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Charge density is $i{蠄}^{鈭�}\frac{鈭�}{鈭�t}蠄鈭�i蠄\frac{鈭�}{鈭�t}{蠄}^{鈭�}$.

The Klein-Gordon equation is given as,

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

The Schr枚dinger Equation is given as,

$\mathbf{鈭�}\frac{{\mathbf{鈩�}}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}}\frac{{\mathbf{鈭�}}^{\mathbf{2}}}{\mathbf{鈭�}{\mathbf{x}}^{\mathbf{2}}}\mathit{唯}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{=}\mathit{i}\mathit{鈩�}\frac{\mathbf{鈭�}}{\mathbf{鈭�}\mathbf{t}}\mathit{唯}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$

Charge density is given as,

$\mathit{i}{\mathit{蠄}}^{\mathbf{鈭�}}\frac{\mathbf{鈭�}}{\mathbf{鈭�}\mathbf{t}}\mathit{蠄}\mathbf{鈭�}\mathit{i}\mathit{蠄}\frac{\mathbf{鈭�}}{\mathbf{鈭�}\mathbf{t}}{\mathit{蠄}}^{\mathbf{鈭�}}$

The positive wave functions are given as follows:

${唯}_1(x,t)=A{e}^{ikx鈭�i蝇t}$ 鈥︹赌�(1)

${唯}_1^{*}(x,t)=A{e}^{鈭�ikx+i蝇t}$ 鈥︹赌�(2)

Differentiate the equation (1) partially with respect to$t$as:

$\begin{array}{l}\frac{鈭�}{鈭�t}{唯}_1(x,t)=A{e}^{ikx鈭�i蝇t}(鈭�i蝇)\\ \frac{鈭�}{鈭�t}{唯}_1(x,t)=鈭�i蝇{唯}_1(x,t)\end{array}$

鈥︹赌︹赌�(3)

Differentiate the equation (2) partially with respect to $t$:

$\begin{array}{l}\frac{鈭�}{鈭�t}{唯}_1^{*}(x,t)=A{e}^{鈭�ikx+i蝇t}\left(i蝇\right)\\ \frac{鈭�}{鈭�t}{唯}_1^{*}(x,t)=i蝇{唯}_1^{*}(x,t)\end{array}$

$\text{颁丑补谤驳别鈥塪别苍蝉颈迟测}=i{唯}_1^{*}\frac{鈭�}{鈭�t}{唯}_1鈭�i{唯}_1\frac{鈭�}{鈭�t}{唯}_1^{*}$

Substitute $\frac{鈭�}{鈭�t}{唯}_1(x,t)=鈭�i蝇{唯}_1(x,t)$ and$\frac{鈭�}{鈭�t}{唯}_1^{*}(x,t)=i蝇{唯}_1^{*}(x,t)$ in the equationand simplify as:

$\begin{array}{c}\text{颁丑补谤驳别鈥塪别苍蝉颈迟测}=i{唯}_1^{*}(鈭�i蝇{唯}_1)鈭�i{唯}_{1}i蝇{唯}_1^{*}(x,t)\\ =鈭�i^2蝇{唯}_1^{*}{唯}_1鈭�i^2蝇{唯}_1^{*}{唯}_1\\ =鈭�i^2蝇\left(\left(A{e}^{鈭�ikx+i蝇t}\right)\left(A{e}^{ikx鈭�i蝇t}\right)\right){唯}_1鈭�i^2蝇\left(\left(A{e}^{鈭�ikx+i蝇t}\right)\left(A{e}^{ikx鈭�i蝇t}\right)\right)\\ =2{蝇}^2{A}^2\end{array}$

The negative energy wave functions are:

${\mathit{唯}}_{\mathbf{1}}^{\mathbf{*}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{=}\mathit{A}{\mathit{e}}^{\mathbf{鈭�}\mathbf{i}\mathbf{k}\mathbf{x}\mathbf{鈭�}\mathbf{i}\mathbf{蝇}\mathbf{t}}$ 鈥︹赌�(4)

${\mathit{唯}}_{\mathbf{1}}^{\mathbf{*}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{=}\mathit{A}{\mathit{e}}^{\mathbf{鈭�}\mathbf{i}\mathbf{k}\mathbf{x}\mathbf{鈭�}\mathbf{i}\mathbf{蝇}\mathbf{t}}$ 鈥︹赌�(5)

Differentiate the equation (4) partially with respect to $t$ as:

$\begin{array}{l}\frac{鈭�}{鈭�t}{唯}_1(x,t)=A{e}^{ikx+i蝇t}\left(i蝇\right)\\ \frac{鈭�}{鈭�t}{唯}_1(x,t)=i蝇{唯}_1(x,t)\end{array}$

Differentiate the equation (5) partially with respect to $t$as:

$\begin{array}{l}\frac{鈭�}{鈭�t}{唯}_1^{*}(x,t)=A{e}^{鈭�ikx鈭�i蝇t}(鈭�i蝇)\\ \frac{鈭�}{鈭�t}{唯}_1^{*}(x,t)=鈭�i蝇{唯}_1^{*}(x,t)\end{array}$

Substitute $\frac{鈭�}{鈭�t}{唯}_1(x,t)=i蝇{唯}_1(x,t)$and $\frac{鈭�}{鈭�t}{唯}_1^{*}(x,t)=鈭�i蝇{唯}_1^{*}(x,t)$ in the equation and simplify as:

$\begin{array}{c}\text{颁丑补谤驳别鈥塪别苍蝉颈迟测}=i{唯}_1^{*}\left(i蝇{唯}_1\right)鈭�i{唯}_1鈭�i蝇{唯}_1^{*}(x,t)\\ =i^2蝇{唯}_1^{*}{唯}_1+i^2蝇{唯}_1^{*}{唯}_1\\ =i^2蝇\left(\left(A{e}^{ikx+i蝇t}\right)\left(A{e}^{鈭�ikx鈭�i蝇t}\right)\right){唯}_1+i^2蝇\left(\left(A{e}^{ikx+i蝇t}\right)\left(A{e}^{鈭�ikx鈭�i蝇t}\right)\right)\\ =鈭�2{蝇}^2{A}^2\end{array}$

This is exactly the negative of the constant for this part.

The quantity is a real constant for positive-energy plane-wave solution, and negative to that for negative-energy solution is proved.