Question

To show that the Klein-Gordon equation has valid solutions for negative values of $\mathit{E}$, verify that equation (12-4) is satisfied by a wave function of the form .$\mathit{\psi }\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}{\mathbf{Ae}}^{\mathbf{\pm }\mathbf{ipx}\mathbf{/}\mathbf{\hslash }\mathbf{\pm }\mathbf{iEt}\mathbf{/}\mathbf{\hslash }}$

Short answer

The equation (12-4) is satisfied by a wave equation of the form$\psi (x,t)=A{e}^{\pm ipx/\hslash \pm iEt/\hslash }$

Step-by-step solution

Significance of the Klein-Gordon equation:

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

Determination of the valid solution of the Klein-Gordon equation

The given wave function is represented as given by,

$\psi (x,t)=A{e}^{\pm ipx/\hslash \pm iEt/\hslash }$

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

$\begin{array}{c}\frac{{\partial }^2}{\partial x^2}\psi (x,t)=\frac{\partial }{\partial x}\left(A{e}^{\pm ipx/\hslash \pm iEt/\hslash }\right)\\ ={(\pm \frac{ip}{\hslash })}^2\left(A{e}^{\pm ipx/\hslash \pm iEt/\hslash }\right)\\ =-\frac{p^2}{{\hslash }^2}\left(A{e}^{\pm ipx/\hslash \pm iEt/\hslash }\right)\\ =-\frac{p^2}{{\hslash }^2}\psi (x,t)\end{array}$

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

$\begin{array}{c}\frac{{\partial }^2}{\partial t^2}\psi (x,t)=\frac{\partial }{\partial t}\left(\frac{\partial }{\partial t}A{e}^{\pm ipx/\hslash \pm iEt/\hslash }\right)\\ ={(\pm \frac{iE}{\hslash })}^2\left(A{e}^{\pm ipx/\hslash \pm iEt/\hslash }\right)\\ =-\frac{E^2}{{\hslash }^2}\left(A{e}^{\pm ipx/\hslash \pm iEt/\hslash }\right)\\ =-\frac{E^2}{{\hslash }^2}\psi (x,t)\end{array}$

The Klein-Gordon equation is expressed as:

$-c^2{\hslash }^2\frac{{\partial }^2}{\partial x^2}\psi (x,t)+m^2{c}^4\psi (x,t)=-{\hslash }^2\frac{{\partial }^2}{\partial t^2}\psi (x,t)$

Substitute $-\frac{E^2}{{\hslash }^2}\psi (x,t)$ for $\frac{{\partial }^2}{\partial t^2}\psi (x,t)$ and $-\frac{p^2}{{\hslash }^2}\psi (x,t)$ for $\frac{{\partial }^2}{\partial x^2}\psi (x,t)$ in the above equation.

$\begin{array}{c}-c^2{\hslash }^2(-\frac{p^2}{{\hslash }^2}\psi (x,t))+m^2{c}^4\psi (x,t)=-{\hslash }^2(-\frac{E^2}{{\hslash }^2}\psi (x,t))\\ p^2{c}^2\psi (x,t)+m^2{c}^4\psi (x,t)=E^2\psi (x,t)\\ p^2{c}^2+m^2{c}^4=E^2\end{array}$

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$\psi (x,t)=A{e}^{\pm ipx/\hslash \pm iEt/\hslash }$ .