Polyhedral symmetry and quantum mechanics

Author(s)
G. S. Anagnostatos

Affiliation(s)

Institute of Nuclear and Particle Physics, National Center for Scientific Research, Demokritos, Greece.

Institute of Nuclear and Particle Physics, National Center for Scientific Research, Demokritos, Greece.

ABSTRACT

A thorough study of regular and quasi-regular polyhedra shows that the symmetries of these polyhedra identically describe the quantization of orbital angular momentum, of spin, and of total angular momentum, a fact which permits one to assign quantum states at the vertices of these polyhedra assumed as the average particle positions. Furthermore, if the particles are fermions, their wave function is anti-symmetric and its maxima are identically the same as those of repulsive particles, e.g., on a sphere like the spherical shape of closed shells, which implies equilibrium of these particles having average positions at the aforementioned maxima. Such equilibria on a sphere are solely satisfied at the vertices of regular and quasi-regular polyhedra which can be associated with the most probable forms of shells both in Nuclear Physics and in Atomic Cluster Physics when the constituent atoms possess half integer spins. If the average sizes of the constituent particles are known, then the average sizes of the resulting shells become known as well. This association of Symmetry with Quantum Mechanics leads to many applications and excellent results.

Cite this paper

Anagnostatos, G. (2014) Polyhedral symmetry and quantum mechanics.*Natural Science*, **6**, 198-210. doi: 10.4236/ns.2014.64024.

Anagnostatos, G. (2014) Polyhedral symmetry and quantum mechanics.

References

[1] Pauling, L. (1990) Regularities in the sequences of the number of nucleons in the revolving clusters for the ground-state energy bands of the even-even nuclei with neutron number equal to or greater than 126. Proceedings of the National Academy of Science USA, 87, 4435-4438, and references therein.

http://dx.doi.org/10.1073/pnas.87.12.4435

[2] Cook, N.D. (1994) Nuclear binding energies in lattice models. Journal of Physics G (Nuclear and Particle Physics), 20, 1907-1917 and references therein.

[3] Rae, W.D.M. and Zhang, J. (1994) Triangular alpha cluster geometries and harmonic oscillator shell structure. Modern Physics Letters A, 9, 599-607 and references therein.

http://dx.doi.org/10.1142/S021773239400383X

[4] Robson, D. (1978) Many-body interactions from quark exchanges and the tetrahedral crystal structure of nuclei. Nuclear Physics A, 308, 381-428 and references therein.

http://dx.doi.org/10.1016/0375-9474(78)90558-4

[5] Naher, U., Zimmermann, U. and Martin, T.P., (1993) Geometrical shell structure of clusters. Journal of Chemical Physics, 99, 2256-2260.

http://dx.doi.org/10.1063/1.465235

[6] Hecht, L. (1988) The geometrical basis for the periodicity of the elements.

[7] White, (1987) New hypothesis shows geometry of atomic nucleus. Executive Intelligence Review, 14, 18.

[8] Anagnostatos, G.S., Giapitzakis, J. and Kyritsis, A. (1981) Rotational invariance of orbital-angular-momentum quantization of direction for degenerate states. Lettere Nuovo Cimento, 32, 332-335.

http://dx.doi.org/10.1007/BF02745301

[9] Anagnostatos, G.S. (1980) The geometry of the quantization of angular momentum (l, s, j) in fields of central symmetry. Lettere Nuovo Cimento, 28, 573-576.

http://dx.doi.org/10.1007/BF02776343

[10] Anagnostatos, G.S. (1980) Symmetry description of the independent particle model. Lettere Nuovo Cimento, 29, 188-192. http://dx.doi.org/10.1007/BF02743377

[11] Anagnostatos, G.S., Ginis, P. and Giapitzakis, J. (1998) α-Planar states in 28Si”. Physical Review C, 58, 33053315. http://dx.doi.org/10.1103/PhysRevC.58.3305

[12] Kakanis, P.K. and Anagnostatos, G.S. (1996) Persisting α-planar structure in 20Ne. Physical Review C, 54, 29963013. http://dx.doi.org/10.1103/PhysRevC.54.2996

[13] Anagnostatos, G.S. (1999) Alpha-chain states in 12C”. Physical Review C, 51, 152-159.

http://dx.doi.org/10.1103/PhysRevC.51.152

[14] Anagnostatos, G.S. (2013) Quantum isomorphic shell model: Multi harmonic shell clustering in nuclei. Journal of Modern Physics, 4, 54-65.

http://dx.doi.org/10.4236/jmp.2013.45B011

[15] Merzbacher, E. (1961) Quantum mechanics. John Wiley and Sons Ltd., New York, p. 42.

[16] Cohen-Tannoudji, C., Diu, B. and Laloe, F. (1977) Quantum mechanics. John Wiley & Sons Ltd., New York, p. 240.

[17] Anagnostatos, G.S., Antonov, A.N., Ginis, P., et. al. (1998) Nucleon momentum and density distributions in 4He considering internal rotation. Physical Review C, 58, 21152119.

http://dx.doi.org/10.1103/PhysRevC.58.2115

[18] Anagnostatos, G.S. and Panos C.N. (1982) Effective twonucleon potential for high-energy heavy-ion collisions. Physical Review C, 26, 260-264.

http://dx.doi.org/10.1103/PhysRevC.26.260

[19] Panos, C.N. and Anagnostatos, G.S. (1982) Comments on a relation between average kinetic energy and meansquare radius in nuclei. Journal of Physics G: Nuclear Physics, 8, 1651-1658.

http://dx.doi.org/10.1088/0305-4616/8/12/007

[20] Hornyak, W.F. (1975) Nuclear structure. Academic, New York, pp. 13, 240, 237.

[21] Anagnostatos, G.S. (1989) Classical equations-of-motion model for high-energy heavy-ion collisions. Physical Review C, 39, 877-883.

[22] Anagnostatos, G.S. and Panos, C.N. (1990) Semiclassical simulation of finite nuclei. Physical Review C, 42, 961965. http://dx.doi.org/10.1103/PhysRevC.42.961

[23] Sherwin, C.W. (1959) Introduction to quantum mechanics. Holt, Rinehart and Winston, New York, p. 205.

[24] Leech, J. (1957) Equilibrium of sets of particles on a sphere. Mathematical Gazette, 41, 81-90.

http://dx.doi.org/10.2307/3610579

[25] Vergados, J.D. (1970) Mathematical methods in physics. Akourastos Giannis.

[26] Anagnostatos, G.S. (2008) A new look at super-heavy nuclei. International Journal of Modern Physics B, 22, 45114523. http://dx.doi.org/10.1142/S0217979208050267

[27] Anagnostatos, G.S. (2008) On the possible stability of tetraneutrons and hexaneutrons. International Journal of Modern Physics E, 17, 1557-1575.

http://dx.doi.org/10.1142/S0218301308010568

[28] Paschalis, S. and Anagnostatos, G.S. (2013) Ground state of 4-7H considered internal collective rotation. Journal of Modern Physics, 4, 66-77.

http://dx.doi.org/10.4236/jmp.2013.45B012

[29] Anagnostatos, G.S. (1991) Fermion/boson classification in micro-clusters. Physics Letters A, 157, 65-72.

http://dx.doi.org/10.1016/0375-9601(91)90410-A

[30] Cundy, H.M. and Rollett (1961) Mathematical models. 2nd Edition, Oxford University Press, Oxford.

[31] Coxeter, H.S.M. (1963) Regular polytopes. 2nd Edition, The Mcmillan Company, New York.

[1] Pauling, L. (1990) Regularities in the sequences of the number of nucleons in the revolving clusters for the ground-state energy bands of the even-even nuclei with neutron number equal to or greater than 126. Proceedings of the National Academy of Science USA, 87, 4435-4438, and references therein.

http://dx.doi.org/10.1073/pnas.87.12.4435

[2] Cook, N.D. (1994) Nuclear binding energies in lattice models. Journal of Physics G (Nuclear and Particle Physics), 20, 1907-1917 and references therein.

[3] Rae, W.D.M. and Zhang, J. (1994) Triangular alpha cluster geometries and harmonic oscillator shell structure. Modern Physics Letters A, 9, 599-607 and references therein.

http://dx.doi.org/10.1142/S021773239400383X

[4] Robson, D. (1978) Many-body interactions from quark exchanges and the tetrahedral crystal structure of nuclei. Nuclear Physics A, 308, 381-428 and references therein.

http://dx.doi.org/10.1016/0375-9474(78)90558-4

[5] Naher, U., Zimmermann, U. and Martin, T.P., (1993) Geometrical shell structure of clusters. Journal of Chemical Physics, 99, 2256-2260.

http://dx.doi.org/10.1063/1.465235

[6] Hecht, L. (1988) The geometrical basis for the periodicity of the elements.

[7] White, (1987) New hypothesis shows geometry of atomic nucleus. Executive Intelligence Review, 14, 18.

[8] Anagnostatos, G.S., Giapitzakis, J. and Kyritsis, A. (1981) Rotational invariance of orbital-angular-momentum quantization of direction for degenerate states. Lettere Nuovo Cimento, 32, 332-335.

http://dx.doi.org/10.1007/BF02745301

[9] Anagnostatos, G.S. (1980) The geometry of the quantization of angular momentum (l, s, j) in fields of central symmetry. Lettere Nuovo Cimento, 28, 573-576.

http://dx.doi.org/10.1007/BF02776343

[10] Anagnostatos, G.S. (1980) Symmetry description of the independent particle model. Lettere Nuovo Cimento, 29, 188-192. http://dx.doi.org/10.1007/BF02743377

[11] Anagnostatos, G.S., Ginis, P. and Giapitzakis, J. (1998) α-Planar states in 28Si”. Physical Review C, 58, 33053315. http://dx.doi.org/10.1103/PhysRevC.58.3305

[12] Kakanis, P.K. and Anagnostatos, G.S. (1996) Persisting α-planar structure in 20Ne. Physical Review C, 54, 29963013. http://dx.doi.org/10.1103/PhysRevC.54.2996

[13] Anagnostatos, G.S. (1999) Alpha-chain states in 12C”. Physical Review C, 51, 152-159.

http://dx.doi.org/10.1103/PhysRevC.51.152

[14] Anagnostatos, G.S. (2013) Quantum isomorphic shell model: Multi harmonic shell clustering in nuclei. Journal of Modern Physics, 4, 54-65.

http://dx.doi.org/10.4236/jmp.2013.45B011

[15] Merzbacher, E. (1961) Quantum mechanics. John Wiley and Sons Ltd., New York, p. 42.

[16] Cohen-Tannoudji, C., Diu, B. and Laloe, F. (1977) Quantum mechanics. John Wiley & Sons Ltd., New York, p. 240.

[17] Anagnostatos, G.S., Antonov, A.N., Ginis, P., et. al. (1998) Nucleon momentum and density distributions in 4He considering internal rotation. Physical Review C, 58, 21152119.

http://dx.doi.org/10.1103/PhysRevC.58.2115

[18] Anagnostatos, G.S. and Panos C.N. (1982) Effective twonucleon potential for high-energy heavy-ion collisions. Physical Review C, 26, 260-264.

http://dx.doi.org/10.1103/PhysRevC.26.260

[19] Panos, C.N. and Anagnostatos, G.S. (1982) Comments on a relation between average kinetic energy and meansquare radius in nuclei. Journal of Physics G: Nuclear Physics, 8, 1651-1658.

http://dx.doi.org/10.1088/0305-4616/8/12/007

[20] Hornyak, W.F. (1975) Nuclear structure. Academic, New York, pp. 13, 240, 237.

[21] Anagnostatos, G.S. (1989) Classical equations-of-motion model for high-energy heavy-ion collisions. Physical Review C, 39, 877-883.

[22] Anagnostatos, G.S. and Panos, C.N. (1990) Semiclassical simulation of finite nuclei. Physical Review C, 42, 961965. http://dx.doi.org/10.1103/PhysRevC.42.961

[23] Sherwin, C.W. (1959) Introduction to quantum mechanics. Holt, Rinehart and Winston, New York, p. 205.

[24] Leech, J. (1957) Equilibrium of sets of particles on a sphere. Mathematical Gazette, 41, 81-90.

http://dx.doi.org/10.2307/3610579

[25] Vergados, J.D. (1970) Mathematical methods in physics. Akourastos Giannis.

[26] Anagnostatos, G.S. (2008) A new look at super-heavy nuclei. International Journal of Modern Physics B, 22, 45114523. http://dx.doi.org/10.1142/S0217979208050267

[27] Anagnostatos, G.S. (2008) On the possible stability of tetraneutrons and hexaneutrons. International Journal of Modern Physics E, 17, 1557-1575.

http://dx.doi.org/10.1142/S0218301308010568

[28] Paschalis, S. and Anagnostatos, G.S. (2013) Ground state of 4-7H considered internal collective rotation. Journal of Modern Physics, 4, 66-77.

http://dx.doi.org/10.4236/jmp.2013.45B012

[29] Anagnostatos, G.S. (1991) Fermion/boson classification in micro-clusters. Physics Letters A, 157, 65-72.

http://dx.doi.org/10.1016/0375-9601(91)90410-A

[30] Cundy, H.M. and Rollett (1961) Mathematical models. 2nd Edition, Oxford University Press, Oxford.

[31] Coxeter, H.S.M. (1963) Regular polytopes. 2nd Edition, The Mcmillan Company, New York.