\(\cfrac { \partial \, \psi _{ 1 } }{ \partial \, t_c }=\sqrt{2}\cfrac { \partial \, \psi _{ 1 } }{ \partial \, t }e^{ i\pi /4 } \)
\(\cfrac { \partial \, \psi _{ 2 } }{ \partial \, (-t_c) }=-\sqrt{2}\cfrac { \partial \, \psi _{ 2 } }{ \partial \, t }e^{ i\pi /4 } \)
given, \(t_c=\cfrac{1}{\sqrt{2}}t.e^{-i\pi/4}\)
When they are exchanging energy along the two orthogonal space dimension,
\(\cfrac { \partial \, \psi _{ 1 } }{ \partial \, t _c} =\cfrac { \partial \, \psi _{ 2 } }{ \partial \left( -t _c\right) } \)
\(\cfrac { \partial \, \psi _{ 1} }{ \partial \, t } =-\cfrac { \partial \, \psi _{ 2 } }{ \partial \, t } \)
Here \(\psi\) is a wave, oscillating between two space dimensions and traveling down a third dimension that is the time dimension, \(t_g\). The wave exist in a second time dimension, \(t_c\) by which velocities and other rate of change in time are measured.
With these relationships we formulate,
\(\ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } =\ddot { x } .\left(-\cfrac { \partial \, \psi _{ 2 } }{ \partial \, t }\right) =-\ddot { x } \cfrac { \partial \, \psi _{ 2 } }{ \partial \, t } \)
Since,
\(\ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } =-\cfrac { c^{ 3 } }{ \sqrt { 2 } } \cfrac { \partial ^{ 2 }\psi _{ 1 } }{ \partial \, x^{ 2 } } e^{ -i\pi /4 }\)
and
\( -\ddot { x } .\cfrac { \partial \, \psi _{ 2 } }{ \partial \, t } =\cfrac { c^{ 3 } }{ \sqrt { 2 } } \cfrac { \partial ^{ 2 }\psi _{ 2 } }{ \partial \, x^{ 2 } } e^{ -i\pi /4 }\)
We have,
\(-\cfrac { c^{ 3 } }{ \sqrt { 2 } } \cfrac { \partial ^{ 2 }\psi _{ 1 } }{ \partial \, x^{ 2 } } e^{ -i\pi /4 }=\cfrac { c^{ 3 } }{ \sqrt { 2 } } \cfrac { \partial ^{ 2 }\psi _{ 2 } }{ \partial \, x^{ 2 } } e^{ -i\pi /4 }\)
and so,
\( -\cfrac { \partial ^{ 2 }\psi _{ 1 } }{ \partial \, x^{ 2 } } =\cfrac { \partial ^{ 2 }\psi _{ 2 } }{ \partial \, x^{ 2 } } \)
The gradient of the forces which are indicative of their directions,
\(\cfrac { \partial \, F_{ 1 } }{ \partial \, x } =\cfrac { \partial ^{ 2 }\psi _{ 1 } }{ \partial \, x^{ 2 } } =-\cfrac { \sqrt { 2 } }{ c^{ 3 } } \ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } e^{ i\pi /4 }\)
\(\cfrac { \partial \, F_{ 2 } }{ \partial \, x } =\cfrac { \partial ^{ 2 }\psi _{ 2 } }{ \partial \, x^{ 2 } } =\cfrac { \sqrt { 2 } }{ c^{ 3 } } \ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } e^{ i\pi /4 }\)
These forces satisfy the change in energy required of both particles as \(\psi\) oscillates in \(t_c\) time. \(\psi\) however is not oscillating in \(t\). In the diagram above, we consider a point on \(\psi_2\) along the line joining the two centers of the particles. The resultant force \(F\) is towards \(\psi_1\) drawing the particles closer and we see that the forces are attractive; that particles in opposing directions on the time axis \(t_c\) attract each other. It is assumed that the force decreases monotonously with \(x\).
In a similar way,
when both particle are traveling in the same direction,
\(\cfrac { \partial \, F_{ 1 } }{ \partial \, x } =\cfrac { \partial ^{ 2 }\psi _{ 1 } }{ \partial \, x^{ 2 } } =-\cfrac { \sqrt { 2 } }{ c^{ 3 } } \ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } e^{ i\pi /4 }\)
\(\cfrac { \partial \, F_{ 2 } }{ \partial \, x } =\cfrac { \partial ^{ 2 }\psi _{ 2 } }{ \partial \, x^{ 2 } } =-\cfrac { \sqrt { 2 } }{ c^{ 3 } } \ddot { x } \cfrac { \partial \, \psi _{ 1 } }{ \partial \, t } e^{ i\pi /4 }\)
The resultant force \(F\) is away from \(\psi_1\) pushing the particles closer and we see that the forces are repuslive; that particles in the same directions on the time axis \(t_c\) repel each other. It is assumed that the force decreases monotonously with \(x\); that \( \cfrac { \partial \, \psi }{ \partial \, t }\) does not flip sign.
It is better to solve for the force,
\(F_{\psi}=-\cfrac { \partial\,\psi }{ \partial \, x} \)
directly. Just from the above analysis, attractive force is greater than the repulsive force.
