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- There are five examples below to show how to run a reversed conversion from natural unit to cgs units (see Table) :
- Electromagnetic Wave Equation -

In natural unit of [cm] (the natural unit is enclosed within the bracket [ ]) this equation is in the form :

^{2}**E**-_{}**E**= 0

Conversion back to cgs units in this case is very simple by substituting the natural unit [cm] with the cgs conversion factor c for the time variable. There is no conversion involved with the spatial variables. Then the equation in cgs units is recovered as:

^{2}**E**- (1/c^{2})_{}**E**= 0 - Total Energy -

E = (m^{2}+**p**^{2})^{1/2}

In natural unit of [erg], m is converted back to cgs by c^{2}, while the conversion for**p**is c. Therefore the formula in cgs is :

E = (m^{2}c^{4}+**p**^{2}c^{2})^{1/2} - Schrodinger Equation -

In natural unit it has the form : -(1/2m)_{}= i_{}

In converting from the natural unit of [erg], the left-handed side produces a factor of (1/c^{2})(c)^{2}=^{2}. On the right-handed side the converting factor is . Combining everything together, we recovered the equation in cgs :

-(^{2}/2m)_{}= i_{}

Incidentally, for all cases, the same form in cgs units will be recovered whether the natural unit is in [cm] or [erg]. - Traveling (quantum) Wave -

In natural unit with the energy E in [erg] :_{}= exp[i(kx-Et)]. Backward conversion to cgs becomes_{}= exp[i(kx-Et/)]. - Scales of the Universe - The formulas for the various scales (see table) are expressed in natural units. The following demonstrates the procedure of converting them back to the cgs units and in the process reproduces the numerical values quoted in the table.
- Planck Scale - E
_{Pl}= (8G)^{-1/2}which has a dimension of [erg] in natural unit. Substituting the numerical values and converting the result from erg to ev yields E_{Pl}= 0.8x10^{3}[erg] = 5x10^{14}[ev] which is incorrect. The proper way is to perform a reversed conversion by multiplying (c^{5})^{1/2}(see Table) to arrive at the correct answer of E_{Pl}= 2x10^{27}ev. This is the "Reduced Planck Energy" with a factor of (8)^{-1/2}difference from the conventional definition (see "Planck Scale"). The natural unit can also be expressed in term of [cm^{-1}] so that E_{Pl}= 0.8x10^{3}[cm^{-1}]. It becomes E_{Pl}= 0.8x10^{3}x(c^{4}/c)^{1/2}cm^{-1}= 1.5x10^{32}cm^{-1}by reversed conversion, the reciprocal of which is the length scale L_{Pl}= 0.7x10^{-32}cm (see "Scale of the Universe"). - Fermi Scale - E
_{F}= 1/(G_{F})^{1/2}= [Gev]/(1.17x10^{-5})^{1/2}= 3x10^{11}ev. The length scale is obtained by converting ev to erg, and then divided by c (changing the unit to cm^{-1}), the reciprocal of which is L_{QCD}= 0.6x10^{-16}cm. The conversion for this one and the next is straight forward because the natural unit is in unit of energy already. - QCD Scale - The natural unit for this one is just the coupling constant E
_{QCD}=_{QCD}= 2x10^{8}ev. The length scale is obtained by converting ev to erg, and then divided by c to change the dimension to cm^{-1}, the reciprocal of which is L_{QCD}= 10^{-13}cm. - Vacuum Scale - E
_{vac}= (_{vac})^{1/4}[erg]. Conversion back to cgs units involves multiplying (c^{5}^{3}) to_{vac}= 0.7x10^{-29}, taking the 1/4 root, and translating to ev. The result is E_{vac}= 1.5x10^{-3}ev. The length scale is obtained by multiplying_{vac}(in the alternate natural dimension of [cm^{-1}]) with (c/), then takes the 1/4 root, the reciprocal of which yields L_{vac}= 0.7x10^{-2}cm. - Hubble Scale - E
_{H}= (8G)^{1/2}in natural unit of [erg]. Conversion to cgs units involves taking the 1/2 root of 8G (with = 10^{-29}), multiplying , then translating to ev. The result is E_{H}= 2.5x10^{-33}ev. The multiplication factor is 1/c (when it is in natural unit of [cm^{-1}]) in deriving the inverse length scale, and then takes the reciprocal L_{H}= 0.8x10^{28}cm.

BTW, the Hubble equation is originally derived from General Relativity, i.e., H =(dR/dt)/R = (8G/3)^{1/2}for k = 0 in cgs unit of sec^{-1}. It has nothing to do with , which comes from quantum theory. However, in an effort to express this equation in unit of energy, it is artificially multiplied by and declared that (8G)^{1/2}is in natural unit of [erg] (also taking away a factor of 1/3). It is rather doubtful whether the uncertainty principle is applicable to such macroscopic object. Anyway, in a similar vein, it is multiplied by 1/c to pretend that it is in natural unit of [cm^{-1}].

- Planck Scale - E