The energy of a photon is given as,
1) Ε = h ν
where h is Planck's constant and ν is the frequency.
If we model a photon as a lossless, resonant LC circuit, the energy stored in its equivalent capacitance would be,
C V**2 2) Ε = ------ 2
where C is the capacitance and V is the voltage across it. Capacitance is the charge divided by the voltage, so this can be rewritten as,
q V 3) Ε = --- 2
where for a photon, q is the charge of an electron. From Ohms Law, the voltage is the current, I, times the resistance, Ζ (in this case its really an impedance), and current is the charge times the frequency. Substituting we get,
q I Ζ 4) Ε = ----- 2 Ζ q**2 ν 5) Ε = --------- 2
By setting equations 1 and 5 equal to each other and solving for Ζ, we get,
2 h 6) Ζ = ---- q**2
This represents the required impedance, based on energy considerations, of the equivalent LC circuit which models a photon. At resonance, the capacitor has a negative imaginary resistance (negative reactance) equal to Ζ and the inductor has a positive reactance of the same magnitude. Resonance is defined when these two are equal and opposite. Note how the concept of resonance is related to Conservation of Curvature.
The fine structure constant α (alpha), is expressed as,
c μ0 q**2 7) α = -------- 2h
where c is the speed of light and μ0 is the permittivity of free space. Substituting Ζ0 for c μ0, where Ζ0 is the impedance of free space, which can also be computed as sqrt(μ0/ε0), we can rewrite the fine structure constant as,
Ζ0 q**2 8) α = ------ 2h
Rearranging to put Ζ0 on the left hand side, we get,
2 h α 9) Ζ0 = ----- q**2
substituting equations 6 and 9, we get an impedance relation as derived from the energy equation of,
Ζ0 10) Ζ = -- α
Thus based on energy considerations, the reactance at resonance of the equivalent LC circuit which models a photon is equal to the intrinsic impedance of free space divided by the fine structure constant. If Ζ0 is the driving impedance of an LC circuit and Ζ is the reactance of the C and L at resonance, this ratio is called the Q of the circuit. Therefore, the fine structure constant is 1/Q of the equivalent LC resonant circuit required to model a photon. Converting to numbers, the equivalent circuit of a photon has a Q of about 137, which while a 'high Q circuit', is not at all an unreasonable value.
By setting the capacitive reactance, 1/ωC, equal to the inductive reactance ωL (the resonance condition), and solving for capacitance and inductance, we arrive at these relationships for the L and C of the equivalent LC circuit of a photon,
α q**2 q**2 11) C = ---- = ----- = ----- Ζ0 ω 2 h ω 4 π Ε Ζ0 2 h h**2 12) L = --- = ------ = ----------- α ω ω q**2 ω q**2 π Ε
where ω is the radian frequency or 2π times the photons frequency.
A resonant circuit with less C and more L has a higher Q than one with less L and more C. The resonant frequency is given by 1/sqrt(LC) and the Q of a resonant circuit is given by the reactance at resonance, sqrt(L/C) divided by the driving impedance, which for a photon is the intrinsic impedance of free space, or Ζ0.
sqrt(L/C) 1 13) Q = --------- = - Ζ0 α
Now we have all the characteristics of an LC circuit which accurately models a photon based its energy considerations. Next, Part 2 of the proof calculates the L and C from purely geometric considerations.
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