\(\Delta PE_{n}= { m }_{ e }c^{ 2 }\left\{1 -\cfrac { 1 }{ { {n}^{ 2 } } } \right\}\)
It is not related to photon frequency, but it behaves like light in that the electron move in a helical path characterized by frequency, \(f\), a wavelength \(\lambda\), and a radius of circular motion, \(r\) and a speed lower than light speed, \(v\)
The total energy of such a particle is,
\(v=\lambda.f\), if the electron circular velocity is also \(v\) then \(v=2\pi r.f\)
\(E_e=\cfrac{1}{2}{m}_{e}{v_e}^2+\cfrac{1}{2}{m}_{e}{v_e}^2\)
\(E_e={m}_{e}{v_e}^2\)
So,
\( { m }_{ e }{v_e}^2={ m }_{ e }c^{ 2 }\left\{ 1-\cfrac { 1 }{ { { n }^{ 2 } } } \right\}\)
\( {v_e}^2=c^{ 2 }\left\{ 1-\cfrac { 1 }{ { { n }^{ 2 } } } \right\} \)
If we formulate,
\(\cfrac{c}{{v_e}} = \begin{matrix} 1 \\ \overline { \sqrt{ 1-\cfrac { 1 }{ { n }^{ 2 } }}} \end{matrix}=n_n\)
then we see that "this pretending to be light ray" is travelling at a low speed as if light traveling in a medium with reflective index given by the expression above. We know that refraction depends on the relative speed of the light before entering and after entering the medium boundary. The greater the difference in speed the greater the deflection. Equivalently, the greater the difference in refractive index between the two mediums the greater the deflection on screen. The sign of the differences determine whether the refraction is away or towards the normal at the point of incident. A plot of this \(n_n\) is given below, n is discrete, \(n=2,3,4,5...\)
So, given a prism of refractive index, \(n_p\), all spectra lines with \(n_n\) less than \(n_p\) will refract towards the normal and all spectra lines with \(n_n\) greater than \(n_p\) will reflect away from the normal. Since \(n\) is discrete, we then see discrete lines spaced apart, all deflected to different angles depending on \(n_n\).
It also means, according to this model, that sometimes, part of spectra lines (corresponding to higher values of \(n\)) are positioned before other lines with lower \(n\), because \(n_n\) is now smaller than the prism refractive index \(n_p\). All such lines with smaller \(n_n\) will be deflected backwards and overlap the lines that comes before them. The direction of deflection on screen reverses as \(n_n\) decreases with increasing \(n\) below \(n_p\), the refractive index of the prism used.
The question is; do we sense such electrons as light with all its colors? This Heavy Light are electrons not photons. It is too heavy to be diffracted on a diffraction grating. The channels formed in space by the grating will not channel the electrons at high speed to form patterns on screen.
Heavy light and we are not on square one.
