Klein--Gordon equation

On the one hand, is the equation of a free scalar field in a relativistic context. See Relativistic Field Theory:

\frac{\partial^{2} ϕ}{\partial t^{2}} - \frac{\partial^{2} ϕ}{\partial x^{2}} - \frac{\partial^{2} ϕ}{\partial y^{2}} - \frac{\partial^{2} ϕ}{\partial z^{2}} + m^{2} ϕ = 0.

By rescaling the constants:

◻ ϕ + m^{2} ϕ = 0,

where $◻ = \frac{\partial^{2}}{\partial t^{2}} - c^{2} \nabla^{2}$ is the d'Alembertian operator in one spatial dimension.

Also, it comes from the continuous limit of several harmonic oscillators.

This equation could be also obtained from the Hamiltonian point of view, see this note.

On the other, it appears when one try to quantize the four-momentum equation in special relativity, in a attempt to create a relativistic version Schrodinger equation for the wavefunction of a relativistic free particle. We have the equation

E^{2} = (p_{x}^{2} + p_{y}^{2} + p_{z}^{2}) c^{2} + m_{0}^{2} c^{4} . $ $ w h i c h r e f l e c t t h e l e n g t h o f t h e [[♾ ️ C O N C E P T S / f o u r - m o m e n t u m ∥ f o u r - m o m e n t u m]] v e c t o r . I f y o u t r y t o q u a n t i z e t h i s e q u a t i o n y o u a r r i v e a t

(i\hbar\partial_t)^2 \psi-c^2(-i\hbar\partial_x)^2\psi-c^2(-i\hbar\partial_y)^2\psi-c^2(-i\hbar\partial_z)^2\psi=m^2c^4 \psi,

-\frac{\hbar^2}{c^2}\partial_t^2 \psi+\hbar^2\partial_x^2\psi+\hbar^2\partial_y^2\psi+\hbar^2\partial_z^2\psi=m^2c^2 \psi,

a n d f i n a l l y

\frac{1}{c^2}\partial_t^2 \psi-\partial_x^2\psi-\partial_y^2\psi-\partial_z^2\psi=-\frac{m^2c^2}{\hbar^2} \psi.