Bohr-formula

We shall use these results in Section VII to obtain the Bohr formula for the energy levels of the hydrogen atom. 

Waerden, 1968) and can be used to rederive the Bohr formula for the energy levels. Corresponding to each energy level a unirrep of so(4) is obtained. The basis functions for each such unirrep are just the hydrogenic wave functions belonging to the energy level. 

The Born-Oppenheimer approximation was developed in 1927 by the physicists Max Born (German) and J. Robert Oppenheimer (American), just one year after Schrodinger presented his quantum treatment of the hydrogen atom. This approximation method is the foundation for all of molecular quantum mechanics, so you should become familiar with it. The basic idea of the Born-Oppenheimer approximation is simple because the nuclei are so much more massive than the electrons, they can be considered fixed for many periods of electronic motion. Let s see if this is a reasonable approximation. Using H2 as a specific example, we estimate the velocity of the electrons to be roughly the same as that of an electron in the ground state (w = 1) of the hydrogen atom. From the Bohr formula, V = we calculate an electron velocity of 2.2 X 10 m sec . We... 

Figure 1. Graphs of cross sections of electron capture (B and C) and loss (A) by carbon ions in water as a function of velocity. For capture, a bare nucleus is considered whereas for loss the ion considered has one electron bound to it in the K-shell before the occurrence of the process. Bohrs formulas are used with some modifications (see text). Notice that when the velocity is less than the K-or-bital velocity, capture may only take place at higher orbitals...

Using the Bohr formula for the energy levels, calculate the energy required to raise the electron in a hydrogen atom fi-om n= 1 to n = 00. Express the result for 1 mol H atoms. Because the n = 00 level corresponds to removal of the electron from the atom, this energy equals the ionization energy of the H atom. 

Calculate the ionization energy of the He" ion in kJ/mol (this would be the second ionization energy of He). See Problem 8.85. The Bohr formula for the energy levels of an ion consisting of a nucleus of charge Z and a single electron is... 

This has the same type denominator as the Bahner formula and when the other numbers are compared, it is found that the Bohr equation is essentially the same as the Balmer equation. There is only a slight difference due to the fact that the nucleus in the Bohr model is fixed at die center of the atom while the real spectra include the fact that the electron and proton both orbit around the center-of-mass (the see-saw balance point) of the two particles. That is really very close to the position of the proton because it is much more massive than the electron. When this correction is made to the Bohr formula, the agreement with the experimental spectra is essentially exact. 

The abbreviation SEM indicates a scanning electron microscope that can use an x-y scan to form a pixel image due to changes in transmitted intensity of the beam. However, in the last few years it has been realized that the same device can be used with a lower voltage of the electron beam of typically 20,000 V. The lower voltage is still sufficient to cause Ka x-rays for elements up to Z = 48 (Cd) but the same LiF diffraction monochrometer and detector can then be used to measure Let and Lp XRF (x-ray fluorescence) since for elements with Z> 48, the L shell is much deeper in energy. We can use the Bohr formula to estimate the L wavelength also ... 

Using the Bohr formula, equation (1.8), and the modern value of the Rydberg constant. 

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