Atomic Units

Atomic Units

The natural units for describing atomic systems are obtained by setting to unity the three fundamental constants that appear in the hydrogen Hamiltonian, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar=m=e=1} . One thus arrives at atomic units, such as

• length: Bohr radius = $a_{0}={\frac {\hbar ^{2}}{me^{2}}}={\frac {1}{\alpha }}{\frac {\hbar }{mc}}=0.53A$
• energy: 1 hartree = ${\frac {e^{4}m}{\hbar ^{2}}}=({\frac {e^{2}}{c\hbar }})^{2}mc^{2}=\alpha ^{2}mc^{2}=27.2\ {\textrm {eV}}$
• velocity: $mv^{2}={\frac {e^{4}m}{\hbar ^{2}}}\Rightarrow v={\frac {e^{2}}{\hbar }}=\alpha \cdot c=2.2\times 10^{8}\ {\textrm {cm/s}}$
• electric field: ${\frac {e}{a_{0}^{2}}}=5.142\times 10^{9}~{\rm {V/cm}}$

Note: This is the characteristic value for the $n=1$ orbit of hydrogen.

As we see above, we can express atomic units in terms of $c$ instead of $e$ by introducing a single dimensionless constant

$\alpha ={\frac {e^{2}}{\hbar c}}\approx {\frac {1}{137}}.$

The fine structure constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} obtained its name from the appearance of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha^2} in the ratio of fine structure splitting to the Rydberg; it is the only fundamental constant in atomic physics. As such, it should ultimately be predicted by a complete theory of physics. Whereas precision measurements of other constants are made in atomic physics for purely metrological purposes , $\alpha$, as a dimensionless constant, is not defined by metrology. Rather, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} characterizes the strength of the electromagnetic interaction, as the following example will illustrate. If energy uncertainties become become as large as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E=mc^2} , the concept of a particle breaks down. This upper bound on the energy uncertainty gives us, via the Heisenberg Uncertainty Principle, a lower bound on the length scale within which an electron can be localized (before e.g. spontaneous pair production may occur) $\Delta \simeq mc^{2}\Rightarrow \Delta p=mc$, so (uncertainty principle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow <\math>) <math>\Delta x=\frac{\hbar}{mc}=\lambda_c} Even at this short distance of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_c} , the Coulumb interaction---while stronger than that in hydrogen at distance $a_{0}$ --- is only:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_c=\frac{e^2}{\lambda_c}=\frac{e^2mc}{\hbar}=\frac{e^2}{\hbar c}mc^2=\alpha mc^2 \,, }

i.e. in relativistic units the strength of this "stronger" Coulomb interaction is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} . The fact that $\alpha ={\frac {1}{137}}$ implies that the Coulomb interaction is weak.