From the post "Discreetly, Discrete \(\lambda\) And Discrete Frequency, \(f\)", we have an expression for discrete \(\lambda\),
\(x=e^{ -\cfrac { Ax+C }{ n^{ 2 } } }\)
From the post "Standing Waves, Particles, Time Invariant Fields", ψ is a circular standing wave about a center,
\( 2\pi x=n\lambda \)
where \(n=1,2,3...\) when \(x=0\), \(n=0\).
Substitute for \(x\),
\( 2\pi e^{ -\cfrac { An\lambda }{ 2\pi n^{ 2 } } -\cfrac { C }{ n^{ 2 } } }=n\lambda \)
Since, \( f\lambda =c\)
\( 2\pi e^{ -\cfrac { Anc }{ 2\pi fn^{ 2 } } -\cfrac { C }{ n^{ 2 } } }=n\cfrac { c }{ f } \)
\( f=\cfrac { nc }{ 2\pi } e^{ \cfrac { Anc }{ 2\pi fn^{ 2 } } +\cfrac { C }{ n^{ 2 } } }\)
\( f=\cfrac { nc }{ 2\pi } e^{ \cfrac { A }{ 2\pi \, n } \cfrac { c }{ f } }.e^{ \cfrac { C }{ n^{ 2 } } }\)
From the post "Folding Up, Peekapoo, What's Under \(a_\psi\)?", where \(a_\psi\) is the radius of the particle mass,
\( C=-n_{ o }^{ 2 }ln(a_{ \psi })-Aa_{ \psi }\)
Substitute this into \(C\),
\( f=\cfrac { nc }{ 2\pi } e^{ \cfrac { A }{ 2\pi \, n } \cfrac { c }{ f } }.e^{ \cfrac { -n_{ o }^{ 2 }ln(a_{ \psi })-Aa_{ \psi } }{ n^{ 2 } } }\)
\( f=\cfrac { nc }{ 2\pi } e^{ \cfrac { A }{ 2\pi \, n } \cfrac { c }{ f } }.a_{ \psi }^{ -\cfrac { n_{ o }^{ 2 } }{ n^{ 2 } } }e^{ -\cfrac { Aa_{ \psi } }{ n^{ 2 } } }\)
\( ln(\cfrac { 2\pi f }{ nc } )=\cfrac { A }{ 2\pi \, n } \cfrac { c }{ f } +ln(a_{ \psi }^{ -\cfrac { n_{ o }^{ 2 } }{ n^{ 2 } } })-\cfrac { Aa_{ \psi } }{ n^{ 2 } } \)
\( n^{ 2 }ln(\cfrac { 2\pi f }{ nc } )=n^{ 2 }\left\{ \cfrac { A }{ 2\pi \, n } \cfrac { c }{ f } -\cfrac { n_{ o }^{ 2 } }{ n^{ 2 } } ln(a_{ \psi }) \right\} -Aa_{ \psi }\)
\( n^{ 2 }ln(\cfrac { 2\pi f }{ nc } )=A\cfrac { nc }{ 2\pi f } -n_{ o }^{ 2 }ln(a_{ \psi })-Aa_{ \psi }\)
If we let \( x=\cfrac { 2\pi f }{ nc } \)
\( n^{ 2 }ln(x)=A\cfrac { 1 }{ x } -n_{ o }^{ 2 }ln(a_{ \psi })-Aa_{ \psi }\)
where each \(n\) has a corresponding \(f\) from the spectrum of the particles.
So, a plot of \( n^{ 2 }ln(x)\) verses \(\cfrac { 1 }{ x }\) will give \(A\) as the gradient and \(-n_{ o }^{ 2 }ln(a_{ \psi })-Aa_{ \psi }\) as the y-intercept. from which we obtain the size of the particle \(a_\psi\).
If we assume that \(f\) is the minimum at \(a_\psi\) then \(n_o=1\).
Various such plots can be obtained by varying \(A\) by changing temperature \(T\). Since \(f\) can be folded upwards as shown in the posts "Discreetly, Discrete \(\lambda\) And Discrete Frequency, \(f\)" and "Another Take On Discrete", \(T\) should be high such that \(A\) is low and does not fold the values of \(f\) corresponding to lower values of \(n\) upwards. The assignment of \(f\) to \(n\) is then simply 1 to 1, starting from the lowest value of \(f\), \(f_{min}\) to \(n_o=1\).
We are assuming that \(T\) does not change \(a_\psi\). Which would be very interesting if it does.
\(T\) does effect \(a_\psi\). Increasing \(T\) reduces the resistance to \(\psi\), more of a particle is manifested as \(\psi\) and its mass decreases, thus \(a_\psi\) decreases.
