Difference between revisions of "Atomic Units"
imported>Ichuang (New page: The natural units for describing atomic systems are obtained by setting to unity the three fundamental constants that appear in the hydrogen Hamiltonian (Eq. \ref{eq:coulomb}), <math>\hbar...) |
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* velocity: <math>m v^2=\frac{e^{4}m}{\hbar^2}\Rightarrow v=\frac{e^2}{\hbar}=\alpha\cdot | * velocity: <math>m v^2=\frac{e^{4}m}{\hbar^2}\Rightarrow v=\frac{e^2}{\hbar}=\alpha\cdot | ||
c=2.2\times 10^8\ \textrm{cm/s}</math> | c=2.2\times 10^8\ \textrm{cm/s}</math> | ||
| − | * electric field: | + | * electric field: <ref>This is the characteristic value for the <math>n=1</math> orbit of hydrogen.</ref><math>\frac{e}{a_0^2}=5.142\times 10^9~{\rm V/cm} </math> |
\end{itemize} | \end{itemize} | ||
As we see above, we can express the atomic units in terms of <math>c</math> instead of <math>e</math> by introducing a single dimensionless constant | As we see above, we can express the atomic units in terms of <math>c</math> instead of <math>e</math> by introducing a single dimensionless constant | ||
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interaction is <math>\alpha</math>. That <math>\alpha=\frac{1}{137}</math> says that the | interaction is <math>\alpha</math>. That <math>\alpha=\frac{1}{137}</math> says that the | ||
Coulomb interaction is weak. | Coulomb interaction is weak. | ||
| + | |||
| + | <references/> | ||
Revision as of 02:08, 6 February 2009
The natural units for describing atomic systems are obtained by setting to unity the three fundamental constants that appear in the hydrogen Hamiltonian (Eq. \ref{eq:coulomb}), 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 \begin{itemize}
- length: Bohr radius =
- energy: 1 hartree = 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 \frac{e^4 m}{\hbar^2}=(\frac{e^2}{c\hbar})^2mc^2=\alpha^2 mc^2=27.2\ \textrm{eV}}
- velocity: 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 m v^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: [1]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 \frac{e}{a_0^2}=5.142\times 10^9~{\rm V/cm} }
\end{itemize} As we see above, we can express the atomic units in terms 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 c} instead 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 e} by introducing a single dimensionless 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=\frac{e^2}{\hbar c}\approx\frac{1}{137}. }
The \textit{fine structure constant} (footnote: The name "fine structure constant" derives from the appearance of in the ratio of fine structure splitting to the Rydberg.} 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} 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 (see Appendix \ref{app:metrology) ), 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} , as a dimensionless constant, is not defined by metrology. Rather, 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: 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\simeq mc^2\Rightarrow \Delta p=mc} 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 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 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 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} . That 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=\frac{1}{137}} says that the Coulomb interaction is weak.
- ↑ This is the characteristic value for the 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 n=1} orbit of hydrogen.
