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Chaotic Behaviour of Atomic Energy Levels

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Chaotic Behaviour of Atomic Energy Levels Chaotic Behaviour of Atomic Energy Levels A. YILMAZ a, G. HACIBEKIROGLU a, E.l BOLCAL a and Y. POLATOGLU b aDepartment of Physics, TC Đstanbul Kultur University, 34156 Đstanbul, Turkey bDepartment of Mathematics and Computer Science, TC Đstanbul Kultur University, 34156 Đstanbul, Turkey Abstract: Theauthors of this paper studied Schrodinger waveequation to investiagatethe chaotic behavior of atomic energy levels in relation with threequantum numbers n, l, m by means of derived inequality. It could give rise to thesiplitting of atomic spectral lines. Keywords: Chaos, Schrödinger waveequation, atomic energy levels Introduction: In an atomic spectra that measureradiation absorbed or emitted by electrons "jumping" from one"quantum state" to another, a quantum stateis represented by values of n, l, and m. So called "selection rules" limit what "jumps" arepossible. Generally a jump or "transition" is only allowed if all thesethreenumbers changein the process. This is becausea transition will beableto causetheemission or absorption of electromagnetic radiation if it involves a changein electromagnetic dipoleof theatom. On theother hand, closeexamination reveals that someof thespectral lines display splitting as thefollows, • So-called finestructure splittting occurs becauseof an interaction between thespin and and motion of theoutermost electron ( spin-orbit coupling ). • Someatoms can havemultipleelectron configurations with thesameenergy level, thus appears a s singleline. Theinteraction of themagnetic field with theatom shifts theseelectron configurations to slightly different energies, resulting in multiplespectral lines (Zeeman effect ). • Thepresenceof an external electric field can causea comparablesplitting and shifting of spectral lines by themodifying theelectron energy levels (Stark effect ). It was theeffort to explain this radiation that led to thefirst successful quantum theory of atomic structure, developed by Niels Bohr in 1913. Bohr’s one-dimensional model used onequantum number to describethe distribution of electrons in theatom. Theonly information that was important was thesize of theorbit, which was described by then quantum number. Although theBohr theory does a good job of predicting energy levels for thehydrogenic ( one-electron ) atom , and fails even for helium, it predicts nothing about transition rates between levels. In 1916 the fine-structure constant was introduced into physics by Arnold Sommerfeld as a measureof the relativistic deviations in atomic spectral lines from the predictions of theBohr model. It appears naturally in Sommerfeld's analysis and determines thesizeof thesplitting or fine-structure of thehydrogenic spectral lines. However theproof of Sommerfeld is not effective in a alkali metals. Finestructuresplitting indicates that up to two electrons can occupies a singleorbital. Hovewer, two electrons can never havethe sameexact quantum numbers according to Hund’s rule, which addresses thePauli’s exclusion principle. In 1925 thediscoveries of Goudsmit and Uhlenbeck suggested that theelectron itself might havean intrinsic angular momentum that was (somehow) half as large as the smallest allowable nonzero orbital angular momentum - what wenow call “ spin 1/2 “. In 1926 Schrödinger extended thedeBroglieconcept of matter waves to thewavefunction concept, by providing a formal method of treating thedynamics of physical particles in terms of associated waves. to describequatum stateof a singleelectron bound to theatomic nucleus by means of quantum numbers to thethree-dimensional wavefunction model of theatom. Theprincipal quantum number (n), arosein thesolution of theradial part of thewaveequation, theazimuthal quantum number (l), arosein thesolution of thepolar part of thewaveequation, and themagnetic quantum number(m) arosein thesolution of theazimuthal part of thewaveequation. Physical meaning of thequantum numbers can beexplained as thefollowings: Theprincipal quantum number ( n) ( n = 1, 2, 3, 4 ... ) describes thesizeof theorbital. Thesets of orbitals with thesamen-valueareoften referred to as electron shells or energy levels. Theangular quantum number ( l) (l = 0, 1 ... n−1 ) describes theshapeof theorbital and for an atomic orbital Thevarious orbitals relating to different values of l aresometimes called sub-shells. Themagnetic quantum number ( m),. (ml = −l, −l+1 ... 0 ... l−1, l) represents thenumber of possiblevalues for availableenergy levels of a subshell l. It determines theenergy shift of an atomic orbital dueto an external magnetic field, hencethenamemagnetic quantum number (Zeeman effect). Thespin quantum number (s) ( ms = −1/2 or +1/2 ), theintrinsic angular momentum of theelectron to explain the existence of two electrons in thesameorbital. It determines spin-orbit coupling effect to result in fine structuresplitting. [1] [2] All thesefour quantum numbers quantum numbers n, l, m, and s specify thecompleteand uniquequantum state of a singleelectron in an atom called its wavefunction or orbital. In thenext section, alternatively, weproposed that thechaotic behavior of theSchördinger wavefunction related to theelectron in an atom would deform thestructureof energy levels from singlelineto multiple lines. In turn this would explain the splitting or shifting of atomic spectral lines in relation with threequantum numbers n, l, m through the derived inequality. Calculations: To get thenessary condition for the chaotic behavior of energy levels of an hydrogen atoms we utilize theradial m M 2 Schrödinger equation of a particleof a reduced mass µ = moving at thepotential V (r)= − Ze / r in m + M threedimensional spaceis d dR 2µ 2r 2 Ze 2 (ll +1)h 2 2 + + − = (1.1) r 2 E 2 2 0 dr dr h r 2µ r Thepotential V (r) is attractive (V ≤ )0 , then only theexistenceof linked states arepossiblewhen ( E < )0 . For this reason, in this paper wewill study (investigate) theconnected states. If ρ is a variableand λ is energy parameter, then ρ = ar Ze 2 µ (1.2) λ = − , a 2 = −8µE / h 2 , E < 0 h 2E by using thetransformations is defined by theequation (1.2), thedifferential equation (1.1) is dimensionless then it can bedefined as d ρ 2 dR + λρ + 1 ρ 2 − ()− = (1.3) ll 1 R 0 dρ dρ 4 Otherwise(by theway), theasymptotic behaviaour of theradial function is thefollowing form for small ρ values , lim R(r) ≅ r l (1.4) r → 0 d dR 1 for big ρ values , ()ρ → ∞ (ρ 2 ) − ρ 2 R ≅ 0 dρ dρ 4 r → 0 thegeneral solution of differential equation (1.4) is −ρ / 2 ρ / 2 R(ρ )= A.e + B.e (1.5) When thesolution function is divergent at infinity, then B must beequal to zero. If weconsider this asymptotic behaviours, for thewaveequation R(ρ ) wetakethefollowing solutions into account as l −ρ / 2 R(ρ )= ρ e L(p) (1.6) by using theequation (1.6) in theradial differential equation (1.3), weobtain thefollowing differential equation d 2 R dL ρ + ()2l + 2 − ρ + ()λ −l −1 L = 0 (1.7) dρ 2 dρ If weuseseries method to solvethedifferential equation (1.7), for thesolution wecan definea following divergent series ∞ ()ρ = + ρ + ρ 2 + ρ 3 + = ρ k L a0 a1 a2 a3 ... ∑ck (1.8) k =0 and dL = + ρ + = ()+ ρ k ρ dL = ρ + ρ 2 + = ρ k a1 2a2 ... ∑ k 1 ak+1 , a1 2a2 ... ∑k ak dρ k dρ k (1.9) d 2 L d 2 L = 2.1 a + 3.2 a ρ +... = (k +1 )(k + 2 )a ρ k , ρ = 2.1 a ρ + 3.2 a ρ 2 +... ρ 2 2 3 ∑ k +1 ρ 2 2 3 d k d = ()+ ρ k ∑k k 1 ak +1 k To writetheequation (1.9) in thedifferential equation (1.3), wecan find k [ ( + ) ( + ) + − + (λ − − ) ] ρ = (1.10) ∑ k k 1 ak +1 2l 2 (k )1 ak +1 ka k l 1 ak 0 From theequation (1.10), therecurrancy relation is k + l +1− λ a + = a (1.11) k 1 (k +1 )(k + 2l + 2 ) k and given by theequation (1.8) that wetakeD’alembert criteria for theconvergency of thesolution wherethe coefficients provides therecurrancy relation C + k 1 lim k 1 → = 2 (1.12) Ck k k k → ∞ then from theequation (1.12) 2 k ρ ρ ρ e =1 + ρ + + ... = ∑ 2 ! k k ! by this series expansion wecan obtain a + k ! 1 1 lim k 1 = = ≅ ()+ + a k k 1 ! k 1 k k → ∞ ρ This shows us, our solution function behaves like( e ) series expansion. So for theinfinity terms, theseries ρ solution ( ρ → ∞ ) situated for L(ρ) is divergent like( e ). It is connected to this, thewaveequation R(ρ ) ρ ρ ρ l − l − l R()ρ = ρ e 2 L()ρ = ρ e 2 ≅ ρ e 2 will bedivergent. Weremovetheconditions of to bedivergent, for thesolution function has a finiteterms which means it is polinomial. Consequently in therecursion relation (1.11), thenumerator has been zero after by the = definite( kmax ) index. This means for a given l, for some k kmax , thefact = ( ) + + = n kmax l 1 and kmax 2,1,0 ,... (1.13) = − − takes every valueon theequation (1.13), theorbital quantum number l n 1 kmax just taken l = n − ,1 n − ,2 ..., 0,1,2 values or it must be l ≤ n −1. When weusetheequation bounding n parameter to theenergy Ze 2 µ n = − (1.14) h 2E theatomic energy levels areobtained, theseare ( α )2 1 2 Z E = − µ c (n= 3,2,1 ,.... ) (1.15) n 2 n 2 If weset λ = n , then therecursion relation (1.11) after somecalculations is k+1 n − (k + l +1) n − (k + l) n − (l +1) a + = ()−1 ⋅ ⋅⋅⋅ a (1.16) k 1 (k +1 )(k + 2l + 2 ) k ()k + 2l +1 1⋅ ()2l + 2 0 and using the power series expansion for (1.8), weobservethat this equation is called as associated Laguerre polynomials [3] (ρ )= 2l +1 (ρ) H Ln−l−1 (1.17) Therefore, thesolution of (1.17) is given as −ρ ()ρ = ρ l 2 2l +1 ()ρ Rnl N nl e Ln+l (1.18) This radial wavefunction solution what welook for radial providing a relation between theorbital and angular quantum numbers. Up to now, theanswer of thefollowing question was not given, “ What kind of relation between thequantum numbers will definea chaotic behaviour of an energy levels?“.
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