91影视

The Klein-Gordon equation is described as the relativistic equation of wave. This equation is the quantized version of the energy-momentum relation.

The given wave function is represented as given by,

$蠄(x,t)=A{e}^{卤ipx/\mathrm{鈩�}卤iEt/\mathrm{鈩�}}$

Double differentiating the above equation of the wave function with respect to$x$.

$\begin{array}{c}\frac{{鈭�}^2}{鈭�x^2}蠄(x,t)=\frac{鈭�}{鈭�x}\left(A{e}^{卤ipx/\mathrm{鈩�}卤iEt/\mathrm{鈩�}}\right)\\ ={(卤\frac{ip}{\mathrm{鈩�}})}^2\left(A{e}^{卤ipx/\mathrm{鈩�}卤iEt/\mathrm{鈩�}}\right)\\ =鈭�\frac{p^2}{{\mathrm{鈩�}}^2}\left(A{e}^{卤ipx/\mathrm{鈩�}卤iEt/\mathrm{鈩�}}\right)\\ =鈭�\frac{p^2}{{\mathrm{鈩�}}^2}蠄(x,t)\end{array}$

Double differentiating the above equation of the wave function with respect to the time.

$\begin{array}{c}\frac{{鈭�}^2}{鈭�t^2}蠄(x,t)=\frac{鈭�}{鈭�t}\left(\frac{鈭�}{鈭�t}A{e}^{卤ipx/\mathrm{鈩�}卤iEt/\mathrm{鈩�}}\right)\\ ={(卤\frac{iE}{\mathrm{鈩�}})}^2\left(A{e}^{卤ipx/\mathrm{鈩�}卤iEt/\mathrm{鈩�}}\right)\\ =鈭�\frac{E^2}{{\mathrm{鈩�}}^2}\left(A{e}^{卤ipx/\mathrm{鈩�}卤iEt/\mathrm{鈩�}}\right)\\ =鈭�\frac{E^2}{{\mathrm{鈩�}}^2}蠄(x,t)\end{array}$

The Klein-Gordon equation is expressed as:

$鈭�c^2{\mathrm{鈩�}}^2\frac{{鈭�}^2}{鈭�x^2}蠄(x,t)+m^2{c}^4蠄(x,t)=鈭�{\mathrm{鈩�}}^2\frac{{鈭�}^2}{鈭�t^2}蠄(x,t)$

Substitute $鈭�\frac{E^2}{{\mathrm{鈩�}}^2}蠄(x,t)$ for $\frac{{鈭�}^2}{鈭�t^2}蠄(x,t)$ and $鈭�\frac{p^2}{{\mathrm{鈩�}}^2}蠄(x,t)$ for $\frac{{鈭�}^2}{鈭�x^2}蠄(x,t)$ in the above equation.

$\begin{array}{c}鈭�c^2{\mathrm{鈩�}}^2(鈭�\frac{p^2}{{\mathrm{鈩�}}^2}蠄(x,t))+m^2{c}^4蠄(x,t)=鈭�{\mathrm{鈩�}}^2(鈭�\frac{E^2}{{\mathrm{鈩�}}^2}蠄(x,t))\\ p^2{c}^2蠄(x,t)+m^2{c}^4蠄(x,t)=E^2蠄(x,t)\\ p^2{c}^2+m^2{c}^4=E^2\end{array}$

The above equation is the equation of the special relativity. Hence, the equation is proved.

Thus, the equation (12-4) is satisfied by a wave equation of the form$蠄(x,t)=A{e}^{卤ipx/\mathrm{鈩�}卤iEt/\mathrm{鈩�}}$ .