S. M. Blinder University of Michigan Ann Arbor, 48104
Wave Mechanics without Waves
Schrodinger's formulation of nonrelativistic quantum mechanics has provided the most convenient framework for~ the theorv of atoms. molecules. and solids. ~ - modern ~ Originally, of course, the Schrodinger equation was pronosed to account for the underlying wave nature of matter bn an atomic scale. One can readiG produce the 1-particle Schrodinger equation by putting the de Broglie wavelength relationship A = h / p = h/(2m[E
- V(r)l)"'
are the N orbits of lowest energy, singly occupied a t most in accord with the exclusion principle. In the theory of determinants one can find a remarkable mathematical analog of the exclusion principle. Suppose r$a(~),6a(~), . . . &fx) represent linearly independent functions of the possible arguments x ~ x z ., . . XN and we set up the N X Ndeterminantal function
(1)
into the classical time-independent wave equation
The physical significance of the wavefunction is also suggested by analogy with classical wave theories (electromagnetism, accoustics, etc.). In many instances, the square of the wave amplitude $(r) provides the most direct measure of the intensity of an effect. I t is thereby suggested that the physical significance of the quantummechanical wavefunction resides in the quantity l$(r)lz, which was interpreted by Born as the position probability density. The preceding arguments presuppose some fair degree of sophistication in mathematical physics which few heginning students in chemistry possess. It would therefore appear desirable to have available some alternative presentation of the Schrodinger equation which makes no explici t reference to classical wave theory. The approach to be outlined in this article is not without its mathematical demands. However. this mathematics must subsequently be introduced in any event as part of the apparatus of quantum chemistry. One can not develop the ~ r i n c i ~ l of e squantum chemistry wholly on the basis of LhemLal concepts. Still a heainnina alona these lines is possible if one pursues the lopical i ~ p l i c a ~ o of n sthe periodic structure 6f the elements. Bohr and Stoner (1921-24) developed a picture of the atom within the framework of the old quantum theory. They proposed that (1)each electron could move in one of a set of allowed Bohr-Sommerfeld orbits; (2) each orbit had a maximum possible occupancy. On the basis of the resulting electronic "shell structure," the periodic table could he rationalized. Pauli in 1925 made these ideas more explicit in enunciating his exclusion principle. He postulated that (1) each electron is described by a set of 4 quantum n u m b e r s 3 determining its orbital motion, the fourth its spin: . . .(2). no two electrons in an atom can 00ssess the identical set of quantum numbers. Pauli proposed in addition that (3) there can he no unique association between a set of quautum numbers and a particular electron-in other words, that the electrons in an atom are indistineuishable. This last asuect of the exclusion principle " governs the enumeration of possible states in a many-particle svstem. It is of central importance in statistical mechanics, resolving, for example,'the Gibhs paradox regarding the spurious entropy of mixing for identical molecules. Let the symbols a, b . . . each stand for a possible set of 4 quantum numbers describing an electron's orbit and its spin. Then the N electrons in an atom must occupy N distinct sets a,b . . . n. In the ground state of the atom, these ~
Should any two of the functions ... 6" coincide then two columns of the determinant would he identical and the determinant would vanish. This means that (xl,xz, . . . XN) will exist (not vanish) only when a, b, . . . n are all different. This mathematical realization of the exclusion principle suggests that Q can represent in some way the state of an N-electron atom. Evidently x ~ x z ., . . XN refer to the orbital and spin coordinates of electrons 1,2, . . . N a n d a, b, . . . n to the associated sets of quantum numbers. Indistinguishability is also provided for by virtue of the fact that the N! terms of the expanded determinant contain every possible permutation of the indices a,b, . . . n among the electron numbers 1,2, ... N. Determinantal functions of the above form were, in fact, first introduced by Slater (1929) after the Schrodinger equation was well established. We have thus far inferred the existence of a set of functions &(x),&,(x), . . . describing the individual orbits of the electrons in an atom. The spin quantum number has, for each electron, just two possible values, corresponding to two spin states designated a and @. Since two electrons of opposite spin can stilj occupy the same distribution in space (orbit in the old quautum theory), it is reasonable to suppose that each 1-electron function @(xi is factorizable in the form C( or dx) = +(r){@ (4) in which r represents the positional (or orbital) coordinates. One-electron functions of the type $(r) have since been dubbed "orbitals" (Mulliken, -1933). The complete functions @(xi are now termed "spinorhitals." It is convenient, incidentally, in order to assure linear independence of the spinorhitals, to select them from among a complex orthonormal set, viz.,
~~
J&*(xMdx)dx
= Bad.a,@ =
a,b, ...a
(5)
The determinantal function (3) will in that case be normalized according to
JJ ...flQ(x,,x
~ , , ) I ~ d x , d x ~ . . . d x=~ 1
(6)
We have inferred, more or less intuitively, that an orbital function $(r) describes in some way the distribution Volume 51. Number 8 , August 1974
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515
in space of an electron. To deduce, more precisely, the physical significance of +(r), we shall connect the description of the atom represented by the Slater determinant (eqn. (3)) with that pictured in the Rutherford model. According to Rutherford's model, by now amply verified by the experimental results of physics and chemistry, an atom consists of a diffuse distrihution of electronic charge surrounding a highly concentrated nuclear charge. The distribution in space of the electronic charge "cloud" can be described by a density function p(r). For an atom with N electrons (equal to the nuclear charge Z for a neutral atom), the charge density function must he normalized according to the condition Jp(r)d3r = -Ne
(7)
If atomic electron clouds behaved according to the laws of classical causality, we should expect the total density to he expressible as a sum of individual electronic contributions N
dr)= Ipj(r)
(8)
/-I
the equation determining the wavefunction +(r) and the associated energy E. Let us specialize our consideration to a l-electron system such as a hydrogenlike atom. By classical electrostatics, the potential energy of an electron distribution p(r) = -e(+(r)lz in the field of a positive nuclear charge Ze is given by
The kinetic energy of a particle can be expressed T = p.p/2rn (17) in terms of its momentum p. For a diffuse distribution i t might he more appropriate to represent eqn. (17) as an integral over some field quantity n(r), something akin to a momentum d e n ~ i t y In . ~ analogy with eqn. (161, one might therehy propose (18) The total energy of the electron is given by (19)
E = T + V which may be cast as anjntegral relation
Equivalently, in terms of spinorhital designations ~ ( r=) Cp,(r)
(9) ,,(In quantum theory these addition formulas are only approximations, hut they still fulfill our heuristic purpose.) In analogy with eqn. (7), each orbital density function must be normalized to the charge of a single electron, i.e.,
Jp,Jr)d3r = -e
(10)
a = a,b,...n
Let us now attempt to reconcile the alternative descriptions of the atom inherent in eqns. (3) and (7). Any relationship between p and cannot he simple linear since these two functions behave differently under permutation of the indices a, b, . . . n. Whereas p is symmetrical with is antisymrespect to interchange of any two indices, metrical (changes sign). However, the square I+ l2 = a*+ is again a symmetrical function, suggesting that this may be the quantity related t o p . It remains to connect p(r), a function of a single position variable, with +(xl,xz, . . . XN) = +(rl,rz, . . . RV,OI.OZ, . . . O N ) a, function of N position as well as N spin variables. By matching u p the respective normalization conditions of eqns. (6) and (71, lifting an integration over r in each, we are led to postulate the relation p(r) = -NeJJ...Jl+(x,,x9 ,...~ ~ ) 1 ~ d u , d ~ , d ~ ~ . . . d x(11) ,~j,,~,
+
+
By the indistinguishability property of the determinant, any other of the position variables rz,r3, . . . r N could have been singled out on the right-hand side with an exactly equivalent result. Specializing to the 2-electron case (e.g., He atom) for manipulative convenience, we find
and ~ ( r=)
pAr)
+
p&r)
(13)
Applying eqn. (11) in conjunction with the orthonormalization conditions of eqn. (5) and the factorization of eqn. (41, we obtain It is therehy implied, in general, that
It would he nice if r could he expressed in terms of +; for then eqn. (20) would give an equation involving alone. It is reasonable to expect that the n-field vector might he large in regions where the probability amplitude has a large gradient. Let us boldly postulate on this basis that r = %V$ (21) where h is a constant (which later comparison with experiment identifies as Planck's constant/Zz). Thus eqn. (20) becomes
+
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On the further postulate that the iotegrand itself is the vanishing quantity, we arrive at the l-particle Schrodinger equation
Development of the many-particle Schrodinger equation can proceed in the conventional way. It then emerges, of course, that the Slater determinant (eqn. (3)) is just an approximate solution, notwithstanding its admirable utility as a heuristic device. We have thus outlined a novel pedagogical approach to the fundamental principles of quantum mechanics making no explicit reference to classical wave theories. The existence and physical significance of the wavefunction have been inferred from the periodic structure of the elements and the Rutherford atomic model. The Schrodinger equation itself was developed by consideration of the energy density in a hydrogenlike atom.
'This connection is also suggested by comparing the normalization relation eqn. (10)with eqn (5) for0 = a : .f4,*(r,a)4,(r,o)d+d0
From this one is led to the physical significance of an orbital function as a particle density amplitude.' In the final part of our development, we shall arrive a t
+
-
Integrating by parts in the first term, assuming that 0 sufficiently rapidly as r m to eliminate any boundary terms, we find
=
J+e*(r)+e(rM3r= 1
2The momentum density in quantum mechanics is actually given by -iliJ.*v$.Note that=(r) as defined has dimensions of mom(~ol)-'/~ and cannot itself be a momentum density.
