With the function $R{ e }^{ \frac { i }{ \hbar } S }$ one can separate the Schrödinger equation $$i \hbar \frac{\partial \psi}{\partial t}=\left(-\frac{\hbar^{2}}{2 m} \nabla^{2}+V\right) \psi$$ into
$$\begin{aligned} &\rightarrow \frac { \partial \rho }{ \partial t } +\nabla \cdot (\rho v)=0\qquad\qquad\qquad \left(R={ \rho }^{ 2 },\quad v=\frac{1}{m} \nabla S\right)&\\ &\rightarrow \frac{\partial S}{\partial t}=-\left[\frac{|\nabla S|^{2}}{2 m}+V+Q\right]\qquad\qquad\left(Q=-\frac{\hbar^{2}}{2 m} \frac{\nabla^{2} R}{R}\right)& \end{aligned}$$ My question is:
- Is it possible to separate the Klein-Gordon-/Dirac equation with the same function or is there a mathematical or physical reason why it's not possible?
- Is there another function or way to separate these equations to get a better feeling for the real and imaginary part (or the phase an absolut value)?
I tried to separate the Klein-Gordon-equation
\begin{equation} \partial_{t}^{2} \psi-\nabla^{2} \psi+m^{2} \psi=0 \end{equation}
with the function $R{ e }^{ iS }$ but I am stuck with
\begin{equation} R\left[ \left( i{ \partial }_{ t }^{ 2 }{ S }-{ \left( { \partial }_{ t }{ S } \right) }^{ 2 } \right) -i\left( i\left( { S }_{ x }^{ 2 }+{ S }_{ y }^{ 2 }+{ S }_{ z }^{ 2 } \right) +{ \nabla }^{ 2 }S \right) \right] +{ \nabla }^{ 2 }R+{ m }^{ 2 }R+{ \partial }_{ t }^{ 2 }R+2i\cdot \left( { \partial }_{ t }{ S }{ \cdot \partial }_{ t }{ R-{ \nabla }S\cdot { \nabla }R } \right) =0 \end{equation}
Edit: The equation above leads to: \begin{equation} i\left[ R\left( { \partial }_{ t }^{ 2 }S-{ \nabla }^{ 2 }S \right) +2\cdot \left( { \partial }_{ t }{ S }{ \cdot \partial }_{ t }{ R-{ \nabla }S\cdot { \nabla }R } \right) \right] -R\left( { \left( { \partial }_{ t }{ S } \right) }^{ 2 }+{ \left( { \nabla }S \right) }^{ 2 } \right) +{ \nabla }^{ 2 }R+{ m }^{ 2 }R+{ \partial }_{ t }^{ 2 }R=0 \end{equation}
Because $S,R$ are real one gets the following equations:
$$\begin{aligned} &\rightarrow 2\cdot \left( { \partial }_{ t }{ S }{ \cdot \partial }_{ t }{ R-{ \nabla }S\cdot { \nabla }R } \right) =R\left( { { \nabla }^{ 2 }S-\partial }_{ t }^{ 2 }S \right) &\rightarrow R\left( { \left( { \partial }_{ t }{ S } \right) }^{ 2 }+{ \left( { \nabla }S \right) }^{ 2 } \right) ={ \nabla }^{ 2 }R+{ m }^{ 2 }R+{ \partial }_{ t }^{ 2 }R \end{aligned}$$
The left equation yields to
\begin{equation} 2{ \partial }_{ \mu \\ }S{ \partial }^{ \mu }R=R\Box S \end{equation}
