Benedict, M. G.

Gilde, Ferenc J.

English

Benedict M. G.

Gilde Ferenc J.

University of Szeged

0 4 REPRESENTATIONS AND THE SPECTRA OF HYDROGEN AND HELIUM By M. G. BENEDICT A N D F.J. GILDE Institute of Theoretical Physics, Attila József University, Szeged (Received 21 July 1973) The correspondence between some of the representations of the 0 4 group and the hydrogen levels is stated again. The speciality of the occurring representations is pointed out. The helium states are obtained using Oi®Oi®S 2 . The group theoretical classification of hadrons based on the assumption that mesons and baryons form SU3 multiplets has proved very fruitful. But only some of the irreducible representations of Sl/s appear really as physical states. Supposing that quarks exist, the speciality of these representations would be clear. But, as there have not been found any quarks yet, it is still a puzzle why these and not other representations do actually occur in nature. So it may be interesting to study the speciality of the occurring representations in the case of other well-known problems. As it was shown by F O C K [1, 2] , the Schrodinger equation of the hydrogen atom can be transformed into an integral equation in the space of the square integrable functions on the four-dimensional unit sphere S3. The Hamiltonian H(£) becomes an integral operator with a kernel invariant under rotations in the four-dimensional space. These rotations form a group, the group C4 . Let g € 0 4 , then H(g£)=H0f). More generally, for each g<E04 let where T(g) is an operator in the space of the V s . This defines a mapping g—T(g) i.e. a representation of the group, called quasiregiilar representation. It can be seen that T(g) commutes with H for each g £ 0 4 : T ( g ) H ( ! m l ) = H ( g - ' Q W i g - H ) = H ( l ) T f e ) To . find the possible energy levels, we have to decompose our representation T(g) into a direct sum of irreducible components. The four-dimensional homogeneous harmonic polynomials of degree k (k=0, 1, ...) form a complete set on the four-dimensional unit sphere. They are of the form l'\(x) = Z ^«3,3*4 x i 1 * ! 2 * ! 3 ^ 4 «1 + a2+ot3+oi4=fc and i , a 2 198 M. G. BENEDICT AND F. J. GILDE The value of ^ (x ) in any point is uniquely determined by its value on the unit x •* sphere point -pr . The functions /^(¿j) are the four-dimensional spherical harmonics; there exist (& + 1)2 linearly independent such functions for each k. These functions span a space H¡¡ of dimension (k+1)2 which can be shown to be invariant and irreducible under T(g). This means that we have decomposed our representation T(g) into the irreducibles T4(g)-s which act in the subspaces H4. The subspaces H\ are usually characterized by k+1 = n. The dimensionality of B\ is then n2, which means that each level is n2-fold degenerate. The spherical harmonics are the eigenfunctions. Now let us take the three-dimensional rotation group 03, which is clearly a sub-set of 04. Reducing T4(g) into the direct sum of the irreducible representations of 0 4 , we find that TS(g) = ¿ n ( g ) » ¿ r £ 0 3 , 1 = 0 where the T£(g)-s are the 2 /+1 dimensional representations of 03. We see that / = 0 , 1, . . . , « - 1 . Now. the fact which we want to point out is that the representations T4(g), though infinitely many of them exist, do not exhaust all the possible irreducible representations. A general irreducible representation of 04 is labelled by two indices [3]. Thus our representations are very special, as they are labelled only by one integer/:. The characteristic feature of the Tj(g)-s is that they are subrepresentations of the quasiregular representation. in the space where T j acts there is always a vector, namely which is left invariant by all T4(/;) for each h £ 0 3 , because it acts only on x1( x2 and x3 . The interesting fact is that the T4(g)-s are the only irreducible representations of 0 4 which have this property, i.e. restricting them to 03, there exists a vector in their space, invariant under all operators of the representation [4]. In the language of physics this means that, among the representations of 04, only the T4(g)-s possess in their representation space spherically symmetric states in the three-dimensional sense, i.e. s states. The above correspondence between some of the representations of 04 and the energy levels of hydrogen gives the possibility to make a pure group-theoretical classification of the energy levels of helium. The approximate symmetry group is here the direct product group O^O^St. S2 is the two-element permutation group and appears because the system is .invariant under the operation of inter-changing the two electrons. The Coulomb interaction between the two electrons breaks the 04<g>04<g>S2 symmetry and the real symmetry of this system is only the 03<S>S2 group. The appropriate irreducible representations of 04<8>04<S)S2 are the direct product representations T4 '(g)T42®Tj = T'1'12' where the T4'-s are the representations of 04, and Tj is one of the two possible irreducible representations of S2, namely Ts+ or T~. Their invariant subspaces are the symmetric and antisymmetric two-electron functions, respectively. In the case when > 1 > 0 , k2>0, the energy levels of the doubly excited atoms fall into the continuous spectrum of the singly ionized helium atom and so they are optically unobservable. Thus we shall consider only the case k.¿=0. 0 4 REPRESENTATIONS A N D THE SPECTRA OF HYDROGEN AND HELIUM 199 Now if we restrict the T k l 0 ' representations of 04(&04®S2 to 03®S2, they will become reducible. After finding the irreducible components we obtain some levels, but they do not give the true multiplicities. We must take into account the spin of the electrons, too. The one-electron spin states belong to the self-representation of SU2 [5]. This is a double valued representation of 03 and we write it T^2. The two-electron spin states belong to the irreducible parts of T ^ 2 ® ^ ' 2 . These are the TSJ and T3 representations of 03. The functions that span the space of TJ are symmetric, while those of T3 are antisym-metric when interchanging the two electrons. T" and T3 are one- and three-dimensio-nal and the corresponding states are the singlet and triplet states, respectively. The helium states belong to those representations of 03<8>S2 which are irreducible parts of TM+(giT3 and T ' 0 - ® T j . According to the Pauli principle, the symmetric repre-sentations of O 4 ® O 4 0 S 2 with + sign are to be coupled to T3 (parahelium), those with — sign to T3 (orthohelium). The results of the reductions are shown in Table I for parahelium and in Table II for orthohelium, with the respective spectroscopic signs in the right hand column. Table I n Decomposition of Direct Products Spectroscopic Signs 1 T » » + ® T g = T S ® T S = T°+ . 1>5„ 2 ' T ' » + ® T S = ( T § + © T j + )(8iTg = T ° + f f i T » 2/So, 2lPl 3 T 2 0 + ® T S = ( T g + © T J + © T i + ) ® T ? = T°+ © T l + © T 2 + I n t h e c a s e o f T A l 0 _ w h e n kl=0, t h e c o r r e s p o n d i n g s p a c e i s t h e z e r o v e c t o r , s o w e m u s t b e g i n w i t h kx = \ , n=2. Table 11 n Decomposition of Direct Products . Spectroscopic Signs 2 T 1 0 : ® T i = ( T r © T j - ) ® T j = ( T S - ® T J ) © ( T J - ® T 5 ) = = T L © © T i f f i T L f f i T l 2 »Sx 2s P0, 2*Pl,23P1 3 T 2 0 " ® T | = ( T g - © T J - © T l - ) ® T J = ( T g - ® T J ) © ( T J - ® T j ) f f i © ( T f - ® T | ) = T L © © T i © T L © T i © © T L © T i © T l 3 ^ 0 , 3 ' ^ , 3 3 f , I'D!, I'D«, 3 3 D 3 2 0 0 M. G. BENEDICT A N D F. J. GILDE References [1] Fock, V. A.: Z. Physik 98, 145 (1935). [2] Bander, M., C. ltzykson: Rev. Mod. Phys. 38, 330 (1966). [3] Weyl, H.: The Classical Groups, Princeton University Press, Princeton, 1946. [4] Виленкин, H., Я.: Специальные функции и теория представлений групп, Наука, Москва, 1965. [5] Wigner, Е. P.: Group Theory and its Applications in Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959. ПРЕДСТАВЛЕНИЯ ГРУППЫ 04 И СПЕКТРЫ ВОДОРОДА И ГЕЛИЯ . М. Г. Бенедикт, Ф. Й. Гилде Снова определена связь между некоторыми представлениями группы 0 4 и уровнями водорода. Показаны особенности проявляющихся представлений. Состояния гелия могут быть получены также с помощью 0 4 Ф 0„ ® . 

O4 representations and the spectra of hydrogen and helium

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O4 representations and the spectra of hydrogen and helium

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