\(\psi =\psi _{ 1 }+i\psi _{ 2 }\)
is the solution to,
\(\nabla ^{ 2 }\psi +\left( \cfrac { n }{ x } \right) ^{ 2 }\psi =0\)
And,
\(\psi =|\psi |e^{ iwt }\)
which is the vector sum of \(\psi_1\) and \(\psi_2\), rotating about the origin with angular velocity \(\omega\).
But \(\psi\) is not a vector, it is energy that oscillate between two dimensions. What happened?
It is always possible to represent two orthogonal values as one vector with an appropriate phase shift. When considered along with the dimensional axes, \(\psi\) together with the associated dimension is a vector. And so we are back to the dipole model, of a rotating particle with a force field, travelling at light speed, from which we wrote down the expression,
\(B=-i\cfrac{\partial\,E}{\partial\,x^{'}}\)
And, as the diagram on the far right suggests, the force field can be reversed by increasing the value of \(x_z\), in the expression for \(F_\rho\), the force density on the particle,
\(F_{ \rho }=i\sqrt { 2{ mc^{ 2 } } } \, G.tanh\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_{ z }) \right)\)
And the force on an external agent is,
\(F_{i}=-F_{ \rho }=-i\sqrt { 2{ mc^{ 2 } } } \, G.tanh\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_{ z }) \right)\)
Have a nice day.
