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of the Quantum States of an Electron in an Atom |
The analysis of Niels Bohr that is referred to here is not the highly simplified version that is usually associated with his name but a more sophisticated analysis that preceded that planetary model. The version of Bohr's analysis that is utilized is the one presented by David Bohm in his Quantum Theory. It is this analysis that gives a basis for the quantization of the angular momentum of an electron in an atom.
The electron of charge −e is attracted to the point charge of the nucleus of charge +Ze. The nucleus is so massive compared to the electron that it can be consider fixed in position while the electron revolves around it. The potential energy of an electron at a distance r from the nucleus is −αZe²/r, where α is the constant for the Coulomb force. The kinetic energy of the electron moving at a velocity of v is in classical physics ½mv², where m is the mass of the electron. Thus the total energy of the electron is
In a circular orbit the balancing of the Coulombic attraction with the centrifugal force gives:
This latter condition means that
The angular momentum p_{θ} is equal to mvr; i.e. the linear momentum mv times the distance to the center of revolution. From the previous condition it follows that
This means that the total energy is
E = −½m(αZe²)²/p_{θ}²
If the electron goes from angular momentum p_{θ1} to angular momentum p_{θ2} there will be a change in energy ΔE which will go into a photon of frequency ν where
For a change in angular momentum which is small the change in energy may be approximated by
According to classical physics the angular frequency ω associated with a charge revolving in a circular orbit of radius r with a tangential velocity v is ω=v/r. The regular frequency f is related to the angular frequency by the relation f=ω/(2π). This should be the frequency associated with the escape of an electron from the hydrogen atom. Therefore
The expression h/2π is usually denoted as h and called h bar.
Since p_{θ}=mvr the above equation reduces to
Previously the relation αZe²/r = mv² was derived. Squaring both sides of the equation gives
Comparing this equation with the equation
reveals the stunning result that
In words, the angular momentum of an electron in an atom must change by an increment equal to Planck's constant divided by 2π. Or, stated differently the angular momentum of an electron is quantized.
The condition on the incrementatl change means that angular momentum must be of the form
Imposing the condition that the physical relationships must be the same for left-handed and right-handed coordinate sytem requires that k must be either 0 or ½.
A humorous note: David Bohm in his presentation has h in his derivation but he knows the final result should involve h-bar so he shamelessly changes from h in one equation to h-bar in the next equation without a word of justification. The original h's were a mistake stemming from treating angular frequency as being the regular frequency.
(To be continued.)
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