From the Bohr theory to modern atomic quantum theory and the double helix of magnetic field. by Dan Petru Danescu,

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1 From the Bohr theory to modern atomic quantum theory and the double helix of magnetic field by Dan Petru Danescu, Date: July 28,

2 I. Interpretation of Bohr-de Broglie Atom Model a) Bohr s theory From Bohr s postulates we have (simplifyed and Z = 1): h Bohr s condition: L n = n = n [1-4]; 2π Equilibrium equation: F coulombic = F centripetal 2 e m = e v 2 (Gaussian units) [5]; 2 r r Transforming r r n and v v n we get: r n = n 2 = n cm; v n = n = n cm s -1, e 2 m m e e r n where, r 1 = a 0 (Bohr s radius) = cm. 2 The energy of electron on n orbit (E n ) is: m v E n = E kin(n) + E pot(n) = e n 2 e2 e2 1 e4m - = - = - e 2 rn 2rn n The energy of ground state (n=1) is: E 1 = erg (-13.6 ev). The frequency and spectral series is given by the formula: 1 1 e ν n,m = (E m E n ) 4 m = e 1 1 h 4π 3, n 2 m 2 1 e where, 4 m e = Hz (Rydberg constant - R 4π 3 c). Considering helicity of magnetic field [6-11] and knot-like particles structure [12-23], (Fig.1) we have 2

3 or E = hν 1 4π E 1 λ 1 1 = c 4π E 1 λ 1 = c E 2 λ 2 =... c E 2 λ 2 1 =... c 4π E n λ c n = const. (=h), E n λ n = const. (= ). c 2 The wavelength λ 1 associated with constant is : c c λ 1 = 2π 3 1 = 4π = E 1 e4 = cm, me R Fig.1 (interpretation of electromagnetic interaction) and Fig.2 (interpretation of Bohr s model) b) de Broglie hypothesis According to de Broglie hypothesis [24-26], an electron with impulse p = m e v should have a wavelength of λ = h = p h m v e. In the case of an circular motion of electron in the hydrogen atom, from de Broglie hypothesis result h 2π r = nλ = n mev. We observ that Bohr s condition (in the case n=1) are correspondent to the de Broglie hypothesis: L 1 = = 2π h = me vr 2π r = h m v e. In 1927 this hypothesis was confirmed when the diffraction of electrons was observed experimentally by C.J. Davisson [27]. The author s interpretation of the Bohr-de Broglie atom model with circular orbit is shown in Fig.3. 3

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5 Fig.3 Illustration of Bohr-DeBroglie atom model 5

6 References [1] N. Bohr, On the Constitution of Atoms and Molecules Part I, Philosophical Magazine 26, 1-25, (1913). [2] N. Bohr, On the Constitution of Atoms and Molecules Part II Systems Containing Only a Single Nucleus, Philosophical Magazine 26, , (1913). [3] N. Bohr, On the Constitution of Atoms and Molecules Part III Systems containing several nuclei, Philosophical Magazine 26, , (1913). [4] N. Bohr, Atomic structure, Nature 106, , (1921). [5] D.P.Danescu, Consideraţii asupra impedanţei caracteristice a vidului, Buletinul Ştiinţific şi Tehnic al Institutului Politehnic "Traian Vuia",Timişoara, 26, (40), fascicola 2, 1981, pp [6] A.Y.K. Chui, H.K. Moffatt, The energy and helicity of knotted magnetic flux tubes, Proc. R. Soc. Lond. A (1995) 451, [7] H.K. Moffatt, R.L. Ricca, Helicity and the Calugareanu invariant, Proc. R. Soc. Lond. A (1992) 439, [8] V.B. Semicoz, D.D. Sokoloff, Magnetic helicity and cosmological magnetic field, arxiv: astro-ph/ v3, 7 Apr. (2005). [9] M. A. Berger, Introduction to magnetic helicity, Plasma Phys. Control. Fusion, 41, B167-B175 (1999). [10] *** Magnetic helicity Wikipedia, Internet. [11] *** Magnetic helicity Nation Master Encyclopedia, Internet. [12] A.D. Sakharov, The Topological Structure of Elementary Charges and CPT Symmetry, in COLLECTED SCIENTIFIC WORKS (New York: Marcel Dekker, 1982). [13] D.P. Danescu,.Spin of the Electron Interpreted as a Revolution and Translation Motion. Elementary Electric Charge Structure and CPT Symmetry, Preprint IFA - Bucuresti nr. 19 / 25 apr. 1977, pp [14] D.P. Danescu,Transformarea prin inversiune.aplicaţii, Gazeta Matematică, Anul LXXXIII, nr.3, (1978), pp [15] D.P. Danescu, Efectul Compton şi structura sarcinii electrice elementare. Implicaţii epistemologice, Revista Învăţămîntului Superior "FORUM", 11, (1979), pp [16] D.P. Danescu, O interpretare a constantei structurii fine, Buletin de Fizică şi Chimie, Volumul 12-13, ( ), pp [17] D.P. Danescu,.Despre electronul cuantic, Revista de Fizică şi Chimie, Volumul 36, Nr , apriliemai-iunie, (2001), pp [18] D.P. Danescu, Despre norul electronic şi structura electronului dedusă din considerente cuantice, Revista de Fizică şi Chimie, Volumul 36, Nr , iulie-august-septembrie, (2001), pp [19] D.P. Danescu, Electronul ca proces cinematic, Partea I, Revista de Fizică şi Chimie, Volumul 37, Nr , aprilie-mai-iunie, (2002), pp [20] D.P. Danescu, Electronul ca proces cinematic, Partea II, Revista de Fizică şi Chimie, Volumul 37, Nr , iulie-august-septembrie, (2002), pp.1-7. [21] D.P. Danescu, Reprezentarea atomului de hidrogen în starea 1s reunind conceptele cuantice şi relativiste, Revista de Fizică şi Chimie, Volumul 37, Nr , octombrie-noembrie-decembrie, (2002), pp.4-9. [22] Z. Was, Trefoil knot and ad-hok classification of elementary fields in the standard model, Phys. Letters B, vol.416, p.369 (1998). [23] L. Fadeev, A.J. Niemi, Stable Knot-like structures in classical field theory, Nature, 387, 58 (1997). [24] L. de Broglie, Recherches sur la theorie des quanta, Thesis, Paris (1924). [25] L. de Broglie, Waves and Quanta, Comptes rendus 177, (1923). [26] L. de Broglie, Ondes et mouvements, Paris, Gauthier Villars (1926). [27] C.J. Davisson and L.H. Germer, Diffraction of Electrons by a Crystal of Nikel, Phys. Rev. 30, (1927) 6

7 II. Interpretation of Sommerfeld s Atom Model a) Plane quantization A. Sommerfeld proposed elliptical orbits in addition to Bohr s circular ones (Fig.4). In the case of elliptic motion, the electron speed to be decompose in two components: radial speed (v r ) and azimuthal speed (v ϕ ). Corresponding himself introducting two quantum integer numbers: n r - radial quantum number = (n-1), (n-2), 0, and n ϕ - azimuthal quantum number = 1, 2, 3,, with condition n r + n ϕ = n. Sommerfeld introduced quantum condition, 2π 0 L.d ϕ = n ϕ h, p r.dr = n r h, where L (or p ϕ ) is the angular momentum and p r is the linear momentum. The angular momentum of electron (Fig.4a) is: L = rm e vsinα = rm e v ϕ = const. The angular momentum is quantified (exactly of Bohr s model): h L n = n ϕ = n ϕ. 2π The azimuthal quantum number describes the shape of the electron s orbits. For n=1 and n ϕ =1, n r = 0, the orbit can be only circular; for n=2 there are two orbits of different shapes: n ϕ =2, n r = 0 circular shape, and n ϕ =1, n r = 1 elliptic shape; for n=3 there are three orbits of different shapes: n ϕ =3, n r = 0 circular shape, n ϕ =2, n r = 1 elliptic shape; and n ϕ =1, n r = 2 elliptic shape etc., [1-4] The author s interpretation of this model is shown in Fig.4b. 7

8 Fig.4 The Bohr-Sommerfeld atomic model : a) Cicular and elliptic orbits; b) Author s interpretation 8

9 b) Spatial quantization A. Sommerfeld have a crucial contribution by quantized the z-component of the angular momentum [1-4]. In old quantum theory this idea called space quantization (Richtungsquantelung). From general condition of old quantum mechanics (in spherical coordinates r, ϑ,ψ ): p i.dqi = n i h, result and p r.dr = n r h, p ϑ. d ϑ = n ϑ h, p ψ.dψ = n ψ h. p ψ = p cos α = n ϕ cos α. Because we get h p ψ = n ψ = nψ, 2π n ψ = n ϕ cos α. Change notation n ψ m, we have: cos α =m/n ϕ and p ψ =(m/n ϕ )p. Orientation in space of electron s plane orbit is described by quantum integer numbers m, m= n ϕ, n ϕ -1, , - n ϕ. The author s interpretation of this model,with Schrodinger s amendment, L n = n n = 1, 2,... n L n = ( + 1) l = 0, 1, 2...(n-1). l is the orbital quantum number. is shown in Fig.5. References [1] N. Bohr, On the Constitution of Atoms and Molecules Part I, Philosophical Magazine 26, 1-25, (1913). 9

10 [2] N. Bohr, On the Constitution of Atoms and Molecules Part II Systems Containing Only a Single Nucleus, Philosophical Magazine 26, , (1913). [3] A. Sommerfeld, Zur Quantentheorie der Spektrallinien, Ann. d. Physik, 51, 1 (1916). [4] A. Sommerfeld, Atombau und Spektrallinien, Friedrich Vieweg & Sohn, Braunschweig (1919). 10

11 Fig.5 Interpretation of space quantified for n=2 11

12 III. Interpretation of Schrodinger s Atom Model Starting from Fig.5,it is possible tu imagine the atomic structure according to Schrodinger s theory [1]-[4], considering the incertainty relation of Heisenberg [5], q. p > /2. In this case, the electron trajectory lose its significance. The electron cloud is replacing the electron trajectory (Fig.6) References [1] E. Schrodinger, Quantisierung als Eigenwertproblem (Erste Mitteilung) Ann. Phys., 79, pp , (1926) [2] E. Schrodinger, Quantisierung als Eigenwertproblem (Zweite Mitteilung) Ann. Phys., 79, pp , (1926) [3] E. Schrodinger, Quantisierung als Eigenwertproblem (Dritte Mitteilung) Ann. Phys., 80, pp , (1926) [4] E. Schrodinger, Quantisierung als Eigenwertproblem (Vierte Mitteilung) Ann. Phys., 81, pp , (1926) [5] W. Heisenberg, Uber de anschaulichen Inhalt der quantentheoretischen Kinematik unde mechanic, Zaitschrift fur Physik, 43, 172 (1927) 12

13 Fig.6 From Sommerfeld s interpretation to modern quantum theory 13