Natural units

Definitions of speed of light, reduced Planck's constant, and electronvolt in terms of kilograms, meters, seconds:

c = 299792458 m s {\displaystyle c=299792458\ {m \over s}}
ℏ = 1.054571817 × 10 − 34 k g m 2 s {\displaystyle \hbar =1.054571817\times 10^{-34}\ kg{m^{2} \over s}}
e V = 1.602176463 × 10 − 19 k g m 2 s 2 {\displaystyle eV=1.602176463\times 10^{-19}\ kg{m^{2} \over s^{2}}}

Then solve for second:

ℏ e V = 6.582120268 × 10 − 16 s {\displaystyle {\hbar \over eV}=6.582120268\times 10^{-16}s}
s = 1.519267287 × 10 15 ℏ e V {\displaystyle s=1.519267287\times 10^{15}{\hbar \over eV}}

Solve for kilogram:

c 2 = 8.987551787 × 10 16 m 2 s 2 {\displaystyle c^{2}=8.987551787\times 10^{16}\ {m^{2} \over s^{2}}}
e V c 2 = 1.782661731 × 10 − 36 k g {\displaystyle {eV \over c^{2}}=1.782661731\times 10^{-36}\ kg}
k g = 5.609589202 × 10 35 e V c 2 {\displaystyle kg=5.609589202\times 10^{35}\ {eV \over c^{2}}}

Solve for meter:

ℏ c = 3.51767294 × 10 − 43 k g ⋅ m {\displaystyle {\hbar \over c}=3.51767294\times 10^{-43}\ kg\cdot m}
ℏ c = 1.973270014 × 10 − 7 e V c 2 m {\displaystyle {\hbar \over c}=1.973270014\times 10^{-7}\ {eV \over c^{2}}m}
5067730.179 ℏ c c 2 e V = m {\displaystyle 5067730.179\ {\hbar \over c}{c^{2} \over eV}=m}
m = 5067730.179 ℏ c e V {\displaystyle m=5067730.179\ {\hbar c \over eV}}

Summarize results:

k g = 5.609589202 × 10 35 e V c 2 {\displaystyle kg=5.609589202\times 10^{35}\ {eV \over c^{2}}}
m = 5067730.179 ℏ c e V {\displaystyle m=5067730.179\ {\hbar c \over eV}}
s = 1.519267287 × 10 15 ℏ e V {\displaystyle s=1.519267287\times 10^{15}{\hbar \over eV}}
Dimensional conversion:  k g x m y s z ∼ c y − 2 x ℏ y + z e V x − y − z {\displaystyle kg^{x}m^{y}s^{z}\sim c^{y-2x}\hbar ^{y+z}eV^{x-y-z}}

Now omit c and ℏ:

k g = 5.609589202 × 10 35 e V {\displaystyle kg=5.609589202\times 10^{35}\ eV}
m = 5 , 067 , 730.179 1 e V {\displaystyle m=5,067,730.179\ {1 \over eV}}
s = 1.519267287 × 10 15 1 e V {\displaystyle s=1.519267287\times 10^{15}{1 \over eV}}
Dimensional conversion:  k g x m y s z ∼ e V x − y − z {\displaystyle kg^{x}m^{y}s^{z}\sim eV^{x-y-z}}

Based on: http://www.physics.gla.ac.uk/~dmiller/lectures/RQM_2008.pdf, page 12