Ring wave guide
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In quantum mechanics, the ring wave guide starts from the one dimensional, time independent Schrödinger equation:
- <math>-\frac{h}{2m}\nabla^2 \phi=E\phi.<math>
This must be solved under the circularity condition. Let the ring's radius be r, then in one dimension:
- <math>\nabla = \frac{\partial}{\partial x}=\frac{1}{r}\frac{\partial}{\partial \alpha}, <math>
where α is the position angle on the ring. We get
- <math>-\frac{h}{2mr^2}\frac{\partial^2 \phi}{\partial \alpha^2} = E \phi.<math>
The solution of this is
- <math>\phi = \exp\left(\pm i \sqrt{\frac{2mE}{h}} r\alpha\right).<math>
The circularity condition is now, that
- <math>\sqrt{\frac{2mE}{h}} r \alpha = 2 n \pi \alpha<math>
or
- <math>E = \frac{h}{2m}\frac{4n^2\pi^2}{r^2}=\frac{2h\pi^2}{mr^2}n^2.<math>
Now the wave function becomes
- <math>\phi=\exp(\pm i 2 n \pi \alpha).<math>
Quantum states found:
<math>n=0<math>:
- <math>\phi=1<math>,
- a constant function, and <math>E=0<math>. This represents a stationary :particle (no angular momentum spinning around the ring).
<math>n=1<math>:
- <math>\phi=\exp(\pm 2\pi \alpha)<math>
- and
- <math>E= \frac{2h\pi^2}{mr^2}n^2<math>.
- This produces two independent states that have the same energy level (degeneracy) and can be linearly combined arbitrarily; instead of <math>\exp(\pm\cdots)<math> 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 :<math>\pm\frac{h}{2\pi}<math>.
<math>n=2<math> (and higher):
- the energy level is proportional to <math>n^2<math>, the angular momentum to n. There are always two (degenerate) quantum states.
Conclusion: every quantum state is filled by a total of <math>2n + 1<math> particles.
Application
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 <math>2\times(2n+1)<math> electrons, as electrons have additionally two possible orientations of their spins.
The rule that <math>4n+2<math> excess electrons in the ring produces an exceptionally stable ("aromatic") compound, is know as the Hückel rule.