Symmetry Principles and Conservation Laws in Atomic and ...

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GENERAL ARTICLEthe associated symmetries. What is interesting is thatthe elliptic orbit of the Kepler system for bound statesis ¯xed, i.e., the ellipse does not precess (Figure 2).Figure 2. If the eclipse wereto precess it would generatewhat is called a ‘rosette’motion since the trajectoryof the planet would seem togo over the petals of a rose,if seen from a distance.Can we then ¯nd a symmetry that would explain theconstancy of the orbit? It turns out that the orbit itselfremains ¯xed if and only if the potential in which motionoccurs is strictly of the form ¡1=r and the associatedforce is of the form ¡1=r 2 . This is true for both thegravitational and the Coulomb potential, and hence theKepler elliptic orbits remain ¯xed. This is rigorouslyexpressed as the constancy of the Laplace{Runge{Lenz(LRL) vector. The LRL vector is de¯ned as:³~A = ~v £ H ~ ´¡ ê ½ (11)and is shown in Figure 3 [4]. In the above equation ~vis the `speci¯c' linear momentum and H ~ is the `speci¯c'angular momentum. The term `speci¯c' denotes the factthat the physical quantities linear momentum and angularmomentum, which are being referred to, are de¯nedper unit mass. Likewise in the second term of the LRLvector, is the proportionality in the inverse distancegravitational potential per unit mass of the planet. Itcan be easily veri¯ed that the time derivative of the LRLvector vanishes, and the A ~ is therefore a conserved quantity.Its direction is from the focus of the ellipse to theperihelion (Figure 3) [4], which has a direction along themajor axis of the ellipse, thus holding the ellipse ¯xed.Figure 3. Schematic diagramshowingtheLaplace–Runge–Lenz vector, ~A .840 RESONANCE September 2010
GENERAL ARTICLEThe constancy of the LRL vector is a conservation principle,and since the governing criterion involves dynamics(namely that the force must have a strict inversesquare form), the associated symmetry is called `dynamicalsymmetry'. Sometimes, it is also called an àccidental'symmetry. This symmetry breaks down when thereis even a minor departure from the inverse square lawforce, as would happen in a many-electron atom, suchas the hydrogen-like sodium atom. The potential experiencedby the òuter-most' electron goes as 1=r only inthe asymptotic (r ! 1) region. Close to the center,the potential goes rather as ¡Z=r, due to the reducedscreening of the nuclear charge by the orbital electrons,and thus departs from 1=r. This di®erence in the hydrogenatom potential and that in the sodium atom is dueto the quantum analogue of the breakdown of the LRLvector constancy in the sodium atom. Using group theoreticalmethods, Vladmir Fock (1898{1974) explainedthe dynamical symmetry of the hydrogen atom [5].Using the language of group theory, the Fock symmetryaccounts for the (2l + 1)-fold degeneracy of the hydrogenatom eigenstates. This degeneracy is lifted for thehydrogen-like sodium atom due to the breakdown of theassociated symmetry. In atomic physics, this is oftenexpressed in terms of what is called as `quantum defect'¹ n;l which makes the hydrogenic energy eigenvalues dependnot merely on the principal quantum number n butalso on the orbital angular momentum quantum numberl. This enables the use of the hydrogenic formulafor energy with n replaced by n e®ective = n ¡ ¹ n;l . The`quantum-defect theory' has very many applications inthe analysis of the atomic spectrum, including the àutoionizationresonances' [6,7]. As pointed out above,the conservation of angular momentum is due to the rotationalsymmetry, referred to as the symmetry underthe group SO(3). All central ¯elds have this symmetry.However, the inverse-square-law force (such as gravity orRESONANCE September 2010841
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