Radial Probability Density and Normalization in Hydrogenic Atoms Luis Lain, Alicia Toree, and Jose M. Alvarifio,' Departamento de Quimica Fisica Facultad de Ciencias. Universidad de Pais Vasco. Apdo. 644. Bilbao. Spain
In dealing with hydrogenic atoms, most general and physical chemistry textbooks pay some attention to the radial probability with independence of the angular variables. The probability ofthe radiusr to have a value in ( r , r d r ) iscalculated by the "radial density of pn)habilityn (RDP). This concept, however, is not homogeneously formulated in different textbooks. Some physical chemistry ( I ) as well as more specialized (2) and physics ( 3 )textbooks refer to theRDP as R W while the vast majority of general and physical chemistry textbooks define it as 4aR2r2( 4 ) . One well-known texthook ( 5 ) uses both formulations without anv comment. This aoparently trivial problem becomes puzzling for most students and has to our knowledee not heen dealt with ~reviouslv. It is interesting to analyze the role of the 4kfactor and to show the equivalence of both formulations. To make the comparisons of interest we must carefully watch the wavefunction normalization conditions. Quantum mechanics expresses the nrhital wavefunction, +, fin a hydrogenic atom as
+
$ = R(r)Y(ll,ch) (I) where R f r )depends on the radial cwrdinate, r, and Y(H.6) are The normalized function, $, can the spherical harmonics (6). he expressed as the product of c&(r) and CAY(O,$),where clx and c~ are the constants of normalization of the radial, Rir), and angular Y(H,$) functions. With this formalism, theR and Y functions depend just on the variahles r and (8,4), respectively, and they must contain no numerical constants.
So, if CAY(B,@) is normalized to unity
From a physical point of view, eqn. ( 7 ) represents the probability that the nucleus-electron distance lies in a spherical shell of thickness d r a t radius r. Formula ( 7 ) normalized to unity reads c'
p
=
Jhr.;~2r2dr
(41
This formula has been obtained using the total wavefunction of the system On the other hand, as the nucleus-electron distance does not depend on the angular variables some authors ( 4 )do not start with $, hut directly with R ( r ) and express the prohahility, d p , to find the nucleus-electron distance in a volume d r as
+.
4sR2r2dr
(81
= 1
and so, the prohability of the radius r being in the range [a.b] is
The way to arrive a t eqn. (9)was rather intuitive. We did not use the total wavefunction of the system +,as we did to get eqn. (4). However, both expressions mean the same thing, so it must hold c$ = c24a or CR = 2cJ;; (10) The curves c:R'r2 versus r and c24aR+' versus r are then idmtical and the two expressions for calculating the RDP are only apparently different. Notwithstanding it must be stressed that eqn. (4) has been more rigorously derived although eqn. (9)shows more clearly the physics of the problem. Generally speaking, it can be said that the more rigorous hooks (1-3) use the first formulation while the more pictorial ones work the second way. Because the normalization constants are related bv the formulas (10) . . both formulations are identical. In conclusion the common R2r2versus r d o t can be uoheld if one implicitly assumes that the radial function is already normalized, i.e., if it contains the pertinent numerical constants. This should be explicitly written out, i.e., one should plot R6,,,, r 2 versus r where R .,,,, must hold cA
The curve that gives the RDP normalized to unity will he the function cKR'r2. The probability, p , that the nucleuselectron distance has a value, r, a < r 8 b will then be
ie
RZrZdr=
i=
Xi,,.,r'dr
=1
(11)
R,,,,, is exactly that function which is brought in tables. On the other hand, if one uses 4aR2r'as the probability density, which mathematically must he normalized to the unity, this R is not exactly the radial function as taken from tables, but This point is not a t all mentioned in hooks R = R.,,,,/2\/;;. which use 4aR2r2as the RDP ( 4 ) . Literature Cited F., r re per^, N. W. Hn1sey.C. D..and Rehinovit.
(5) dp = RZ(r)dr Due to the spherical symmetry of the problem all angular positions are equivalent, and d~ can be written
dr = d14/:,rr")= 4 d d r and so, formula (5)is transformed into dp = 4rRzr'dr
(61 (7)
I
Author to whom all correspondence should be addressed. Volume 58 Number 8 August 1981
617
