Symmetry Principles and Conservation Laws in Atomic and ...

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GENERAL ARTICLEConservation ofenergy followsfrom the symmetryprinciple that theLagrangian isinvariant withrespect to time.From the above, it immediately follows that when theLagrangian depends on time only implicitly through itsdependence on q and q, : then:¸d @L :q ¡ L = 0 ; (10)h³which implies@L@ : qdt@ q:´ i :q ¡ L is a conserved quantity. Thisquantity is called the Hamiltonian, or Hamilton's principalfunction, of the system, which for a conservativesystem is essentially the same as the total energy of thesystem. This can be seen easily by identifying the generalizedmomentum and substituting T ¡ V for the Lagrangian.We thus see that conservation of energy followsfrom the symmetry principle that the Lagrangianis invariant with respect to time.These results illustrate an extremely powerful theoremin physics, known as the Noether's theorem, which canbe stated informally as:If a system has a continuous symmetry property, thenthere are corresponding quantities whose values are conservedin time [1].This theorem is named after Noether (1882{1935), ofwhom Einstein said:If a system has acontinuoussymmetryproperty,then there arecorrespondingquantities whosevalues areconserved in time.In the judgement of the most competent living mathematicians,FrÄaulein Noether was the most signi¯cantcreative mathematical genius thus far produced since thehigher education of women began [2].4. Symmetry Principles and Physical LawsWe have now seen that both the equation of motionand the conservation principles result from the singleprinciple of least action. Moreover, the same principleprovides for the connection between symmetry and conservationlaws, exalted by Noether to one of the mostprofound principles in contemporary physics. We now838 RESONANCE September 2010
GENERAL ARTICLEask if the conservation principles are consequences of thelaws of Nature, or, rather the laws of Nature are consequencesof the symmetry principles that govern them?Until Einstein's special theory of relativity, it was believedthat conservation principles are the result of thelaws of Nature. Since Einstein's work, however, physicistsbegan to analyze the conservation principles asconsequences of certain underlying symmetry considerationsfrom which they could be deduced, enablingthe laws of Nature to be revealed from this analysis.Wigner's profound impact on physics is that his explanationsof symmetry considerations using `group theory'resulted in a change in the very perception of just whatis most fundamental, and physicists began to regard`symmetry' as the most fundamental entity whose formwould govern the physical laws. Wigner was awardedthe 1963 Nobel Prize in Physics for these insights [3].The conservation of linear and angular momentum weillustrated above are consequences of invariance undercontinuous displacements and rotations respectively inhomogenous and isotropic space. Likewise, the conservationof energy is a consequence of invariance undercontinuous temporal displacement.A detailed exposition of the governing symmetry principlesrequires group theoretical methods, and is clearlybeyond the scope of this article, but we continue to dwellon some other kinds of symmetries now and examinetheir connections with conservation principles.Figure 1. Masters of symmetry.5. Dynamical Symmetry: Laplace{Runge{LenzVectorIt is well known that in the classical two-body Keplerproblem (gravitational Sun{Earth system, or the Coulombicproton{electron planetary model of the old-quantumtheoryof the hydrogen atom), both energy and angularmomentum are conserved. We have already discussedRefer to Resonance issues on:Einstein, Vol.5, March and April2000.Noether, Vol.3,September 1998.Wigner, Vol.14, October 2009.RESONANCE September 2010839
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