Quantization of Angular Momentum

Bohr, recognizing that such a collapse does not occur, was emboldened to impose an arbitrary quantum condition to stabilize his model of the atom. Bohr's postulate of 1913 was of the quantization of the angular momentum L of the electron of mass m circling the nucleus, in an orbit of radius r and speed v:

Here n is the arbitrary integer quantum number n= 1,2 Note that the units of Planck's constant, J-s, are also the units of angular momentum. This additional constraint leads easily to the basic and confirmed properties of the "Bohr orbits" of electrons in hydrogen and similar one-electron atoms:

En = -kZe2/2rn, rn = n2a0/Z, where a0=h2/mke2 = 0.053 nm. (4.3)

Here, and elsewhere, k is used as a shorthand symbol for the Coulomb constant fe = (4jt£0)-1.

The energy of the electron in the nth orbit can thus be given as En = -E0Z2/n2, n = 1,2,..., where

All of the spectroscopic observations of anomalous discrete light emissions and light absorptions of the one-electron atom were nicely predicted by the simple quantum condition hv= hc/X= EQ(l/ni2-l/n22). (4.4)

The energy of the light is exactly the difference of the energy of two electron states, nlj n2 in the atom. This was a breakthrough in the understanding of atoms, and stimulated work toward a more complete theory of nanophysics which was provided by Schrodinger in 1926 [2].

The Bohr model, which does not incorporate the basic wavelike nature of microscopic matter, fails to precisely predict some aspects of the motion and location of electrons. (It is found that the idea of an electron orbit, in the planetary sense, is wrong, in nanophysics.)

In spite of this, the electron energies En = -E0Z2/n2, spectral line wavelengths, and the characteristic size of the electron motion, aQ= h2/mke2 = 0.053 nm, are all exactly preserved in the fully correct treatment based on nanophysics, to be described below.

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