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##

Quantized orbits

Although it appears to be a radical step to consider the electron
as a wave just to get a stable orbit, this model of the the atom
with an electron wave has some further surprising successes.
The condition for a stable orbit is that an integral number of
wavelengths of the wave fit around the orbit:

where *n* is any integer. Now, according to de Broglie's hypothesis,
the wavelength of the wave is related to the speed of the
associated particle. Furthermore, for a particle going around a circle,
the speed of the particle is related to the force - remember that a force
exists because the direction of motion is constantly changing. This force,
which is the electrical force of attraction between the electron and
proton, in turn depends on the radius of the orbit. Ultimately, what
happens is that the condition for a standing wave pattern is a condition
on the radius of the orbit which can be satisfied
only by certain values. Numerically, these orbits are given by

where again *n* is an integer. The orbitals are thus **quantized** -
they occur in discrete steps. Relating the values of the radii back to
an energy through de Broglie's relation, one finds
the energy of the electron also is quantized:

where 1 eV = 1 electron Volt = 1.6 x 10^{-19} Joules (the kinetic
energy gained by an electron when accelerated through 1 Volt of
potential difference). This energy of the electron includes the
kinetic energy as well as the electrical potential energy, and
by convention is negative - this means that higher energy
states have larger values of *n*.

** Next:** Emission spectrum
**Up:** Bohr model
** Previous:** Electrons as de Broglie
*modtech@theory.uwinnipeg.ca *

1999-09-29