The Radial Equation for Hydrogen-Like Atoms Carl Peterson
Ohio Wesieyan University Delaware, 43015
The Polynomial versus the factorization technique
The radial equation for the hydrogen-like atoms is normally treated in introductory courses in quantum chemistry using the polynomial method of Sommerfeld.' In advanced undergraduate and beginning graduate quantum chemistry this is a most difficult undertaking for the large majority of chemistry students who must simultaneously learn the mathematical as well as the physical concepts. One could argue that the required mathematics should have been taken, but unfortunately most chemistry curricula do not at present require more than one year of calc11lus. The purpose of this article is t o compare the polynomial method with the factorization method,z a method ignored by virtually all textbooks on quantum chemistry,% to assess which is more heuristically advantageous in light of the aforementioned difficulties. Radial Equation The radial equation for the hydrogen-like atoms can be written as
where V f r ) instead of -Ze2/r has been used to emphasize that the equation is equally valid for all central fields. The problem is to solve the radial Schrbdinger equation for the allowed energies and eigenfunctions for the electron in this field. Before doing this let us make the transformation R(r) = f(r)/r, and make some comments about the radial equation. Inserting f(r) = rR(r), eqn. ( 1 ) reduces to
and seek a solution. The Sommerfeld polynomial approach is to investigate the limiting behavior in the regions of very small and very large r. This is investigating the so-called asymptotic behavior of r. Sommerfeld noticed that in a large number of problems the eigenfunction of the Schmdinger equation can be separated into a product of a transcendental factor Q ( X ) and a polynomial factor P(X). The factor Q ( X ) accounts for the behavior which bound the regions of validity of the Schrodinger equation, that is the asymptotic regions, and the polynomial factor accounts for the nodes of the eigenfunction. Using this procedure for very large r, eqn. (4) may be approximated by
where
The solutions are
The function with the plus sign is not square integrable and is physically inadmissible. The acceptable solution is
For E > 0, the energy spectrum is continuous and for E < 0, the states are bound. In the small r limit, the dominating term in eqn. (4) is 1(1 l)h2/2@rz; SO eqn. (4) can be approximated by
+
The one-dimensional Schrodinger equation is
A solution of the form rmleads to the equation Comparison of eqns. (2) and (3)points out their similarities.
In eqn. (2) the quantity in brackets is the "effective potential" experienced by the particle (electron). If 1 # 0,there is the so-called "centrifugal harrier," which the particle (electron) mrlst surmount as it approaches the origin (nucleus). These remarks are valid for all central fields. The special case of a coulomb field in which an electron is attracted to a nucleus, hydrogen-like atom cases, is the one to wh~chwe will direct our attention. Sommerfeld Polynomial Method We write eqn. (2) in the form
m(m
-
The asymptotic solution is r-1 or ++I. r-1 is ill-behaved a t the o r i n so one takes as the solution f ( r ) = i+' ,.- ,,,,
for small r. The solutions that remain to he found are those for the middle region. Knowledge of this region is obtained by use of the polynomial factor. According to most texts, it is now convenient to change to dimensionless form by intmducing a dimensionless variable p = 2sr. This may he seen as follows
' Sommerfeld, A., "Wave Mechanics," Dutton, New York, 1929,
p. 11.
Infeld, L., and Hull, T.E.,Xeo. Mod. Phys., 23.21 (1951). 31t was hmught to the attention of the writer that a quantum chemistry text, to appear soon by Goddard and Dunning, uses the factorization method. 92
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I)=/(/ + I )
This equation has the solutions
so the dimensions for s are
Therefore p = cm-1 x cm (no dimensions) using this variable and the resuit
One should also note that
4sYf"(p)= f Y r ) k -h2 -
Eqn. (4) becomes
411 ('I The difference between eqn. (9)and eqn. (7)is that the numerators of the 1/+ term have different signs for the term in parenthesis on the left-hand side of each equation. Taking into consideration this difference, the raising and lowering operators may he defined, respectively, as R + = -d - - +l - k ' d r r U and l - hR - =d + d r r 2 1 These definitions are not arbitrary as may be seen by the following operation: Operating on eqn. (8) from the left with R,-, one obtains r
where n = Ze%2/2s = Z&/h%.
One may also write
and i t only remains to determine the form of n. This is accomplished in our intermediate region by setting up a power series solution so as to obtain a polynomial. Therefore
or we can construct our total solution for eqn. (5) as which gives
In order to determine the coefficients, one must differentiate eqn. (5) to obtain the recurrence formula k + l + l - n =
(k
+
1x21
The above may be written as
+ k + aa*
The conditions under which this infinite power series reduces to a kth order polynomial requires that ah+* = 0. This occurs when k = n - (1 I), where k = 0,1,2, . . . The quantity n is known as the principal quantum number. One may note that n is allowed t o assume all integral values except zero. For a particular value of n, 1 may vary from 0 to n - 1. The solutions ohtained are the energy eigenfunctions and one may easily obtain the eigenvalues for a given solution. Almost every quantum chemistry text follows this approach or some variation thereof.
+
Factorization Method A generalized treatment of the factorization method for eigenvalue problems can he found in footnote 2. The treatment used by this miter resembles more closely that of Salshurg.4 Equation (4), the radial equation for the hydrogen-like atoms, may be written
where k = 2wZez/h2 and w = 2w[El/hz. Operators for this equation may be ohtained by notingthat
If one adds a constant term - b to the factor llr, one ohtains
where f i _ ~= (d/dr + l/r - k/2l)fi. If in eqn. (6) 1 is replaced by (1 - I), one can easily see why we say d/dr + l/r - kt21 is the lowering operator for the quantum number 1. This implies f i - , = Rr-f, In operator notation the procedure is
where the energy is independent of the quantum numher 1. A similar operation can he carried out with the raising operator on the state f i - l . One should note that this formalism does not yield the ladder operators for 1 = 0, although one can generate all the solutions by using the R* operators. The procedure of operating with RL* can, in principle, be carried on forever; but in order to determine the radial function for a given energy, there should he an upper bound on the quantum number 1. This can he shown4 hut the roof is not s i m ~ l e . There is then a maximum value of 1 = lmar such that (11)
R+'-+'f,m., = 0 Operating on the left by R-l,,,,,one obtains + k" Rb.+& Rims.+, fk""= 4(lmax+ l)fjm,,s
By setting b = k/21, eqn. (6) may be written for 1 # 0 as
The energy eigenvalue is in comparison with eqn. (8)
4
Salsburg, 2.W., Amer J Phys., 33, 36 (1965).
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2fiZezlhz and ao = h*/fiez to obtain R d r ) = Ar exp(Zr/200) the familiar 2p-orbital; for the 2s-orbital one gets using eqn. (14)
where 1.".
+
1
is characteristic of the energy level. This implies that one I. Using this one obtains W = k2/4n2, may let n = l,,, and this may be used to evaluate the energy as E = Z2e4fi/2hzn2. The eigenfunction
+
f.-1
-
fr.,.,.
Making the substitutions for k and ao and factoring out a constant. one has
which gives
can be obtained in a straightforward manner from eqn. (ll),which written out is the familiar L-orbital Comparisons and Comments
where l,,, + 1 has been replaced by n. This equation is easily integrated to give
where A is the normalization constant. All the eigenfunctions fi for 1 < lmax can be ohtained by successive applications of the lowering operator. This can be shown as follows
hut since eqn. (13)gives f I , , ,
, one can write
.
,
This equation gives
One may continue in a similar manner to generate all the f, for a given n, and using the substitution made earlier, f/rl, = rR(r1. one can set the radial eieenfunctions in ,. terms o f kt;) which are somewhat mar; familiar. It is shown here that the radial function clearly d e ~ e n d son 1 and that the numher of 1 solutions is n. -~he;efore, one may denote the radial eigenfunctions as f d r ) or in terms of the more familiar functions RnL(r).For example, for n = 2 and from eqn. (13), f d r ' = Ar2 exp(-Rr/4), which gives on substitution Rzl = A r exp(-kr/4). One may use k =
94 1 Journalof Chemical Education
The major disadvantage of the polynomial method is that it is a long drawn out mathematical exercise, which affords the students the opportunity to get so involved with the mathematics that they lose sight of the physical concepts presented by the radial equation. The polynomial method as outlined in this article leaves out a lot of the thought Drocesses and middle stem involved so that the pres&taiion is considerably shokened. The instructor mav note that the ~olynomialmethod is unlikely to be us& again, and in ihis sense its applications are rather limited. On the other hand, in the study of atomic and molecular structure, more often than not one uses the factorization technique. For some unknown reason this technique is always saved for more advanced courses in which angular momentum commutation relations are discussed. In such courses, the factorization technique seems to be more abstract and less useful which is definitely not the case, as shown by the examples. Due to the techniques value in the areas of atomic and molecular structure, i t seems to this writer that its introduction at the beginning level would give the potential users of quantum theory more comprehensibility for some of the concepts ohtained by its application. Also, the factorization procedure is outlined in its entirety, which is to say that it is immensely shorter than the polynomial method. Acknowledgment
Thanks are due to Professors R. M. Pitzer and R. D. McQuigg for helpful criticisms.
