GROUP PROPERTIES OF RADIAL WAVEFUNCTIONS

HAL Id: jpa-00213858

https://hal.archives-ouvertes.fr/jpa-00213858

Submitted on 1 Jan 1970

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

GROUP PROPERTIES OF RADIAL WAVEFUNCTIONS

L. Armstrong, Jr.

To cite this version:

L. Armstrong, Jr.. GROUP PROPERTIES OF RADIAL WAVEFUNCTIONS. Journal de Physique

Colloques, 1970, 31 (C4), pp.C4-17-C4-23. �10.1051/jphyscol:1970403�. �jpa-00213858�

GROUP PROPERTIES OF RADIAL WAVEFUNCTIONS (*)

L. L. A R M S T R O N G , JR.

The Johns Hopkins University, Baltimore, Maryland, U. S. A.

Résumé. — Le groupe non-compact O (2, 1) est utilisé pour les fonctions d'ondes radiales hydrogénoïdes : on peut montrer que ces fonctions forment les bases des représentations des dimen- sions infinies de l'algèbre d e O ( 2 , l).On trouve également que /v et r Y( A' positif) se transforment dans cette algèbre comme des tenseurs. L'analyse des éléments de matrice de rN et r~x montre que le théorème de Wigner-Eckart est valable pour ce groupe et que les coefficients de Clebsch-Gordan cor- respondants sont proportionnels aux coefficients de Clebsch-Gordan du groupe RQ) qui sont très bien connus. Cette proportionnalité fournit des explications simples des règles de sélection, trouvées par Pasternack et Sternheimer, sur les éléments de matrice radiaux hydrogénoïdes. On démontre par ailleurs que les éléments diagonaux hydrogénoïdes de rx et r--v sont proportionnels à des symboles 3 — /.

Abstract. — The non-compact group O (2, 1) is used in an investigation of hydrogenic radial wavefunctions. These radial functions are shown to form bases for infinite dimensional repre- sentations of the algebra of O (2, I). It is found that /-v and r v {/V positive) transform as tensors with respect to this algebra. Analysis of matrix elements of rx and /• v shows that the Wigner- Eckart theorem is valid for this group and that the pertinent Clebsch-Gordan coefficients are pro- portional to the familiar R(3) Clebsch-Gordan coefficients. This proportionality provides simple explanations of the selection rules for hydrogenic radial matrix elements noted by Pasternack and Sternheimer, and the proportionality of hydrogenic expectation values of r-v and r~-v to 3 — / symbols.

1. Introduction. — Although the hydrogen atom has been studied in great detail for many years, many of its properties are not yet fully understood. For example, several years ago it was noted by Pasternack and Sternheimer [1] that hydrogenic radial matrix elements satisfy certain selection rules :

where the radial wavefunction is R„ijr. In addition, expectation values of powers of r calculated with hydrogenic functions [2] can be seen to be propor- tional to 3 — j symbols in which / plays the part of the angular momentum, and the principal quantum number, n, plays the part of a magnetic quantum number, e. g.

which is clearly proportional [3] to the 3 — j symbol

That properties of the radial function such as these are not well understood emphasizes the lack of interest which has traditionally been shown in this portion of the total wavefunction. The atomic physicist has concentrated his attention on the angular portion of the wavefunction, the spherical harmonics. G r o u p theory has been vigorously applied to this study of angular wavefunctions, and has proved to be inva- luable [4]. The high-energy theorist, on the other hand, has been interested in the group properties of the entire wavefunction of the hydrogen atom, and so has considered this atom in the light of such groups as 0 ( 4 ) , 0 ( 4 , 1) and 0 ( 4 , 2) [5], [6], [7].

In the commonly used central field model of the atom, one considers separately the radial and angular parts of the wavefunction. Thus, a study of the group properties of the radial functions alone is of great interest to the atomic theorist. In the remainder of this paper, we propose to carry out such a study for the hydrogen atom. Application to other systems will be discussed in the final section.

2. The Group Algebra. — We find that our study can be carried out using a three-dimensional algebra composed of the operators J+, / _ , and /3, havingthe commutators

( I ) (*) This work was partially supported by the United States

Atomic Energy Commission.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970403

C4-18 L. L. ARMSTRONG JR.

The desired realization of these operators in two

variables (z and z) is given by [8] ⁽⁶⁾

where dl2 = z-' d.t dz. One can easily show that, in tliis space, thef,, are othonormal

(2) (fin

I

^{f,,n,> =} 6(1, 1') 6(n, n')

.

⁽⁷⁾

Using (2), (6) and integrating by parts, one obtains Bases for representations of this algebra can be (fin

1

⁵³

1

fi'n') = (53

fm 1

fitn,) ,

formed by the functions

(An

I

^Ji

I

h n , ) =

-

( J T f i n

I

fi'n')

.

⁽⁸⁾

A n =

['

ⁿ

^-

^{1 -} 1 ) ! ( 2 1

+

I)]"

x These results imply that the representations of O(2, l), (n

+

^{1) !3 2 n} formed by the states f,, are unitary [lo]. We shall refer

e i n r e - z / ~ I + ' 21+1 to irreducible representations of this type as 9:.

z n - 1 (3)

where Li is the Laguerre polynomial of Morse and 3. Operators. - We wish to consider matrix ele- Feshback [9]. This function is simply related to ments of operators of the type r-keiqr ( = T:~)) hydrogenic radial functions by and rk eiqr (=

P:)),

where k 3 0 and integer in both cases. The manner in which these operators trans- ((2 1

+

^{1) (2 n))}

"(

ⁿ

)"

(fin)z= ^2Zr/n = einr ^- R,,

,

(4) form with respect to the operations of the group is 2 n 2 Z determined by their commutators with the group

algebra : where Z e is the charge of the nucleus.

Using (2) and (3), one obtains [J,, T,""] =

-

(k T q) Ti:),

J3 f i n = nfin [ J 3 , Tik)] =

q ~ q ( k )

J*hn

=

+

^{[(n T} 1) (n f I

+

^{I)]%hn,tr (5)}

[J*, p;)] = ( k

f

4)

P F ~

^I

The Casimir operator for this algebra, which is given

by G = J+ J-

+

^J:

^-

J,, can be shown, using (2) [J3, P;)] = qp:)

.

and

(9,

to satisfy the relationship It is convenient to also define states G,, and Hkq which satisfy the relations

J*

Gkq = - (k T q) Gkq* 1

Using ( 5 ) , one finds that _{5 3 Hkq =} qHkq

.

J - f i 1 + 1 = 0 , The transformation properties of the quantity

produced by tlie action of T;" on fl,, are then the same implying that the basis has a lower bound when

as the transformation properties of the product

There is, however, no upper bound, since J+ f,,, # 0 for all n

>

⁷ⁱ ; the representation is thus infinite dimensional. I t is also clearly irreducible. This algebra is thus the complexification of an algebra isomorphic to the algebras of the three-dimensional non-compact groups (e. g. SU(1, I), O(2, 1)). Operators very similar to these have, in fact, been used by Barut and Kleinert [5] in a study of a subgroup of O(4, 2), an O(2, 1) group which they called the ^(< transition group ⁾⁾ of the hydrogen atom. As we shall see later, the algebra of Barut and Kleinert [5] does not appear to be useful in the context of the problem we wish to consider.

Finally, we define a Hilbert space as the space of functions f,, with inner product

A corresponding relationship liolds, of course, bet- ween ~ ? ) f , , and Hkq,fln. If the Wigner-Eckart theo- rem liolds in this space, therefore, we can study tlie effeets of operators by carrying out a more straight- forward investigation of tlie coupling of two states.

In following tliis procedure we are, of course, imitating the traditional techniques used i11 R(3). We shall assume that the Wigner-Eckart theorem is valid for tliis non-compact group ; this assumption will later be shown to be consistent with our results.

One notes immediately that the states G,, form a representation of the algebra wliicli is not fully redu- cible. However, one can fonn an irreducible represen- tation in the subspace with

1

q

1 <

k. We shall lience- forth consider only this subspace, and shall denote

this 2 k f 1 dimensional representation by Y)(k).

One can also form a (not-fully) reducible representa- tion using the states H,,. Irreducible representations can, in this case, be formed in tlie subspaces with

However, we shall find it convenient to deal with the entire reducible representation, which we denote as 9,.

A representation of O(2, 1 ) can be unitary only if, among other things, the eigenvalues of J + J - and J - J+

are real and negative definite [ l o ] . Clearly, this is not the case for representations having as bases either C,, or H,, with

I

q

1 <

k. Thus neither 9 ( k ) nor 9, is unitary. This result is in agreement with that of Barut and Fronsdal [ l o ] , who found no finite dimensional unitary representations of O(2, 1) (except for the trivial 1-dimensional representation).

4. The Product Representation B(k) x fi):

We now consider the product representation

with basis formed by the functions Gkq,fl,,. We must first decompose this product representation into irreducible representations of the algebra (2). By considering the possible eigenvalues of J 3 , one finds immediately that

In order to carry out any actual calculations, of course, we must obtain the Clebsch-Gordan coeffi- cients for this decomposition. In the general case, these coefficients will be difficult to find because of the complexity of (I 1 ) when k 3 1. We shall see, however, that we can restrict ourselves to the range I' 2

I

k - I 1 for the purposes of this problem ; in this range, the coefficients can be obtained in a straight-forward manner using the techniques of Racah [ I l l .

Before proceeding, it is advantageous to renorma- lize the states C,, such that tlie new states, called g,,, satisfy the equations

We can also renormalize our operators in order to keep the transformation properties of tlie states and

( h )

operators tlie same. The renormalized operators, t , , can be written as

One can easily verify that these operators satisfy the commutation relations

[J+,rr)]

^{= [ ( k}

³

^{q ) ( k}

^{5 +}

i ) ] ~ t ~ ~ l

[ J 3 ,

$!"]

⁼ qf?'. (14)

With tlie definition (12), one has

which again emphasizes that the gkq do not form a unitary representation of the algebra.

We now proceed to determine the coefficients B(kq, In

I

I' n')

which appear in tlie expansion

I

F1.n.) =

1

B(kq, 111

I

1' 12')

I

g,,)

If,,)

(16)

q,n

where the states F,.,. form a unitary basis for

a : ,

i. e.

and I' 2

I

I

-

k

1 .

This final condition assures the simple form of (16).

The calculation proceeds in a straightforward manner : we first operate on both sides of (16) with J- and obtain the recursion relation

-

[ ( k

-

q ) ( k

+

^q

+

I)]" B(kq

+

^{1 , 111}

I

I' ^11')

+ +

^[(I?

-

I ) ⁽¹⁷

+

I

+

I)]" B(kq, ^It1

+

1

I

1' 11') ⁼

= [(11' - I' ^- 1 ) (11'

+

I t ) ] % B(kq, In

I

I' 11'

-

1). ( 1 80) Operation on both sides of (16) with J+ leads to another set of recursion relations :

[ ( k

-

q

+

^{1) ( k f} q)]" B(kq

-

1 , 111

I

1' 11')

+ +

^{[ ( I ?}

^-

¹

^-

^{I ) (12}

+

I ) ] % B(kq,

h -

1

I

I' n') =

= [(Iz'

-

1') (11'

+

1'

+

I ) ] % B(kq, 111

I

1' 11'

+

1). (19a) In order to avoid square roots, we define a new variable f (q, n, n') :

B(kq, In

I

I' n') = ( - ])"+'+"[(k - q) ! / ( A

+

q ) ! ( I

+

n ) ! ( I '

+

^{I ? ' )} ! ( n - 1 - 1) ! (11' - 1'

-

1) ! ] % x

E q . (18a) then becomes

(ii'

+

1') (12'

-

l' - 1 )

f

(q, n, n'

-

1) =

=

-

f ( q f l , n , n f )

+

f ( q , n

+

1,n') (186) and (19a) becomes

f (q, 11, n'

+

^{1 ) =}

= ( k

-

q

+

I)

(k +

q ) f ( q

-

1 , n, n')

+

+

^{(n -} I - I ) ( n

+

I ) f ( q , n

-

1,n'). (19b)

Wlien 11' = 1'

+

^{1 =} 2 , (18b) becomes

C4-20 L. L. ARMSTRONG JR.

i. e., f ( q , n, n') must be independent of q and n. We may therefore write f ( q , 11, n') = B, where B is a function of only k, I, and 1'. We can then proceed, using (19b), to obtain the coefficients f'(q, n, E'

+

^I),

f ( q , n , E'

+

2), etc., in terms of the constant B. One obtains in this manner the general formula :

=

1 --

^x ( 0 ' - I f - I ) ! B

( k - n + l l + l + t ) ! ( k - q ) ! ( n + l - t ) ! x

where we have also used the obvious equality

Because of the difference between ( 8 ) and (15), the expansioncoefficients for the bras will not be the same as the expansion coefficients for the kets obtained above. We must therefore consider the coefficients

A(kq, In

I

1' n') which appear in the expansion

Thcse coefficients can be obtained by operating on both sides of (22) with the operators (J,)t and fol- lowing the procedure given above. If we impose one further condition, that is,

we find the different coefficients are simply related by

The inverse transformations, that is, expansions of the product ( g,,)

1

fin), cannot be so easily carried out because of the complexity of the second of eq. ( I I). We shall not consider these inverse transfor- mations any further.

Finally, using eqs (16), (22) and (23) we obtain the orthogonality relationship

1

A ( k [ / , I n ( I' 11') A(liq, 111 ( 1" 11") ( - I)""'" -

-

4.11

=

a(/',

^I") ~ ( I I ' , 11")

.

₍₂₄₎

We see that the inverse transformation to the A's is not obtained simply by taking the conjugate trans- pose. This result demonstrates that the transforma- tions are not unitary - a result of the representa- tion t!)(k) not being unitary itself.

We can use eq. (24) to obtain the constant B.

Using (20) and (21) to obtain A(kq, 111

I

I' n'), specia- lizing to the case 11' = i ~ ' , and writing n = l'

+

I

-

q, we have

x(1' - 1

-

q ) ! ] B~ = 1

.

This sum can be easily carried out using the tech- niques described in Appendix I of Edmonds [3]. One obtains

Finally, before writing out the complete coupling coefficient, it is convenient to change the summation parameter in (21) using the technique of Racah [ l l ] and Edmonds 131 (p. 45). We write

Collecting all of these results, we have

( n ' - 1 ' - l ) ! ( l + n ) ! ( 2 1 ' + 1 ) ^%

X

[ ( k + q ) ! (I1+n1) ! (n-1-1) ! ( k - q ) !

I

where the sum over p is over all integers for which all factorials are positive, and A(/, I', k ) is defined as

5. The Coefficients A(kq, In

1

I' n'). - Before proceeding to the calculation of matrix elements, let us consider the coefficients derived in the previous section. It is convenient at this point to re-write (26) in terms of binomial coefficients

n' - 1'- 1) ! ( I '

+

n') ! (I<

-

q ) ! ( 2 1 f + 1 ) ] ( l < + q ) ! ( / + I ? ) ! ( n - / - I ) !

In order for (28) to be equal to (26), we must assume for the moment that the binomial coefficient

( 1

^is

defined only for n 2 0. We can also write the R(3) Clebsch-Gordan coefficient obtained by Wigner [12]

in terms of binomial coefficients

[(I-

n ) ! ( L f + n ' ) ! ( k - q ) ! ( 2 1 f + 1 ) %

( k + q ) ! (I+n) ! ( I f - n 1 )

--I

with the same assumption being made for the binomial coefficients. In this case, of course,

I n

1

< I and I n ' [ d l ' .

The Clebsch-Gordan coefficients given above for O(2, 1) and R(3) are clearly closely related. The most obvious difference between the two coefficients lies in the first binomial coefficient in each summation.

However, if we recall that the binomial coefficient

i;)

^{\ ,}

is defined for negative 11, and is given by [3]

we see that these two binomial coefficients can be expressed in terms of the same algebraic product, with one being defined for I' - n'

+

^p 2 0, the other for I'

-

n'

+

p

<

0. That is

and

Thus, we see that each term in the sum over p is algebraically the same function of I', TI', k , q, I and n for both coefficients.

The remaining difference between the coefficients lies in the square root multiplying the summation. As we shall see in the next section, our only interest lies in the coefficient A(k0,Iti

I

I' 17). In this case, one can easily show that the radical appearing in the O(2, 1) coefficient differs from the radical in the R(3) coeffi- cient by a factor of

((-

I)'-'')".

We have the very interesting result, therefore, that

and (kO, br

1

l ' i ? ) are given by the same algebraic functions of the variables k , I, ^11, and 1'. This can be a very valuable relationship, for there exist several tables of R(3) Clebsch-Gordan and 3 - j symbols [3], [13] in which k has been fixed at some numerical value, and the resulting coefficient is given as a function of I, n , and I' only. These tables can be used directly as a source of the coefficients A(k0, In

I

1' 11) through use of the relation

A(k0, ¹¹¹

I

I' t i ) = a(k0, In

I

I' n ) (30) where a = 1 if I - I' is even,

-

i if I - I' is odd.

Equation (30) is valid, of course, only if the algebraic forms of both coefficients are involved.

6. Matrix Elements of t$'. - If the Wigner-Eckart theorem does indeed hold in this space, we sllould find that the matrix element

where l', k, and 1 satisfy triangular conditions, is proportional to the coefficient A(kq, In

I

1' 11'). We can easily show that this proportionality does, indeed, exist. This can be shown by considering a matrix ele- ment of the commutator [ J * , ti"] ; one obtains the rela- tionship

f [(n' f 1') (n' T 1' f l ) ] % ( f , . , , , ,

I

^t:'

IJ',.)

=

where the t o p sign results from the commutator with J + , the bottom from the commutator with J - . Clearly, these recursion relations will be satisfied if

(f,.,. I fa)

^{If,,) = A(kq, In}

I

I' n ' ) ( I '

11

^t'"

11

I ) , , (32) where (I'

11

^{t ( k )}

11

is a reduced matrix element which is independent of q, 17, and n'.

Using (4), (13), and (32), we can write the matrix element of l / r k evaluated with hydrogenic functions in the form

The selection rules noted by Pasternack and Sternhei- mer [ I ] are seen t o be a natural consequence of the equality (33) : the coeficient A(k - 2 0, 111

1

1' 11)

vanishes unless k - 2, 1, and I' satisfy the triangular conditions, i. e., unless 1

+

1' 2 k

-

2 3 1 1 - I'

I

. I n addition, the discussion of the previous section.

C4-22 L. L. ARMSTRONG JR.

taken in conjunction with (33), makes obvious the proportionality of expectation values of l / r k to 3 - j symbols (Sec. 1).

7. Matrix Elements of

P!). -

The product repre- sentation 9, x a): cannot be decomposed into irre- ducible representations as simply as was the product representation 9 ( k ) x 3'): (eq. 1 I). The added comple- xity in this case arises because 9, is itself not irredu- cible. It therefore seems simplest to consider directly the matrix elements of pi", without studying first the Clebsch-Gordan series.

Proceeding as in the previous section, we consider matrix elements of the commutators [J,, P : ' ) ] ; the resulting equations are

[(n'

i

_{1') (n'} T 1' T I)]' (flr,l,r ¹

I

P!)

1

fill) =

where (I'

11

p ( k )

(1

I ) , is independent of n, and

The coefficient C ( k 0 , In

I

I' n ) has, of course, the same dependence on ^IZ as has A(k0, In

1

I' n). There is in this case, however, no triangular limitation on k ,

I,

and 1'.

Using (4), (37), and (38), we can write matrix ele- ments of r k evaluated with hydrogenic functions in the form

where, again, the top sign refers to the commutator C ( k + 1 0, lrz

1

1' r z ) with I + , the lower, to the commutator with J - . Let us =('li2 ')'--

2 1 [ ( 2 l + 1 ) ( 2 1'+ I ) ] % (1'

I1

P ( ~ + I )

I1

4 2

-

now define the coefficient g(q, n, IT') by the equation ( 4 0 )

( f i , , ,

I

pa' Ifin) = (- 1)"'" g(4, 12, n') b x The discussion of section 5, in conjunction with ( 4 0 ) , makes obvious the proportionality of expectation

x [ ( I

+

n ) ! (1'

+

^{n') !} ( n - 1

-

1 ) ! (n' - 2' - 1)

!I-%

values of r k to 3 - j symbols in which 12 plays the part

where b = ( k

+

q

-

1) ! if - k

<

^q

Inserting (35) into (34), we obtain the recursion rela- tions

(n'

+

1') (n'

-

1'

-

1) g ( q , n , n l

-

1 ) =

= - g(q

+

1, n , n ' )

+

g ( q , n

+

] , a ' ) ,

g(q, n, 11'

+

^{1) =}

= ( k

-

^{g )} ( k

+

^q

-

l ) g ( q

-

1, n, ^H')

+

+

⁽ⁿ

+

^{I ) (n}

^-

^{I - 1) g(q, iz}

^-

^{1 , n')}

,

(36)

valid for all q . These recursion relations can be seen to be exactly equivalent to those satisfied by f ( q , n, 12') (eqs 186 and 19b) if we replace k by k

+

1 . That is, if t ( k ) is defined as a (( tensor ⁾⁾ of rank k with respect to this algebra, P(" transforms like a ⁽⁽ tensor D of rank k - 1. We therefore define the tensor operator

JI"'' by

of a magnetic quantum number.

8 . Reduced Matrix Elements. - In order to fully utilize (33) and ( 4 0 ) , one must be able to evaluate the reduced matrix elements (1'

11

t(')

11

I ) , and

The traditional method for determining reduced matrix elements is to find one easily evaluated matrix elements, and to extract from that matrix element the reduced matrix element. This is the procedure we shall follow here.

We asiume, first, that I' 2 I. Matrix elements in which tlie bra is tlie state ( f l r l . +

, ^I

are easily evaluated, since

The matrix element

It is difficult to obtain a general expression for then becomes simply an integral over one Laguerre g ( q , n, n') valid for all q ; it is, however, clear that g polynomial. Such an integral can be carried out in a ( 0 , n , n) must be equal to f ( 0 , n , n ) to within a factor very straightforward fashion using the generating depending only on Ii, 1 and 1'. Using (21), therefore, we function for Laguerre polynomials [9]

find that we can write e - ~ t ~ ( l -0 a t n

- =

1

(1

-

t l a + l rt=o ( 1 1

+

^{a ) ! t:l(z>}

( 1

P I f ) ⁼ ( 0 , 1

1

^{1 1 (1'}

I

p k

11

1 ) (38)

The same statements apply equally well, of course, t o the matrix element

Finally, the coefficients A(k0, 11'

+

1 11' I'

+

1 ) and C(kO,II1

+

1 11' 1'

+

¹⁾ are easily obtained from (26) and (39). Using the calculated matrix elements a n d these coefficients in eqs (32) a n d (38) allows one t o make t h e identifications

( 2 1+ 1 ) (1+11-k) ! (I' ! ) 2

=(- 1 ) ~

[

( l + k - l ' ) ! ( k + l + l l + l ) ! ( l l + k - I ) !

I"

and

( I +

k +

1 ' + l ) ! ( l t + k - I ) !

k ! ( l l - l ) ! (42)

These results can now be used with (33) and (40) to obtain matrix elements of any power of r .

9. Discussion.

-

We have shown that hydrogen- like radial functions can be used t o form bases for representations of the lion-compact group O(2, 1).

Powers of the radius were found to transform simply with respect t o the group algebra. Tlie Clebscll- Gordan coefficients for a partici~lar product represen- tation were determined, and matrix elements of powers of the radius were shown to be simply expressible in terms of tliese coefficients. Tlie coefficients the~nselves were discussed, and found to be very similar to tlie familiar R(3) Clebscli-Gordan coefficients. In this way, we have been able t o explain in a new and interest- ing manner many of the properties of liydrogenic radial wavefunctions. W e must note, however, that

[I] PASTERNACK (S.) and STERNHEIMER (R. M.), J.

Marlz. Pllys., 1967, 3, 1280.

[2] BETHE (H. A.) and SALPETER (E. E.), Quantum Mechanics of One- and Two-Electron Atoms, 1957, Academic Press, New York.

[3] EDMONDS (A. R.), Angular Momentum in Quant~un Mechanics, 1957, Princeton University Press, Princeton.

the reduction of the Kronecker product of a unitary and a non-unitary representation is very co~nplicated a n d much work reniains to be done in tliis area.

O u r discussion was concerned with functions which have as a variable r. Hydrogenic functions have, of course, the variable

rln.

We were limited, therefore, t o considering matrix elements diagonal in 11. Barut a n d Kleinert [5] have discussed a n algebra which actually has as a basis the liydrogenic radial functions.

Their algebra differs from ours, basically, by the inclusion of a dilatation operator

Do,

which has tlie property

Unfortunately, powers of r d o not transform so nicely with respect t o this algebra as they d o with respect t o the algebra discussed in this work. Never- theless, future work must consider matrix elements off-diagonal in iz.

Questions of possible application of group tlieore- tical techniques t o the radial functions of more complicated atonis naturally arise. Two possible approaches t o tliis more complicated problem would appear p r o m ~ s i n g . First, it is known that Inany properties of a t o ~ n s can be reasonably well predicted using the hydrogenic functions as a first order nppro- ximation t o the real wavefunctions. One l ~ a s . ii)r example, the work of Layzer [I41 in this area ; and one might attempt a fresh look a t Layzcr's tcchniqucs and results using a group theoretical approach. A second avenue of investigation would be a search li)r apprcl- ximate potentials such tliat the r a d ~ a l cqi~ation could be written in ternis of a Casiniir operator I'or sonic group. T h e same general approach as carried out above could tile11 be carried out for the gro~113 01' the new potential. However, whatever approach turns out t o be the best, tlie suggestions that both the radial and angular wavei'unctions can be s t ~ ~ d i e d using group theory opens many fascinating avenues of conjecture.

ences

[8] MILLER (W.), Lie Theory and Special Functions, 1968, Acadeniic Press, New York, has considered an algebra closely related to the one discussed in this paper.

[9] MORSE (P. M.) and FESHBACK (H.), Methods of Theoretical Physics, Vols. I and 11, 1953, McGraw- Hill Book Co., Inc., New York.

[4] JUDD (B. R.), operator Techniques in Atomic Spec- [lo] BARUT (A. 0.) and FRONSDAL (C.), PWC. ROY. SOC.

troscopy, 1963, McGraw-Hill Book Co., Inc.,

New York. ^{A ,} 1965,287, 532.

[5] BARUT (A. 0.) and KLE~NERT (H.), P/IYS. Rev., 1967, [ I l l RACAH (G.), PISYS. Rev., 1942, 62, 438.

156, 1541 ; 157, 1180. [12] WIGNER (E. P.), Group Theory, 1959, Academic [6] SWAMY (N. V. V. J.), KULKARNI (R. G.) and BIEDEN- Press, New York. Translated by J. J. Griffin.

HARN (L. C.), J. Math. Pl~ys., 1970, 4 , 1165.

[71 M u s ~ o (R.), Pllys. Rev., 1966, 148, 1274. PRATT [I31 FALKOFF (D. L.), HOLLADAY (G. S.) and SELLS (R. E.), (R. H.) and JORDAN (T. F.), P11y.y. Rev., 1966, ^{Catt. Jorirll. Pl~ys.} 1952, 30, 253.

148, 1276. [14] LAYZER (D.), Atitials of Pl~ys. (N . Y . ) , 1959, 8, 27 1.