\(\cfrac { \partial B }{ \partial \, t } =-i\cfrac { \partial \, E }{ \partial x^{ ' } } \)
from the post "Scent Of Beauty...".
Using \(E=\cfrac { \partial \, V }{ \partial \, x } \) where \(V\) is the electric potential.
\( 2B\cfrac { \partial B }{ \partial \, t } =\cfrac { \partial B^{ 2 } }{ \partial \, t } =-i2\int { -i\cfrac { \partial \, E }{ \partial x^{ ' } } } \partial \, t\cfrac { \partial \, }{ \partial x^{ ' } } \left\{ \cfrac { \partial \, V }{ \partial \, x } \right\} \)
\( \cfrac { \partial B^{ 2 } }{ \partial \, t } =-2\int { \cfrac { \partial \, t }{ \partial x^{ ' } } \cfrac { \partial \, E }{ \partial x^{ ' } } \cfrac { \partial x^{ ' } }{ \partial t } } \partial \, t\cfrac { i }{ i } \cfrac { \partial \, }{ \partial x^{ ' } } \left\{ \cfrac { \partial \, V }{ \partial \, x } \right\} \)
\( \cfrac { \partial B^{ 2 } }{ \partial \, t } =-i2\int { \cfrac { \partial \, t }{ \partial x^{ ' } } } \partial \, E\left\{ \cfrac { \partial ^{ 2 }\, V }{ \partial \, x^{ 2 } } \right\} \)
\( \cfrac { \partial B^{ 2 } }{ \partial \, t } =-i\cfrac { 2 }{ c } E \left\{ \cfrac { \partial ^{ 2 }\, V }{ \partial \, x^{ 2 } } \right\} \)
\( \cfrac { 1 }{ 2\mu _{ o } } \cfrac { \partial B^{ 2 } }{ \partial \, t } =-i\cfrac { 1 }{ 2\mu _{ o } } \cfrac { 2 }{ c } E \left\{ \cfrac { \partial ^{ 2 }\, V }{ \partial \, x^{ 2 } } \right\} \)
\(Z_o=\mu_o\,c\)
Therefore,
\( \cfrac { 1 }{ 2\mu _{ o } } \cfrac { \partial B^{ 2 } }{ \partial \, t }=-\cfrac { \varepsilon_o }{ 2 } \cfrac { \partial E^{ 2 } }{ \partial \, t } =-i\cfrac { 1 }{ Z_{ o } }E \left\{ \cfrac { \partial ^{ 2 }\, V }{ \partial \, x^{ 2 } } \right\} \)
(cf. post "What Is This?")
Let \(\psi=\cfrac{1}{2\mu_o}B^2=-\cfrac{1}{2}\varepsilon_oE^2\),
\(i\cfrac{\partial\,\psi}{\partial\,t}=\cfrac { 1 }{ Z_{ o } } E \left\{ \cfrac { \partial ^{ 2 }\, V }{ \partial \, x^{ 2 } } \right\}\)
But,
\(\cfrac { \partial \, V }{ \partial \, x } \cfrac { \partial ^{ 2 }\, V }{ \partial \, x^{ 2 } } =\cfrac { 1 }{ 2 } \cfrac { \partial \, }{ \partial \, x } \left\{ \cfrac { \partial \, V }{ \partial \, x } \right\} ^{ 2 }\)
So,
\( i\cfrac { \partial \, \psi }{ \partial \, t } =\cfrac { 1 }{ Z_{ o } } E\left\{ \cfrac { \partial ^{ 2 }\, V }{ \partial \, x^{ 2 } } \right\} =\cfrac { 1 }{ Z_{ o } } \cfrac { \partial \, V }{ \partial \, x } \cfrac { \partial ^{ 2 }\, V }{ \partial \, x^{ 2 } } \)
\( i\cfrac { \partial \, \psi }{ \partial \, t } =\cfrac { 1 }{ Z_{ o } } \cfrac { 1 }{ 2 } \cfrac { \partial \, }{ \partial \, x } \left\{ \cfrac { \partial \, V }{ \partial \, x } \right\} ^{ 2 }\)
\( i\cfrac { \partial \, \psi }{ \partial \, t } =\cfrac { 1 }{ Z_{ o } } \cfrac { 1 }{ 2 } \cfrac { \partial E^{ 2 }\, }{ \partial \, x } \\ i\cfrac { \partial \, \psi }{ \partial \, t } =-\cfrac { 1 }{ \varepsilon _{ o }Z_{ o } } \cfrac { \partial \psi \, }{ \partial \, x } =-c\cfrac { \partial \psi \, }{ \partial \, x } \)
And we DON'T have Schrodinger Wave Equation. We have wave,
\( \cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 } } =ic\cfrac { \partial }{ \partial \, t } \cfrac { \partial \psi \, }{ \partial \, x } =ic\cfrac { \partial }{ \partial \, x } \cfrac { \partial \psi \, }{ \partial \, t } =ic\cfrac { \partial }{ \partial \, x } \left\{ ic\cfrac { \partial \psi \, }{ \partial \, x } \right\} =(ic)^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, x^{ 2 } } \)
where \(ic=i\cfrac{d\,x^{'}}{d\,t}=\cfrac{d\,x}{d\,t}\), \(ic\) is in the \(x\) direction.
Equally probable,
\( i\cfrac { \partial \, \psi }{ \partial \, t } =-c\cfrac{-i}{-i}\cfrac { \partial \psi \, }{ \partial \, x } \)
\(i\cfrac { \partial \, \psi }{ \partial \, t } =ic\cfrac { \partial \psi \, }{ \partial \, x^{'} } \)
\(\cfrac { \partial \, \psi }{ \partial \, t } =c\cfrac { \partial \psi \, }{ \partial \, x^{'} } \)
\(\cfrac { \partial ^{ 2 }\psi }{ \partial \, t^{ 2 } } =c\cfrac { \partial }{ \partial \, t } \cfrac { \partial \psi \, }{ \partial \, x^{ ' } } =c\cfrac { \partial }{ \partial \, x^{ ' } } \cfrac { \partial \psi \, }{ \partial \, t } =c\cfrac { \partial }{ \partial \, x^{ ' } } \left\{ c\cfrac { \partial \psi \, }{ \partial \, x^{ ' } } \right\} =c^{ 2 }\cfrac { \partial ^{ 2 }\psi }{ \partial \, { x^{ ' } }^{ 2 } } \)
A wave in the \(x^{'}\) direction.
