The Klein-Gordon equation and its application
The Einstien's mass-energy dispersion relation be$$E^2=c^2P^2+{m_0}^2c^4\space\space(1)$$
Now in Quantum mechanics
$$E\rightarrow i\hbar\frac{\partial}{\partial t} \Rightarrow E^2 \rightarrow -\hbar^2\frac{\partial^2}{\partial t^2}$$and
$$P\rightarrow -i\hbar\nabla\Rightarrow P^2\rightarrow -\hbar^2 \nabla^2$$
In this fashion we can write
$$ -\hbar^2\frac{\partial^2}{\partial t^2}\psi=-\hbar^2c^2 \nabla^2\psi+{m_0}^2c^4$$
$$\Rightarrow \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi-\nabla^2\psi+\frac{{m_0}^2c^2}{\hbar^2}\psi=0\Rightarrow \Box\psi+\frac{{m_0}^2c^2}{\hbar^2}\psi=0$$
This is the Klein-Gordon equation. This equation solutions is $$Ae^\frac{i p^\mu x_\mu}{\hbar}=Ae^\frac{i \left(Et-\mathbf{p\cdot x}\right)}{\hbar}$$
Where A is a normalization constant.
Now substituting this solution imto Klein-Gordon equation gives $$E^2=c^2P^2+{m_0}^2c^4\Rightarrow E=\pm\sqrt{c^2P^2+{m_0}^2c^4}$$Therefore Klein-Gordon equation gives negative energy solution as well as positive energy.
Now in Quantum mechanics
$$E\rightarrow i\hbar\frac{\partial}{\partial t} \Rightarrow E^2 \rightarrow -\hbar^2\frac{\partial^2}{\partial t^2}$$and
$$P\rightarrow -i\hbar\nabla\Rightarrow P^2\rightarrow -\hbar^2 \nabla^2$$
In this fashion we can write
$$ -\hbar^2\frac{\partial^2}{\partial t^2}\psi=-\hbar^2c^2 \nabla^2\psi+{m_0}^2c^4$$
$$\Rightarrow \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi-\nabla^2\psi+\frac{{m_0}^2c^2}{\hbar^2}\psi=0\Rightarrow \Box\psi+\frac{{m_0}^2c^2}{\hbar^2}\psi=0$$
This is the Klein-Gordon equation. This equation solutions is $$Ae^\frac{i p^\mu x_\mu}{\hbar}=Ae^\frac{i \left(Et-\mathbf{p\cdot x}\right)}{\hbar}$$
Where A is a normalization constant.
Now substituting this solution imto Klein-Gordon equation gives $$E^2=c^2P^2+{m_0}^2c^4\Rightarrow E=\pm\sqrt{c^2P^2+{m_0}^2c^4}$$Therefore Klein-Gordon equation gives negative energy solution as well as positive energy.
