ATOMIC UNITS

'Atomic units' ('au') form a system of units convenient for atomic physics, electromagnetism, and quantum electrodynamics, especially when the focus is on the properties of electrons. In 'au', the numerical values of the following six physical constants are all unity by definition:

★ Two properties of the electron, its mass and charge;

★ Two properties of the hydrogen atom, its Bohr radius and the absolute value of its electric potential energy in the ground state;

★ Two constants, Dirac's and that for Coulomb's Law.

Contents

Fundamental units

Some derived units

Comparison with Planck units

Quantum mechanics and electrodynamics simplified

See also

References

External links

Fundamental units

Fundamental Atomic Units
'Quantity'	'Name'	'Symbol'	'SI value'	'Planck unit scale'
length	Bohr radius	''a''0	5.291 772 108(18)×10-11 m	10-35 m
mass	electron rest mass	''m''e	9.109 3826(16)×10-31 kg	10-8 kg
charge	elementary charge	''e''	1.602 176 53(14)×10-19 C	10-18 C
angular momentum	Planck's constant	$hbar\; =\; h/2\; pi$	1.054 571 68(18)×10-34 J s	(same)
energy	Hartree energy	''E''h	4.359 744 17(75)×10-18 J	109 J
electrostatic force constant	Coulomb's constant	1/(4πε0)	8.9875516×109 C-2 N m2	(same)

These six quantities are not independent; to normalize all six quantities to 1, it suffices to normalize any four of them to 1. The normalizations of the Hartree energy and Coulomb's constant, for example, are only an incidental consequence of normalizing the other four quantities.

Some derived units

{| class="wikitable"
! colspan="4" | Derived Atomic Units
|-
|'Quantity'
|'Expression'
|'SI value'
|'Planck unit scale'
|-
|time||||2.418 884 326 505(16)×10^-17 s||10^-43 s
|-
|velocity||||2.187 691 2633(73)×10⁶ m s^-1||10⁸ m s^-1
|-
|force||||8.238 7225(14)×10^-8 N||10⁴⁴ N
|-
|current|||| 6.623 617 82(57)×10^-3 A||10²⁶ A
|-
|temperature|||| 3.157 7464(55)×10⁵ K||10³² K
|-
|pressure|||| 2.942 1912(19)×10¹³ N m^-2||10¹¹⁴ Pa
|}

Comparison with Planck units

Both Planck units and 'au' are derived from certain fundamental properties of the physical world, and are free of anthropocentric considerations. To facilitate comparing the two systems of units, the above tables show the order of magnitude, in SI units, of the Planck unit corresponding to each atomic unit. Generally, when an 'atomic unit' is "large" in SI terms, the corresponding Planck unit is "small", and vice versa. It should be kept in mind that 'au' were designed for atomic-scale calculations in the present-day Universe, while Planck units are more suitable for quantum gravity and early-Universe cosmology.
Both 'au' and Planck units normalize the Dirac constant and the Coulomb force constant to 1. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity and cosmology: the gravitational constant ''G'' and the speed of light in a vacuum, ''c''. Letting α denote the fine structure constant, the 'au' value of ''c'' is α^-1 ≈ 137.036.
'Atomic units', by contrast, normalize to 1 the mass and charge of the electron, and ''a''₀, the Bohr radius of the hydrogen atom. Normalizing ''a''₀ to 1 amounts to normalizing the Rydberg constant, ''R''_∞, to 4π/α = 4π''c''. Given 'au', the Bohr magneton μ_B=1/2. The corresponding Planck value is ''e''/2''m''_e. Finally, 'au' normalize a unit of atomic energy to 1, while Planck units normalize to 1 Boltzmann's constant ''k'', which relates energy and temperature.

Quantum mechanics and electrodynamics simplified

The (non-relativistic) Schrödinger equation for an electron in SI units is
:
abla^2 psi(mathbf{r}, t) + V(mathbf{r}) psi(mathbf{r}, t) = i hbar rac{partial psi}{partial t} (mathbf{r}, t).
The same equation in 'au' is
:
abla^2 psi(mathbf{r}, t) + V(mathbf{r}) psi(mathbf{r}, t) = i rac{partial psi}{partial t} (mathbf{r}, t).
For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is:
:$hat\; H\; =\; -\; \{\{\{hbar^2\}\; over\; \{2\; m\_e\}\}$
abla^2} - {1 over {4 pi epsilon_0}}{{e^2} over {r}},
while 'atomic units' transform the preceding equation into
:$hat\; H\; =\; -\; \{\{\{1\}\; over\; \{2\}\}$
abla^2} - {{1} over {r}}.
Finally, Maxwell's equations take the following elegant form in 'au':
:$$
abla cdot mathbf{E} = 4pi
ho
:$$
abla cdot mathbf{B} = 0
:$$
abla imes mathbf{E} = -lpha rac{partial mathbf{B}} {partial t}
:$$
abla imes mathbf{B} = lpha left( rac{partial mathbf{E}} {partial t} + 4pi mathbf{J}
ight)
(There is actually some ambiguity in defining the atomic unit of magnetic field. The above Maxwell equations use the "Gaussian" convention, in which a plane wave has electric and magnetic fields of equal magnitude. In the "Lorentz force" convention, a factor of α is absorbed into 'B'.)

See also

Planck units

References

★ H. Shull and G. G. Hall, Atomic Units, Nature, volume 184, no. 4698, page 1559 (Nov. 14, 1959)

External links

★ CODATA Internationally recommended values of the Fundamental Physical Constants.