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Such an electron will have a kinetic energy,

KE = mv^{ 2},
| (3) |

PE = k = k = - k ,
| (4) |

E = mv^{ 2} - k .
| (5) |

k = m mv^{ 2} = k ,
| (6) |

E = - k .
| (7) |

Up until this point the analysis has been classical, and as such the model suffers from the instability problem mentioned previously. What Bohr did was to now consider the electron not as a particle but as a wave, with wavelength given by de Broglie's relation:

= = . | (8) |

then only certain wavelengths would be allowed for a stable orbit:

2r = n n = 1,2,3,...
| (9) |

mvr = n n,
| (10) |

If one combines Eqs.(28.6,28.10), one sees

v^{ 2} = = r_{n} = n^{ 2} a_{ }n^{ 2},
| (11) |

a_{0} = = 0.053 nm
| (12) |

E_{n} = - - eV .
| (13) |

Thus, the electrons in this model are confined to certain orbits with
definite energies. Let us consider what happens when an electron in
a high energy orbit *n* = *n*_{i} falls to a lower energy state with
*n* = *n*_{f} , where *n*_{i} > *n*_{f} . Such a process in itself would not conserve
energy; let us postulate that the excess energy is carried away by a
photon of energy
*hf* = *hc*/ . Energy conservation then tells us

+ E_{ni} = E_{nf} =
| (14) |

R_{H} = = 1.097 x 10^{7} m ^{- 1}.
| (15) |

10/9/1997