Ladder-Operator Treatment of the Radial Equation for the Hydrogenlike Atom Donald D. Fins University of Pennsylvania. Philadelphia, PA 19104-6323 The technique of applying raising- and lowering-ladder operators to ohtain eigenvalues and the corresponding eigenfunctions of quantum-mechanical operators is widely used in current textbooks on quantum mechanics to solve the angular-momentum wave equation and often the harmonic-oscillator wave equation. There are several advantages to using this technique over the older procedure of solving a second-order differential equation by the seriessolution method. Ladder operators provide practice for the student i n operations t h a t a r e used i n more-advanced quantum theory. Moreover, they yield the eigenvalues and eigenfundions more simply and more directly without the need to introduce generating functions and recursion relations and to consider a s" vm. ~ t o t i cbehavior and convergence. The application of ladder operators to the radial equation for the hydrogenlike atom was introduced by Schrodinger ( I ) and was presented in a now out-of-print textbook by Harris and Loeb (2).However, their treatments are not complete because they ohtain the quantized energy levels from a n arhitrary termination of the "lowering" process rather than from a mathematically rigorous argument. Further, their raising and lowering operations do not maintain the normalization of the resulting wave functions. Consequently, the application of the ladder-operator technique to the radial equation for the hydrogenlike atom i s not found i n current textbooks; the series-solution method i s used instead. In this article we present a new approach to the solution of the radial equation for the hydrogenlike atom using the ladder-o~eratormethod. Quantization of the enerm levels and the generation of ort~onormaleigenfunctionsBre obtained rigorously, without any arhitrary assumptions. The Radial Equation When the wave function y1(r,0,9) for a hydrogenlike atom is expressed as the product
the Schrodinger equation separates into two differential equations; one involves only the radial variable r , and the other involves only the angular variahles 0 and 9. The solutions of the equation in 0 and 9 are members of a n infin i t e s e t of orthogonal functions Y,,(0,1p), k n o w n a s
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spherical harmonics, with 1 and m integers. In particular, the so-called azimuthal quantum number 1 takes on the values 0, 1 , 2 , ... . The radial function R(r) satisfies the differential equation
where is the reduced-mass of the two-particle system; E is the total energy; + is the electronic charge; Ze is the nuclear charge; and fiis Planck's constant divided by 2n. The many solutions ofthe differential equation (eq 1)depend not only on the value of 1, but also on the value of E. Therefore, the solutions will he designated a s R ~ i r )Be. cause the potential energy -Ze 2 i r is always negative, we are interested in solutions with negative total energy, that is, where E < 0. We make t h e following conventional change of variables (3.4b
is, according to the earlier Bohr theory, the radius of the circular orbit of the electron in the ground state of the hydrogen atom (Z = 1). We also make the substitution
Equation 1now takes the form
To be a suitable wave function, Su(p)must be well-behaved; i t must be continuous, single-valued, and quadratically integrable. Thus, pSu vanishes when p-1- because SNmust vanish sufficiently fast. Because Su is finite eve-
rywhere, pSu also vanishes a t p = 0. Further, the customary requirement that the fuuctionsR~l(r)be normalized in spherical coordinates yields m
~ ( S A I ( P ) ) ~ P=~1~ P
(6)
0
Ladder Operators A A We next define the operators Ax and Bn as
Equation 5 can then he written i n either of two forms:
T us, the functions Su(p) a r e also eigenfunctions of + 1) and of&(& + 1).Also,
&.(fin
when h i s replaced by h - 1and h + 1, respectively, in eq 5. If we operate on both sides of eq 9 with the operator A (An + I), we obtain
Comparison of this result with eq 11leads to the conclusion that (21 + 1)Suand Sx-I,,are, except for a multiplicative constant, the same function. We implicitly assume here that Su i s uniquely determined by only two parameters: h and 1. Accordingly, we may write
of the eigenfunctions Su(p) and of the raising and lowering operators. Hermitian Operators and Orthogonal Eigenfunctions As discussed in nearly every textbook on quantum mechanics, the eigenfunctions of a Hermitian operator are orthogonal. Alinear operator &is Hermitian if
where f and g are arbitrary well-behaved functions; fY is the complex conjugate off; and d r is the volume element in the appropriate coordinate system. However, only a few of these textbooks point out explicitly that, in the general case, dr may have a function of the relevant variables a s a factor. For functions of a single variable y , the general form for dr is wCy)dy, where wCy) is a weighting factor. Equation 16 then takes the form
where we have specialized to real functions and a real operator for our particular application. The differential equation (eq 1)for the functions R ~ l ( r ) may he written i n the form
with a suitable definition off&. If the weighting factor w ( r ) i s rZso that d.r = P d r , then we can show that
through a n explicit integration by parts. Thus, the radial functions R&) constitute a n orthonormal set, so that
where 6EEis the Krouecker delta where a u is a numerical constant, dependent in general on the values of h and 1, to be determined by the requirement that S u and SA-~,I be normalized. Without loss of generality, we can take a 3 to he real. The function (AA+ 1)Su is a n eigenfunction of the operator i 2 eq 5 with eigenvalue decreased by 1.Thus, the operator (A, + 1) transforms the eigenfunction SXJ determined by h, 1 into the 9genfunction SA-I,~ determined by h - 1 , l . So,the operator (Ah + 1) is a lowering-ladder operator. A n analogous procedure using eqs 10 and 12 shows that (& + 1)Su and SL+I,~ are proportional,
Proper Orthogonal Relationships
The orthogonal relationshi s do not apply to the set of functions Su(p) with w(p) = p%. Because the variable p introduced in eq 3 depends not only on r , but also on the eigenvalue E, or equivalently on h, the situation is more complex. To determine the proper orthogonal relationships for Su(p), we express eq 5 in the form
By means of iut gration by parts, we can readily show is Hermitian as expressed by eq 17 for that this where b~ is the proportionality constant, assumed real, to a weighting factor w(p) equal to p, thereby implying the be~ , ~ orthogonal relationships1 he determined by the requirement that S u and S X + normalized. The operator (&+ 1) transforms the eigenfunction SUinto the eigenfunction S X + with ~,~ eigenvalue h ~ S U ( P ) S A ~ P ) P=~0P for l t ~ increased by 1.Accordingly, the operator (& + 1)is a rais0 (22) ing-ladder operator. Next we must evaluate the numerical constants a u and 'Schrodinger (ref 1) deduced these relationships. bu, so we first investigate some mathematical properties
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Expressing the Desired Integral In order to complete the characterization of integrals of Su(p), we must consider the case where A = h' for w(p) = p. The functions Su(o)are normalizedfor w(o) . = 02 . as expressed in eq 6. The samk'&ult does not obtain for w(p)= p. w e begin by expressing the desired integral in a slightly different form. Substitution into eq 14 gives
We then integrate the right-hand side by parts; the integrated term vanishes. With the introduction of eqs 8 and 15, the integral takes the form
Because the first integral on the right-hand side vanishes according to eq 22 and the second integral equals unity according to eq 6, the result is
1
SU(P)SU(P)P~V = ~ s h l .
(24)
b~
Theorems ~ e l a t i &to the Operators Ah and For any arbitrary well-behaved functions flp) and g(p), we consider the integral
where eq 7 has been used. Integration by parts of the first term on the right-hand side with the realization that the integrated part vanishes followed by substitution of eq 8 gives the desired result,
Following the same procedure, we can also show that2
Evaluation of the Constants ahrand bkr l b evaluate the numerical constant au, we square both sides of eq 14, multiply through by p, and integrate with respect to p to obtain
-
I P&+
=~&J(s~_~JTP~P
(27)
~)su)((&+~ ) S U ) ~ P
O
h
Application of eq 26 with f = Su and g = p(Ah + 1)Su to the left-hand side and substitution of eq 23 on the righthand side give
Using eqs 8, 14,9, and finally 24, we obtain 2Harris and Loeb (ref 1068
derived this relationship.
Journal of Chemical Education
J
J
(30)
We have arbitrarily taken the negative, rather than the positive, square root. The numerical constant b~ may be determined by a n analogous procedure, beginning with the square of both sides of eq 15 and using eqs 25,23,7,15,and 10. We obtain
so that eq 15 becomes
Taking the positive square root here will turn out to be consistent with the choice in eq 30.
Combining this result with eq 22, we obtain
I
\\
Quantization of the Energy The parameter A is positive; otherwise the radial variable p, which is inversely proportional to A, would be negative. Furthermore, the parameter A cannot be zero so that the transformations i n eqs 24 remain valid. To find further restrictions on h, we must consider separately the cases where 1 = 0 and where 1 t 1. For 1 = 0, eq 30 takes the form
Applying the Lowering Operator If we begin with a suitably large value of A, say €, and continually apply the lowering operator to both sides of eq 33 with A = €,,we eventually produce after k iterations an eigenfunction with a negativevalue for A, that is, A = (€,k) < 0. Thus, the sequence must terminate by the condition ( 1 - h) = 0 in eq 33, so that E, must be a n integer and the minimum value of A for 1 = 0 i s A = 1. For the situations in which 1 t 1, in eqs 29 and 31, the quantities a&and b%,being squares of real numbers, must he positive. Consequently, we require that h t (1 + 1). We then select some appropriately large value €, of the parameter A i n eq 30 and continually apply the lowering operator to both sides of the equation i n the same manner as i n the 1 = 0 case. After k such iterations, we obtain a n eigenfunction with A = (5- k) < (1 + 11, which is not allowed, so that the sequence must terminate. If €, is a n integer, then the coefficient a u vanishes when A = (1 + 1). The minimum value of A is then 1 + 1. Combining the conclusions of both cases, we see that the minimum value of h is 1 + 1 for 1 = 0, 1, 2, ... . Beginning with the value h = 1 + 1, we can apply eq 32 to yield an infinite progression of eigenfunctions S d p ) for each value of 1 (where 1 = 0 , 1 , 2 , ...1, where h can take on only integral values, A = n = 1 + 1 , 1 + 2 , 1 + 3, ... . Because €, i n both cases was chosen arbitrarily and was shown to be a n integer, eq 32 generates all of the eigenfunctions Su(p) for each value of 1. There are no eigenfunctions corresponding to noninte-
gral values of h. Because h is now shown to be a n integer n, we replace A by n. Solving eq 2 for the energy E and replacing h by n, we obtain the quantized energy levels for the hydrogenlike atom,
The fi~nrtionsSu,S,I.... an, ;~utom:iticaIlvnonnalizwl as soecified bv"ea .6. The normalized eicenl'unctions S...J o, , for 1 = 2, 3, 4, ... with n > (1 + 1)are obtained by the same procedure.
-
(34)
These energy levels agree with the values obtained in the earlier Bohr theory. Determination of the Eigenfunctions Equation 33 may be used to obtain the ground state (n = I,= 0 ) eigenfunction S d p ) . Introducing the definition of A, in eq 7, we have
1,
from which i t follows that S - ce-M = 2-@e-Ph 10 -
where the constant c of integration was evaluated by applying eq 6. The series of eigenfunctions Szo, S30, ... are readily obtained from eqs 32 and 8 with h = n,1 = 0
Thus, Szo is
1
= $2
2 2
- p)e*
and so forth ad infiniturn. Each eigenfunction is normalized. The eigenfunctions for 1 = 1 are obtained in a similar manner. For n = 2,1= 1,eqs 30 and 7 are
which integrates to yield
Up and Down the Ladder
Just as eq 32 can be used to go "up the ladder" to obtain Sn+l,lfrom S,r, eq 30 allows one to go "down the ladder" and obtain S,-I,I from S,r. The negative sign on the righthand side of eq 30 maintains the signs of the functions Snl obtained from eq 32. Thus, taking the negative square root in going from eq 29 to eq 30 is consistent with taking the positive square root in going from eq 31 to eq 32. In all cases the ladder operators yield normalized eigenfunctious if the starting eigenfunction is normalized. The radial factors R,l(r) of the hydrogenlike atom total wave functions y r ( r , 8 , ~a) r e r e l a t e d to t h e functions Snr(p)by eq 4. Thus, we have
and so forth. An extensive listing of radial functions R,, appears in Pauling and Wilson ( 5 ) . Discussion The technique of using ladder operators to solve differential equations is a general one, known also a s the "factoring" method, and has be-en thoroughly reviewed by Infeld and Hull (6). For the radial equation of the hydrogenlike atom, they mention two approaches. The first one is presented here and also used by Schrodinger (1)and by H a m s and Loeb ( 2 ) .The second approach is to factor the operator in eq 21 into the product of a raising and a lowering operator and then to apply the result to eq 20. This approach was suggested by Schrodinger ( I )and later developed by Salsburg (7). In his development, Salsburg encountered two major difficulties: The case of 1 = 0 must be excluded, and external arguments must be invoked to terminate the ladder process and thereby quantize the energy. Thus, this second method of factoring the'radial differential equation is not a s viable a s the first one. Literature Cited
where the constant c of integration is evaluated with eq 6. Equations 32 and 8 for 1 = 1 give
1. Schrddinger, E. Prm. Roy fish Acnd 194&1941,46A. 9-16, 183-206. 2. Hartis. L.: Laeb, A. L. Intmduerion to Wme Meehcmks: McGraw-Hill: New York, 1963; Chapter 13. 3. Pauling, L.: Wilson, E. B.Jr.lntroduelion lo Quantum Mechanics: McOraw-Hill:New Ywk, 1935;p 121. 4. Pilar, F L. Elmenroly Quantum Chmuatly, 2nd ed.; McGraw-Hill: New York. 1990; 0 119. 5. Ref3. Table 21-3: pp 135-136. 6. Infeld. L.; Hull, T E. Reus Modern Phys 1951.23, 2 1 4 . 7. Sslsburg. 2. WAm. J Phys. 1365.33.3639.
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