INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XVII, 133-137 (1980)

Nodeless Wave Function Quantum Theory GEOFFREY HUNTER

Department of Chemistry, York University, Toronto, Canada M3JlP3

Abstract

The nodes where the wave function changes sign in conventional quantum theory can alternatively be regarded as places where the wave function descends to zero but rises again without changing its sign. This behavior is accomplished by adding delta function barriers to the potential energy wherever a node occurs in the wave function. This model is supported by the recognition that the wave functions of a subsystem, which is not truly isolated, are really marginal amplitudes which implicitly average out interactions of the system with its environment. The finite effect of these interactions is to replace the delta function barriers by barriers of finite width and height.

Introduction

A fundamental postulate of quantum theory is that the dynamical energy of a physical system is represented by a Hamiltonian operator that is the same operator for all physical states of the system [l]. This postulate is commonly believed to be true despite the fact that the explicit form of the exact Hamiltonian operator is not known; the well-established nonrelativistic, Coulomb-interaction Hamiltonian used in quantum chemistry is, of course, an approximation [2].

The purpose of this paper is to explore the alternative postulate that each distinct state of a physical system is represented by a different Hamiltonian operator. This conjecture arises in the following way.

In previous work a formalism was developed in which a wave function +(x, y ) is factorized into a conditional amplitude 4(xJy) and a marginal amplitude f(y); x and y symbolize complementary subsets of the particle coordinates of the whole system [4]. In the present context the marginal amplitude f(y) is of primary interest because it represents the state of motion associated with the y coordinates after averaging over the motion associated with the x coordinates; the y co- ordinates define the subsystem of interest and the x coordinates its surroundings or environment.

The wave function f(y ) satisfies a Schrodinger equation in which the kinetic energy is the same operator as in the total Hamiltonian defining t,b(x, y) . The eigenvalue of this reduced Hamiltonian [having f(y) as an eigenfunction] is the total energy of the system including the kinetic and potential energy associated with the x coordinates; the reduced Hamiltonian H’(y ) contains the x kinetic and potential energy as part of its own potential energy operator [4].

Wave Functions of a Subsystem

If we were unaware of the existence of the x coordinates, or if we wished to average out their effect, we would simply refer to H ’ ( y ) as the Hamiltonian of our

@ 1980 John Wiley & Sons, Inc. 0020-7608/80/0017-0133$01.00

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chosen system and to f ( y ) as its wave function. Thus in all instances in which we choose to single out a part of the universe for theoretical treatment, we are implicitly working with a reduced Hamiltonian and a marginal wave function. This is so because our chosen part of the universe is never physically isolated from the rest of the universe in an exact sense.

In the original formulation of the marginal-conditional factorization it was recognized that it implied a different reduced Hamiltonian for each distinct state of the whole (x, y ) system [4]. From this recognition arose a concern with how the reduced potential energy U(y ) differed from the “adiabatic approximation” potential in which state-dependent x - y interactions are neglected [ 5 ] .

A definitive answer to this question has been given in recent work by Czub and Wolniewicz [6]. They prove analytically that marginal wave functions f ( y ) of any subsystem that is not exactly separable (i.e., physically isolated) from its environment, are necessarily nodeless; thus they may be taken to be real and non-negative everywhere. This implies that the state dependence of the reduced potential U ( y ) takes the form of barriers located at the nodes of the adiabaticf(y) wave functions. This result leads to a reformulation of the quantum mechanics of a subsystem.

The reformulation can be regarded as proceeding from conventional theory (i.e., with nodes in excited-state wave functions) in two stages. In the first stage a Dirac delta function barrier is added to the conventional potential function wherever the excited-state wave function has a node. The barriers are delta functions with an infinite coefficient; i.e., they have an infinite area. Addition of these barriers does not change the energy of the state because the wave function is zero at their loci.

The effect of the barriers is to prevent the wave function from changing sign. The wave function has exactly the same magnitude as in conventional theory everywhere including inside the barriers. However, within a barrier the wave function has a curvature that causes it to leave zero in the same direction from which it approached it; the gradient of the wave function is not continuous across the infinitesimally narrow barrier. Neither the energy nor any other physical property is changed by the delta function barriers.

In stage two of the reformulation the delta function barriers of stage one become potential barriers of finite height and finite width as a result of interactions of our chosen system with its environment over which we are implicitly averaging. Thus the real wave functions of a theoretically delimited subsystem (which is not physically isolated) proceed through the finite barriers with a continuous gradient and without reaching zero. The expectation value of the finite energy barrier over the nonzero wave function produces a change in the energy of the state caused by the interaction between the system and its environment.

The two stages of the reformulation are illustrated in Figure 1. The first pair of graphs show the parabolic potential energy and the vibrational wave function of the first excited state of an isolated one-dimensional harmonic oscillator; the wave function changes sign at the potential minimum. The second pair of graphs are for the same state (with exactly the same energy) in the equivalent, alternative theory

NODELESS WAVE FUNCTION QUANTUM THEORY 135

/ ConvenNonol Theory with Excited State Nodes

2 Equivalent Theory with Positive Wove Funciians

POTENTIAL ENERGY FUNCTION I VIBRATIONAL WAVE FUNCTION

3 Reality- with Nodeless wave functions

c I

'-+I 01 t

c L

-1

Figure 1. Potential energy and vibrational wave function of a one-dimensional harmonic oscillator: (1) for an isolated oscillator in conventional theory; (2) for an isolated oscillator in the equivalent theory in which the wave function does not change sign because of delta function barriers at its nodes; (3) for the oscillator interacting with its environment, hence the barrier is finite and the wave function is

nodeless.

in which a delta function barrier has been added to the parabolic potential. The barrier forces the wave function to have the same sign on both sides of the barrier; its magnitude is everywhere the same as in the first picture.

The third pair of graphs show the effect of including in the one-dimensional Hamiltonian, the effects of interaction with the oscillator's environment. The barrier has a finite width and height, and the wave function has a nonzero minimum value at the center of the barrier. This third pair of graphs were drawn from a model calculation in which the oscillator's environment is represented by another oscillator to which the motion of the first one is coupled [S]. Although this

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double-welled potential energy has excited eigenfunctions with nodes, these are unacceptable solutions for the total (two-oscillator) Hamiltonian; only the node- less ground state is a proper solution for the whole system.

These properties are exemplified in a real system by Czub and Wolniewicz’s nonadiabatic calculations on the third vibrationally excited state of the hydrogen molecule [6]. The Hamiltonian with three finite barriers produces a lower energy than the unperturbed one with infinite barriers; the nonadiabatic effect is to lower the energy in conformity with the variation principle. Thus finite barriers in principle explain the discrepancy between the adiabatic and experimental vibra- tional spacings in the hydrogen molecule.

Motion in Excited States

This reformulation of quantum mechanics produces an intuitively simple picture of the motion in an excited state. For example, the wave function for the ground state of a harmonic oscillator with its maximum probability at the point of minimal potential energy is what one would expect on intuitive, classical grounds, whereas the first excited state with its node at the point of minimum potential energy is classically implausible. In stage one of the reformulated theory the oscillator moves within a double-welled potential in the first excited state, and on intuitive, classical grounds one would expect the probability to be zero at the central, infinitely high barrier.

In stage two of the reformulation the finite barrier is consistent with describing the motion as being a quasiclassical oscillation within one or the other of the two potential wells, with occasional tunneling through the central barrier between the two wells. Thus the probability maxima on each side of the barrier are as classically plausible as is the single maximum in the ground state. The tunneling is quantum mechanically plausible because the wave function is greater than zero within the finite barrier.

Concluding Discussion

Despite some obviously appealing aspects of this new way of interpreting quantum mechanics, many of its implications remain to be fully understood. An intriguing question is whether the exact Hamiltonian for a finite part of the universe is representable in state-independent form [3]? If the answer is no, then it follows that all exact wave functions are nodeless ground states of a state- dependent Hamiltonian. In this case the square of the wave function (the probability density) contains as much information as the wave function itself, thus lending support to the view that the density can be used as a simpler alternative to the wave function.

If this situation pertains in the real physical world, then much of conventional quantum theory is seen to be an approximation of reality. The approximation is qualitatively atypical in the sense that its structure depends upon the complete set of eigenfunctions of a single state-independent Hamiltonian and upon the complete set of symmetry operators that commute with this Hamiltonian [ 11.

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Acknowledgments

The role of delta functions in the theory was originally suggested by Dr. S. M. L. Prokopenko of York University. This research was supported by the National Research Council of Canada.

Bibliography

[l] P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, Oxford, 1958). [2] H. F. Hameka, Advanced Quantum Chemistry (Addison-Wesley, Reading, Mass., 1965). [3] J. D. Bjorken and S. D. Drell, Relativistic QuantumMechanics (McGraw-Hill, New York, 1964). [4] G. Hunter, Int. J. Quantum Chem. 9,237 (1975). [5] D. Bishop and G. Hunter, Molec. Phys. 30, 1433 (1975). [6] J. Czub and L. Wolniewicz, Molec. Phys. 36, 1301 (1978). [7] D. Rapp, Quantum Mechanics (Holt, Rinehart and Winston, New York, 1971). [8] G. Hunter, Int. J. Quantum Chem.; Quantum Chem. Symp. 8,413 (1974).

Received April 11, 1979