Lateral Position Uncertainty of Electrons in Bohr Hydrogen-like Atoms: An Implication of Heisenberg Uncertainty Principle

Year 2019, Volume: 9 Issue: 2, 417 - 430, 30.12.2019

Serkan Alagöz

https://doi.org/10.37094/adyujsci.593724

Abstract

This paper presents a theoretical investigation on effects of lateral position uncertainty of captivity electrons within spherical electron shells of Bohr hydrogen-like atoms. A captivity electron, which is spatially confined in Bohr orbits, introduces a lateral position uncertainty that can be determined by considering the area of the electron shell. After deriving uncertainty relation for position and kinetic energy, author theoretically demonstrates that, due to the lateral position uncertainties of electrons in spherical shells, Heisenberg uncertainty principle suggests uncertainty bounds in measurement of kinetic energy states of captivity electrons that orbits non-relativistic hydrogen-like Bohr atom. Afterward, these analyses are extended for relativistic hydrogen-like Bohr atom case.

Keywords

Hydrogen-like Bohr atom, Heisenberg uncertainty principle, position uncertainty, kinetic energy uncertainty

Bohr Hidrojen Benzeri Atomlarda Elektronların Yanal Konum Belirsizliği: Heisenberg Belirsizlik İlkesinin Uygulanması

Abstract

Bu makale, Bohr hidrojen benzeri atomların küresel elektron yörüngelerinde bulunan elektronlarının yanal konum belirsizliğinin etkileri üzerine teorik bir araştırma sunmaktadır. Bohr yörüngelerinde uzamsal olarak hapsolmuş bir elektron, elektron kabuğunun alanı göz önüne alınarak belirlenebilecek bir yanal konum belirsizliği sağlar. Konum ve kinetik enerji için belirsizlik ilişkisini türettikten sonra, yazar teorik olarak, küresel kabuklar içindeki elektronların yanal konum belirsizliklerini teorik olarak göstermektedir. Heisenberg belirsizlik ilkesi, göreceli olmayan hidrojen benzeri Bohr atomunun yörüngesinde bulunan esaret elektronlarının kinetik enerji durumlarının ölçümündeki belirsizlik sınırlarını ortaya koymaktadır. Daha sonra, bu analizler göreceli hidrojen benzeri Bohr atomu durumu için genişletilmiştir.

Keywords

Hidrojen benzeri Bohr atomu, Heisenberg belirsizlik ilkesi, Konum belirsizliği, Kinetik enerji belirsizliği

References

• [1] Smith, B., Lecture Notes: Quantum Ideas, Department of Physics University of Oxford, 2012.
• [2] Shankar, R., Principle of Quantum Mechanics, Plenum Press, New York, 1994.
• [3] Adler, R.J., Chen, P., Santiago, D.I., The generalized uncertainty principle and black hole remnants, General Relativity and Gravitation, 33(12), 2101-2108, 2001.
• [4] Beiser, A., Mahajan, S., Choudhary, S.R., Concepts of Modern Physics, McGraw-Hill, New Delhi, 2009.
• [5] Adler, R. J., Santiago. D.I., On gravity and the uncertainty principle, Modern Physics Letters A, 14(20), 1371-1381, 1999.
• [6] Harbola, V., Using uncertainty principle to find the ground-state energy of the helium and a helium-like Hookean atom, European Journal of Physics, 32(6), 1607-1615, 2011.
• [7] Akhoury, R., Yao, Y.P., Minimal length uncertainty relation and the hydrogen spectrum, Physics Letters B, 572, 37-42, 2003.
• [8] Gill, P.M.W., Johnson, B.G., Pople, J.A., A standard grid for density functional calculations, Chemical Physics Letters, 209, 506-512, 1999.
• [9] Wilhelm, H.E., Formulation of the uncertainty principle according to the hydrodynamic model of quantum mechanics, Progress of Theoretical Physics, 43(4), 861-869, 1970.
• [10] Olszewski, S., Bohr’s spectrum of quantum states in the atomic hydrogen deduced from the uncertainty principle for energy and time, Journal of Modern Physics, 5(14), 1264-1271, 2014.
• [11] Deeney, F.A., O’Leary, J.P., The effects of the Pauli exclusion principle in determining the ionization energies of the helium atom and helium-like ions, European Journal of Physics, 33(3), 667-675, 2012.
• [12] Kuo, C.D., The uncertainties in radial position and radial momentum of an electron in the non-relativistic hydrogen-like atom, Annals of Physics, 316(2), 431-439, 2005.
• [13] Bohr, N., On the Constitution of Atoms and Molecules-Part I: The binding of electrons by positive nuclei, Philosophical and Journal of Magazine Science, 26(6), 1-25, 1913.
• [14] Bohr, N., On the constitution of atoms and molecules - Part II: systems containing only a single nucleus, Philosophical and Journal Of Magazine Science, 26(6), 476-502, 1913.
• [15] Serway, R.A., Moses, C.J., Moyer, C.A., Modern Physics, Thomson, New York, 1997.
• [16] Weaver, J.H., The World of Physics, Simon and Schuster, New York, 1987.
• [17] Wheeler, J.A., Zurek, H., Quantum Theory and Measurement, Princeton Univ. Press, New Jersey, 1983.
• [18] Schiff, L.I., Quantum Mechanics, McGraw Hill, New York, 1968.
• [19] Heisenberg, W., Über den anschaulichen Inhalt der quantentheoretischen kinematik und mechanik, Zeitschrift für Physik, 43(3-4), 172-198, 1927.
• [20] Busch, P., Heinonen, T., Lahti, P., Heisenberg’s uncertainty principle, Physics Reports, 452, 155-176, 2007.
• [21] Bohr, N., On the constitution of atoms and molecules, Philosophical and Journal of Magazine Science, 26(6), 1-25, 1913.
• [22] Terzis, A.F., A simple relativistic Bohr atom, European Journal of Physics, 29(4), 735-743, 2008.
• [23] Mandache, N.B., On the physical interpretation of the electromagnetic coupling constant \alpha, Romanian Reports in Physics, 64, 1307-1312, 2012.
• [24] Arjun, T.T., On geometric representation of uncertainty relation, Chinese Journal of Physics, 8(1), 26-27, 1970.