We construct O(4)-invariant hydrogen wave function in coordinate representation

Granovskii, Ya.I.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

ON THE HYDROGEN SYMMETRYYa.I. GranovskiiInstitute of Physics, Szczecin University, Szczecin, Poland e-mail: gran@wmf.univ.szczecin.plWe construct O(4)-invariant hydrogen wave function in coordinate representation.PACS: 03.65Ge, 11.30LyThe symmetry mentioned above is the famous O(4)-symmetry of the nonrelativistic hydrogen atom, which was transformed by V.A. Fock from the hidden form to the explicit one [1]. Fock considered the problem at the wave-function level:  he pointed out  that  Schroedinger equation in omentum representation is nothing else as the  defining  property  of  the  4-dimensional  spherical harmonics.At the operator level this symmetry was well known long  ago:  10  years  before  Fock  the  hydrogen  energy was quantized by W. Pauli [2] just using two conserved operators - orbital momentum L and Laplace vector K . As was shown by V. Bargmann [3] both are generators of the O(4)-symmetry.For more than 60 years still remains without answer the question - how to see this symmetry immediately in coordinate representation? Aside of a simple curiosity this question is also of practical meaning because it is more easy to work with symmetric quantities.Indeed, what is the symmetry? In a simplest case it is an independence of the function on the corresponding variable.  For  example,  an  axial  symmetry means  that function has not angle ϕ among its arguments. Another example, isotropy, means that the mentioned axis may be rotated in an arbitrary mode -- now the angle θ drops out of consideration. The isotropic function depends not on three variables x,y,z but only on the one combination ofthem  222 zyxr ++= .  Just  this  contraction  is  the main cause of most simplifications when working with symmetric entities.Let us turn to the hydrogen problem. Its HamiltonianrZempH /2/ 22 −=            (1)is explicitly isotropic. This property is usually used to separate the angles from radius, factorizing the spherical function Ylm(θ,ϕ) from the radial Rnl(r) one.But  then  the  Fock  symmetry  becomes  completely elusive: indeed, O(4)--symmetry mixes the states with different  values  of  the  quantum number  while  during separation  of  the  angles  this  number  is  kept  fixed. Therefore  our  goal  may be  achieved  only  in  the  one way: to find the wave function of the given energy state without  expanding  in  the  partial  waves  --  it  must  be some linear combination of the ''standard'' radial-angular solutions.First  we  separate  the  asymptotic  factor  exp(-λr) bearing in mind that state is bound, E<0, and λ=(2mE/2 )1/2. Schroedinger equation takes on the form 0)()1(222 =Φ−−∇−∇ RiRRR ξ,       (2)where  we  have  introduced  the  dimensionless  radius rR  λ=  and  insert  the  notation  vZe /2=ξ  for  the dimensionless Coulomb parameter.Axially  symmetric  wave  function  )(RΦ  must  not depend on the angle ϕ. Thus, two possible arguments R-z and R+z remain. The wave function, which depends only on one of them u=R+z1, )()( ufR =Φ ,    (3)has a property being constant on the paraboloid u=R+z -- displacements along its surface do not change it.One  of  these  displacements  is  motion  along  the ''parallel'' and is simply a rotation around the z-axis; it generates the axial symmetry mentioned above. But the other  one  --  displacement  along  the  ''meridian''  --  is something new. Parametrizing the paraboloid asR=ucos2τ,    z=usin2τ          (4)we can change the angle τ by arbitrary additive: τ→τ+δ. The  τ is  that  same  third  angle  needed  for  the  4-dim symmetry!To ''liberate'' the z -axis it is sufficient to write the argument  as  RRuµ+=  using  unit  vector  µ.  No property will be lost, only the axis of the paraboloid will change its direction, now pointing along that vector.It is not difficult to treat the positive energies: the asymptotic factor will be exp(ikr) with k=(+2mE/ 2 )1/2 dimensionless radius becomes to be rkR =  and u=ik(r+µr).1 The choice v=R-z is also possible but gives nothing new because of parity conservation.PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 185-186. 185Unlike  to  momentum  representation,  where  Fock symmetry  is  realized  differently:  on  the  sphere  when E<0 and on the hyperboloid when E>0 (see [3,4]), in our case all the wave functions are living on the same paraboloidindependently on the sign of the energy.It  must  be  mentioned  that  in  classical  limit  this paraboloid obtains a simple visual sense:  it  represents the surface, whose interior is unattainable for positively charged particles falling on the nucleus.Another  interesting  and  important  property  is  that homogeneous  flux  of  particles  falling  inside  of paraboloid, after being gathered in its focus is diverging in space according to Rutherford law:dN/dΩ∼[sin(θ/2)]-4 .  (5) This property explains a mysterious coincidence of the quantum  and  classical  cross-sections  for  Coulomb scattering (for more details see [5]).APPENDIX: SOLUTIONS IN EXPLICIT FORMHere some technical questions are sampled together.Equation (2) under assumption (3) takes on the form0)()()1()( ''' =Φ−Φ−+Φ uuuuu η ,      η=1-iξ.     (a.1)Its solution for the bound states ( iξ=n=1,2,…) is simply a Laguerre polynomial       )1;1()()( 111 unFuLu nn −==Φ −         (a.2)where  F11(a;bu) is  degenerate  hyper-geometric function.Despite  of  its  simplicity,  this  expression  contains much  information.  Its  expansion  in  Legendre polynomials∑−==Φ10)()(cos)(nlnllnln RRPCu θ , )!2()!1()!1(llnnCnl−−−=      (a.3)generates  all  the  radial  functions )222;1()2()( 11 RlnlFRRRlnl +−+=  belonging to the n-th energy level. For example, the n=3 wave function2/21)()( 223 uuuLu +−==Φ   (a.4)contain  all  the  three  angular  states  S,P,D: )(cos)(~)(cos)(~)(~ 221103 θθ PRRPRRRR ++=Φ  with well-known radial functions:3/221)(~ 20 RRRR +−= ;)2()(~1 RRRR −= ;3/R(R)R~ 22 = .          (a.5)Fourier-transformation of the considered solutions[ ]∫ +Φ= − rdzreepf nrrpin 3)()( λλ    (a.6)after evaluation1223 11)1(8)(−−++=nn ipippnpfλpi   (a.7)coincides with appropriately transformed expression of V. Fock.REFERENCES1. V Fock. Zur Theorie des Wasserstoffsatoms // Zs. f. Phys. 1935, v. 98, S. 145-154.2. W. Pauli jr. Ueber des Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik //  Zs. f. Phys. 1926, v. 36, S. 336-363.3. V. Bargmann. Zur Theorie des Wasserstoffsatoms // Zs. f. Phys. 1936, v. 99, S. 576-582 (1936).4.  M. Bander,  C. Itzykson.  Group  theory  and  the hydrogen atom // Rev. Mod. Phys. 1966, v. 38, p. 330-345.5. Ya. Granovskii,  A.S. Zhedanov.  ''Hidden''  sym-metry of Coulomb problem as a cause of coincidence of the  quantum  and  classical  Rutherford  formulae  // Ukranian Phys. J. 1987, v. 32, p. 1290-1293 (in Russian).186

On the hydrogen symmetry

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On the hydrogen symmetry

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