Conditional probability amplitudes in wave mechanics

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. IX, 237-242 (1975)

Conditional Probability Amplitudes in Wave Mechanics

GEOFFREY HUNTER* Quantum Theory Project, University of Florida, Gainesville, Florida 326 1 1, USA

Abstracts

The total wave function of a system y ( x , y ) is expressed as a product of a marginal amplitude function f ( y ) and a conditional amplitude function $(x I y). The original Schrodinger equation Hy = Ey is reduced to a Schrodinger equation for the marginal amplitude H ‘ f = Ej: The factorization of y is in principle exact. I t achieves a partial separation of the variables x and y.

La fonction d‘onde totale d’un systtme y ( x , y ) est exprimte comme ttant le produit d’une fonction d’amplitude marginale f ( y ) par une fonction d’amplitude conditionelle $(x I y ) . L’equation originale de Schrodinger Hy = E y est rkduite a une equation de Schrodinger pour l’amplitude marginale H Y = EJ est exacte en principe. Le resultat se traduit par une stparation partielle des variables x et y .

La factorisation de

Die totale Wellenfunktion eines Systems y ( x , y) ist als Produkt einer Grenzampli- tudenfunktion f ( y ) und einer bedingten Amplitudenfunktion $(x I y ) ausgedruckt. Die ursprungliche Schrodinger-Gleichung Hy = E y wird zu einer Schrodinger Gleichung fur die Grenz-amplitude H‘f = Ef reduziert. Die Zerlegung von y ist im Prinzip exakt. Sie fuhrt zu einer partiellen Trennung der Variabeln x und y.

1. Introduction

In this paper the factorization of a wave function as the product of a conditional probability amplitude and a marginal probability amplitude is examined. The squared modulus of each of these amplitudes is the wave mechanical analogue of a conditional and a marginal probability density in probability theory [l]. The Born-Oppenheimer separation for a diatomic molecule will exemplify this factorization, since the electronic and nuclear wave functions in the adiabatic approximation [2] are very good approximations to a conditional and a marginal amplitude, respectively.

Born and Oppenheimer’s view of the dynamics of a diatomic molecule is that the motion of the electrons may be calculated with the two nuclei a fixed distance R apart. The electronic motion, and the electronic wave function +(r 1 R ) (r symbolizes the electronic co-ordinates) is parametrically dependent

* Permanent address: Department of Chemistry, York University, Toronto, Canada. 237

0 1975 by John Wiley & Sons, Inc.

238 HUNTER

on R. The squared modulus $*+ is thus identical in concept with a conditional probability density P(r I R ) , since this is defined as the distribution of r for a fixed value of R [I].

The other aspect of the Born-Oppenheimer theory is that the vibrational motion of the nuclei is governed by a radial potential V ( R ) (considering only rotationless states for simplicity). In the adiabatic approximation [2] V ( R ) is the expectation value ($(r I R)I H l$(r I R ) ) , , where H is the internal Hamil- tonian of the molecule and the subscript r denotes integration over the configura- tion space spanned by the electronic coordinates r. H includes the electronic kinetic energy and the nuclear vibrational kinetic energy operators, as well as the total Coulomb potential energy. The vibrational motion is represented by a radial wave function f (R) , so that f * f is the distribution of internuclear distance averaged over the motion of the electrons. This average distribution f*fis identical in concept with a marginal probability density P(R) in probability theory

The complete multivariate distribution of r and R, P(r, R ) is equal to the product P ( R ) x P(r 1 R) [l]. Similarly the total electronic-vibrational wave function y(r, R) is expressed as the product f [ R ) x $(r [ R) in the adiabatic approximation [2]. This factorization of P(r, R) is exact, whereas the adiabatic factorization of y(r, R ) is only approximate. Since the product form f ( R ) x $(r [ R ) does not in itself imply any restriction on the functional form of y(r, R ) , the approximate nature of the adiabatic wave function must arise because 4(r I R ) is defined to be an eigenfunction of the electronic Hamiltonian [2]. The primary purpose of the development reported here is to demonstrate how such a factorization of a total wave function may be made exact.

P I .

2. Factorization of the Wave Function

The Schrodinger Hamiltonian operator of a particular system is denoted by H E H ( x , y), where x and y symbolize complementary subsets of all of the independent variables (commonly particle coordinates) occurring in H. An eigenfunction of H is denoted by y = y ( x , y); that is

(1) H ( x , Y )Y(X , Y) = Y)

The wave function y is factorized as a product of a conditional probability amplitude function 4 E + ( x 1 y ) and a marginal probability amplitude function f = f f ( y ) .

(2) = Y(X> Y) = f ( y ) 4 ( x I Y ) =f$ For simplicity y will be assumed to be normalized:

y*ydxdy = 1 s (3)

CONDITIONAL PROBABILITY AMPLITUDES 239

If y * y is integrated only with respect to x (over the whole range of x), then the resulting function of y must be the normalized probability density distribution of y averaged over x!, that is, the marginal distribution of yf*(y)f(y) [l] :

(4)

Integration of the product form of y (2) with respect to x thus produces the condition :

This normalization condition on the conditional amplitude 4 must hold for all values of y.

If the total wave function y ( x , y) is known, Equations (2) and (4) determine f (y) and +(x 1 y) except for a phase factor of the form exp (ix(y}), where ~ ( y ) is a real function. In the absence of any criterion for determining this phase factor, the simplest procedure would be to set f(y) = f if(y)/, withf(y) changing sign at each zero of If(y)I. The relation $(x 1 y) = y ( x , y)lf(y) may apparently lead to singularities in +(x 1 y) at the zeros off(y). However Equation (2) shows that y(x, y) is zero at the zeros off(y), so that +(x I y) is not necessarily singular at the zeros off(y). Furthermore, in view of the nature off(y) as a marginal probability function, it is unlikely to contain zeros; this is the case for the one specific example studied so far [9]. I t should also be noted that the choice of the phase function ~ ( y ) does not affect the zeros off(y).

:3. Reduction of the Schrodinger Equation

'The original Schrodinger Hamiltonian H ( x , y) will have the general form:

Individual members of the sets of variables x and y are denoted by x andy. The term in the braces { } is the kinetic energy with K representing an inverse mass factor. Dx symbolizes a first-order differential operator with respect to x . For generality all possible nondiagonal kinetic energy terms are included. In Cartesian coordinates the K ' s are constants ( -h2/2M for a particle of mass M ) and Dx is simply a/&. In curvilinear coordinates K may be a function of some of the coordinates as well. This notation is adopted because the sets x and y do not necessarily contain all of the coordinates of a particular particle, so that a sum of V2 terms cannot be used. The nondiagonal kinetic energy terms will be present if the original Hamiltonian in terms of N particles has been transformed in some way

240 HUNTER

such as is involved in the separation of the centre of mass motion [3]. A, G Az(x, y ) and A, = AY(x, y ) are the components of the velocity dependent or vector potential, and V = V ( x , y ) is the scalar potential function. The only terms of H which are always present are the diagonal kinetic energy terms K,, D: and KYY D: .

When this form of H (6), and the factored form of y ( x , y ) (2), is inserted into the original Schrodinger equation (1) , the resulting equation is:

When (7) is multiplied from the left by +*, and integrated over the range of x employing the normalization condition (5), then the following reduced Schro- dinger equation is produced :

Kf-= c {KYYD,2f + PK?J,JY + BYlD,f 1 + 2 c K,,JXD,f Y X Y

+ 2 K,,j{DYDY,f + JY*DYf + JYDY*f } + uf = Ef (8 ) Y Y ' Z Y

The factors KYY , K,, , and Kyv' have been assumed to be independent of x.

J, = J X ( y ) , J,, J,(y), B, E B,(y), and U = U ( y ) , are the following integrals:

(9)

(12)

The J integrals will be zero in many cases; J , will be zero for real + functions because of (5); J x will be zero for real bound state 4 functions, as can be shown using integration by parts. In cases where A, is only a function ofy, By A, . If J, = J , = 0 and B, = A , , as will commonly be the case, then (8) simplifies to :

CONDITIONAL PROBABILITY AMPLITUDES 24 1

This equation is the Schrodinger equation one of whose eigenfunctions is the marginal amplitude f(y). The scalar potential function in H’, U(y), is the expectation value of the original Hamiltonian H with respect to the conditional amplitude +(x I y). Thus part of what was kinetic energy and velocity-dependent potential energy in the original Hamiltonian H, manifests itself as scalar potential energy in the reduced Hamiltonian H’.

The reduced Schrodinger equation ((8) or (1 3)) is of the same general form as the original Schrodinger equation ( l ) , so that its independent variables y may also be partitioned into two complementary subsets. Thus the process of factorization may be applied to f(y) as to the original y ( x , y), resulting in a further reduction of the Schrodinger equation. Successive factorizations and reductions can be carried out as many times as are appropriate to the particular problem in hand.

4. Discussion The factorization of the total wave function (2-5) is energy dependent. That

is, in general a particular total wave function y z ( x , y) factorizes into a particular conditional amplitude +%(x 1 y) and a particular marginal amplitudef,(y). Thus there will generally be a different potential function U,(y) in the reduced Schro- dinger equation (13), for each particular state yz(x, y). This suggests that such reduced potentials are related to optical potentials [4]. For a particular potential U,(y), there will generally only be one solution of the reduced Schrodinger equation (13), which makes the product f,(y) x dz(x 1 y) an eigenfunction of H. We refer to this state dependence of the marginal and conditional amplitudes, as the inseparability of the eigenvalue spectra associated with y and x.

Born and Oppenheimer’s theory [5] is essentially a perturbational approach to the conditional/marginal factorization of a wave function, for the case where x represents the electronic coordinates, and y the nuclear coordinates, in a molecule. In this context it is a successful method, because the perturbing operator (the nuclear kinetic energy) is small by comparison with the other terms of the internal Hamiltonian. The error in the first-order (adiabatic) potential is of the order of

of the total energy, so that for many purposes the electronic and vibrational eigenvalue spectra may be assumed to be separable. As applied to other systems such as H- [6-8) the perturbational approach is less satisfactory.

The above formalism may lead to useful insights into the dynamical structure of complex systems, insofar as the reduced Schrodinger equation (1 3) provides a way of describing the motion associated with a subset of all of the coordinates. This technique has been applied to the model system of two coupled harmonic oscillators [9]. I t could for example be used to generate in principle exact one- electron wave functions and potential functions for many-electron atoms and mole- cules. This reduction technique depends, of course, upon a knowledge of accurate total wave functions for these systems.

For the formalism to be useful in the derivation of total wave functions, one needs a defining equation for the conditional amplitude +(x I y), which does not

242 HUNTER

depend upon the factorization of the total wave function ($2). Removal of the D, differential operators from H leads to the Born-Oppenheimer-Slater theory [2,5,6], and leaving them in produces the Schrodinger equation for the total wave function y(x, y), so that the conditional amplitude +(x I y) is not apparently defined by a Schrodinger differential equation. It may however be defined by a variational principle, similar to that for the total wave function [lo], except that integration occurs only over x. We leave this interesting possibility for future investigation.

Bibliography

[l] A. W. Drake, Fundamentals of Applied Probability Theory, (McGraw-Hill, New York, 1967),

[2] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Appendices VII and VIII (Oxford

[3] H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, 2nd ed., Section

[4] N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions, (Clarendon Press, Oxford,

[5] M. Born and R. Oppenheimer, Ann. Phys. (Leipzig) 84,457 (1927). [6] J. C. Slater, Proc. Natl. Acad. Sci. 13,423 (1927). [7] B. F. Gray and H. 0. Pritchard, J. Chem. SOC. London 709, 3578 (1957). [8] G. Hunter and H. 0. Pritchard, J. Chem. Phys. 46, 2153 (1967). [9] G. Hunter, “Conditional Probability Amplitude Analysis of Coupled Harmonic Oscillators”,

[lo] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon Press, Oxford, 1958), p. 55.

Chapter 2.

University Press, London, 1954).

11.31 (D. van Nostrand, Princeton, New Jersey, 1956).

1965).

Int. J. Quantum Chem.: Symposium No. 8, 413 (1974).

Received March 7, 1973. Revised September 20, 1974.

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Conditional probability amplitudes in wave mechanics