User:Denevedr/sandbox

In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle $S^{1}$) is

$-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi =E\psi$

Wave function

Using polar coordinates on the 1-dimensional ring of radius R, the wave function depends only on the angular coordinate, and so

$\nabla ^{2}={\frac {1}{R^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}$

Requiring that the wave function be periodic in $\ \theta$ with a period $2\pi$ (from the demand that the wave functions be single-valued functions on the circle), and that they be normalized leads to the conditions

$\int _{0}^{2\pi }\left|\psi (\theta )\right|^{2}\,Rd\theta =1\$,

and

$\ \psi (\theta )=\ \psi (\theta +2\pi )$

Under these conditions, the solution to the Schrödinger equation is given by

$\psi _{\pm }(\theta )={\frac {1}{\sqrt {2\pi R}}}\,e^{\pm i{\frac {R}{\hbar }}{\sqrt {2mE}}\theta }$

Energy eigenvalues

The energy eigenvalues $E$ are quantized because of the periodic boundary conditions, and they are required to satisfy

$e^{\pm i{\frac {R}{\hbar }}{\sqrt {2mE}}\theta }=e^{\pm i{\frac {R}{\hbar }}{\sqrt {2mE}}(\theta +2\pi )}$, or

$e^{\pm i2\pi {\frac {R}{\hbar }}{\sqrt {2mE}}}=1=e^{i2\pi n}$

The eigenfunction and eigenenergies are

$\psi (\theta )={\frac {1}{\sqrt {2\pi R}}}\,e^{\pm in\theta }$

$E_{n}={\frac {n^{2}\hbar ^{2}}{2mR^{2}}}$ where $n=0,\pm 1,\pm 2,\pm 3,\ldots$

Therefore, there are two degenerate quantum states for every value of $n>0$ (corresponding to $\ e^{\pm in\theta }$). Therefore, there are 2n+1 states with energies up to an energy indexed by the number n.

The case of a particle in a one-dimensional ring is an instructive example when studying the quantization of angular momentum for, say, an electron orbiting the nucleus. The azimuthal wave functions in that case are identical to the energy eigenfunctions of the particle on a ring.

The statement that any wavefunction for the particle on a ring can be written as a superposition of energy eigenfunctions is exactly identical to the Fourier theorem about the development of any periodic function in a Fourier series.

This simple model can be used to find approximate energy levels of some ring molecules, such as benzene.

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). This ring behaves like a circular waveguide, with the valence electrons orbiting in both directions. To fill all energy levels up to n requires $2\times (2n+1)=4n+2$ electrons, as electrons have additionally two possible orientations of their spins. This gives exceptional stability ("aromatic"), and is known as the Hückel's rule.

Further in rotational spectroscopy this model may be used as an approximation of rotational energy levels.

See also

• Angular momentum
• Harmonic analysis
• One-dimensional periodic case
• Rigid Rotor

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References[edit]

1. ^ Jump up to:^{a b c d e} Davies, p.4
2. Jump up^ Actually, any constant, finite potential  can be specified within the box. This merely shifts the energies of the states by .
3. Jump up^ Davies, p. 1
4. ^ Jump up to:^{a b} Bransden and Joachain, p. 157
5. ^ Jump up to:^{a b} Davies p. 5
6. ^ Jump up to:^{a b} Bransden and Joachain, p.158
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8. Jump up^
9. ^ Jump up to:^{a b c} Bransden and Joachain, p. 159
10. Jump up^ Davies, p. 15
11. ^ Jump up to:^{a b} Todd Wimpfheimer, A Particle in a Box Laboratory Experiment Using Everyday Compounds, Journal of Laboratory Chemical Education, Vol. 3 No. 2, 2015, pp. 19-21. doi:10.5923/j.jlce.20150302.01.
12. ^ Jump up to:^{a b}
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14. Jump up^ P.J. Mohr, B.N. Taylor, and D.B. Newell, "The 2014 CODATA Recommended Values of the Fundamental Physical Constants". This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: [1]. National Institute of Standards and Technology, Gaithersburg, MD 20899.
15. Jump up^ β-Carotene http://www.sigmaaldrich.com/catalog/product/aldrich/855553?lang=en®ion=us (accessed Nov 8, 2016).
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17. Jump up^ U.S. Patent #3,982,207, issued September 21, 1976, Inventors R. Dingle and C. H. Henry ,"Quantum Effects in Heterostructure Lasers", filed March 7, 1975.
18. Jump up^
19. ^ Jump up to:^{a b}
20. ^ Jump up to:^{a b c}
21. ^ Jump up to:^{a b c d e f g h}
22. ^ Jump up to:^{a b c d e f}
23. ^ Jump up to:^{a b c}
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26. Jump up^ Alberto, P; Fiolhais, C; Gil, V M S (1996). "Relativistic particle in a box" (PDF). European Journal of Physics. 17: 19–24. Bibcode:1996EJPh...17...19A. doi:10.1088/0143-0807/17/1/004.

Bibliography

Levine, I. N. (2000). Quantum Chemistry (5th ed.). Prentice Hall. ISBN 0136855121

Bransden, B. H.; Joachain, C. J. (2000). Quantum mechanics (2nd ed.). Essex: Pearson Education. ISBN 0-582-35691-1.

Davies, John H. (2006). The Physics of Low-Dimensional Semiconductors: An Introduction (6th reprint ed.). Cambridge University Press. ISBN 0-521-48491-X.

Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-111892-7.

External links

• [1]Mathematica Demonstration (Rigid Rotor)
• Quantum mechanics java applet simulates rigid rotor (particle in a ring and particle in a sphere as well as other 1-dimensional and 2-dimensional cases.
• Particle in a ring simulation video