Changes
Added some maths
** n=8 to n=2 - 388.9 nm (violet)
** n=8 to n=2 - 388.9 nm (violet)
* Lymann Series - wavelength in the [[Ultraviolet|UV]] range, when an electron falls from n=2 to n=1
* Lymann Series - wavelength in the [[Ultraviolet|UV]] range, when an electron falls from n=2 to n=1
==Mathematics==
The model is based on [[classical physics]] and ignores radiation emitted by the [[electron]] due to it's acceleration. It also supposes that [[electrons]] are only allowed to have certain values ofr their [[angular momentum, or that [[angular momentum]] is '''quantised'''. It correctly predicts, by chance, the relationship beween different levels and their radius and energy. These radius relationship is:
<math>
r_n = \epsilon_0 \frac{n^2 h^2}{\pi me^2}
</math>
where
:<math>\epsilon_0</math> is the permittivity of free space, equal to <math>8.854187817 \times 10^{-12} \, \mbox{F} \, \mbox{m}^{-1}</math><ref>http://physics.nist.gov/cgi-bin/cuu/Value?ep0</ref>
:<math>n</math> is the n<sup>th</sup> energy level
:<math>h</math> is [[Planck's constant]]
:<math>m</math> is the mass of an [[electron]]
:<math>e</math> is the elementary charge (charge on an electron)
The relationship for the energy levels is:
<math>
E_n = - \frac{hcR}{n^2}
</math>
where
:<math>c</math> is the [[speed of light]]
:<math>R</math> is the Ryberg constant, which is:
<math>
R= \frac{me^4}{8 \epsilon_0^2 h^3 c}
</math>
with all constants being the same as above.<ref>{{cite book
|author=Hugh D. Young and Roger A. Freedman
|title=University Physics with Modern Physics
|publisher=Pearson
|location=San Francisco
|isbn=
|pages=
|quote=
|language=English}}</ref>
===Derivation===
The most important part is the quantisation of [[angular momentum]], <math>L_n</math> as:
<math>L_n = mv_n r_n = n \hbar</math>
where
:<math>v_n</math> is the [[speed]] of the [[electron]] in the n<sup>th</sup> level
:<math>\hbar</math> is the reduced Planck constant
This can be done since [[Planck's constant]] has the same units as [[angular momentum]]
By considering [[Coulomb's law]], the electrostatic [[force]] between the [[electron]] and the [[proton]] and the [[acceleration]] of the electron undergoing circular motion we find:
<math>
\frac{1}{4 \pi \epsilon_0} \frac{e^2}{r_n^2}= \frac{mv_n^2}{r^n}
</math>
Solving this equation with the above determines <math>r_n</math> as above and:
<math>
v_n = \frac{1}{\epsilon_0} \frac{e^2}{2nh}
</math>
for the speed.
The [[energy]] of each level can be found as the sum of the [[kinetic energy|kinetic]] and potential energies:
<math>
E_n = \frac{1}{2} m v_n^2 - \frac{e^2}{2 \pi \epsilon_0 r_n^2}
</math>
Substituting in the above gives the desired result:
<math>
E_n = \frac{me^4}{8 \epsilon_0^2 h^2} \frac{1}{n^2}
</math>
==Limitations==
==Limitations==
(username removed)