Science Fair Project Encyclopedia
Ring wave guide
This must be solved under the circularity condition. Let the ring's radius be r, then in one dimension:
where α is the position angle on the ring. We get
The solution of this is
The circularity condition is now, that
Now the wave function becomes
Quantum states found:
n = 0:
- φ = 1,
- a constant function, and E = 0. This represents a stationary :particle (no angular momentum spinning around the ring).
n = 1:
- This produces two independent states that have the same energy level (degeneracy) and can be linearly combined arbitrarily; instead of one can choose the sine and cosine functions. These two states represent particles spinning around the ring in clockwise and counterclockwise directions. The angular momentum is :.
n = 2 (and higher):
- the energy level is proportional to n2, the angular momentum to n. There are always two (degenerate) quantum states.
Conclusion: every quantum state is filled by a total of 2n + 1 particles.
In organic chemistry, aromatic compounds contain atomic rings, such as benzene rings (the Kekulé structure) consisting of five or six, usually carbon, atoms. So does the surface of "buckyballs" (buckminsterfullerene). These molecules are exceptionally stable.
The above explains why: the ring behaves like a circular wave guide. The excess (valency) electrons spin around in both directions.
Every energy level is filled by electrons, as electrons have additionally two possible orientations of their spins.
The rule that 4n + 2 excess electrons in the ring produces an exceptionally stable ("aromatic") compound, is know as the Hückel rule.
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