frac{\partial ψ}{\partial t} = - \frac{ℏ^{2}}{2 m} \nabla^{2} ψ

the plane wave solution:

\begin{matrix} ψ = A \exp (i (\frac{\vec{p} \cdot \vec{r} - E t}{ℏ})) \\ \vec{p} = ℏ \vec{k} if E = \frac{p^{2}}{2 m} \end{matrix}

this is a classical result
trying to get a wave equation where $E^{2} = p^{2} c^{2} + m^{2} c^{4}$ comes out, ie. the relativistic mechanics result

\begin{matrix} E \to i ℏ \frac{\partial}{\partial t} \\ \vec{p} \to - i ℏ \vec{\nabla} \end{matrix}

- ℏ^{2} \frac{\partial^{2} ψ}{\partial t^{2}} = - ℏ c^{2} \nabla^{2} ψ + m^{2} c^{4} ψ

this is called the klein-gordon equation

\begin{matrix} ψ = A \exp (i (\frac{\vec{p} \cdot \vec{r} - E t}{ℏ})) \\ E^{2} = p^{2} c^{2} + m^{2} c^{4} \\ E = \pm \sqrt{p^{2} c^{2} + m^{2} c^{4}} \end{matrix}

in classical mechanics, the negative energy solutions can be discarded
in quantum mechanics, all the solutions are needed to form a complete set of states
the probability density, $ρ (\vec{r}, t) = ψ^{*} ψ$ , is found to be proportional to the energy, which is unphysical for the negative energy solutions