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## Examples of Natural Unit Conversion

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

1. Electromagnetic Wave Equation -
In natural unit of [cm] (the natural unit is enclosed within the bracket [ ]) this equation is in the form : 2E - 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: 2E - (1/c2) E = 0

2. Total Energy -
E = (m2 + p2)1/2
In natural unit of [erg], m is converted back to cgs by c2, while the conversion for p is c. Therefore the formula in cgs is :
E = (m2c4 + p2c2)1/2

3. 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/c2)(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].

4. 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/ )].

5. 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 - EPl = (8 G)-1/2 which has a dimension of [erg] in natural unit. Substituting the numerical values and converting the result from erg to ev yields EPl = 0.8x103 [erg] = 5x1014 [ev] which is incorrect. The proper way is to perform a reversed conversion by multiplying (c5 )1/2 (see Table) to arrive at the correct answer of EPl = 2x1027 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 EPl = 0.8x103 [cm-1]. It becomes EPl = 0.8x103x(c4/c )1/2 cm-1 = 1.5x1032 cm-1 by reversed conversion, the reciprocal of which is the length scale LPl = 0.7x10-32 cm (see "Scale of the Universe").

• Fermi Scale - EF = 1/(GF)1/2 = [Gev]/(1.17x10-5)1/2 = 3x1011 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 LQCD = 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 EQCD = QCD = 2x108 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 LQCD = 10-13 cm.

• Vacuum Scale - Evac = ( vac)1/4 [erg]. Conversion back to cgs units involves multiplying (c5 3) to vac = 0.7x10-29, taking the 1/4 root, and translating to ev. The result is Evac = 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 Lvac = 0.7x10-2 cm.

• Hubble Scale - EH = (8 G )1/2 in natural unit of [erg]. Conversion to cgs units involves taking the 1/2 root of 8 G (with = 10-29), multiplying , then translating to ev. The result is EH = 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 LH = 0.8x1028 cm.
BTW, the Hubble equation is originally derived from General Relativity, i.e., H =(dR/dt)/R = (8 G /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 (8 G )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].