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Zeitschrift für Naturforschung A

Field Theoretic Functional Calculus for the Anharmonic Oscillator in Low Approximations

Wolfgang Schuler

Published Dec 1, 1968• 1 citation
Publisher

Abstract

The theory of solution for quantum field functional equations as developped in II and III for a suitable test problem of quantum mechanics is investigated in low approximations. In Sect. 1 the functional formulation of the anharmonic oscillator is once more given and in Sect. 2 general translational equivalent functional equations. The expansion of the physical state functional into series of unsymmetrical and symmetrical base functionals and the representation of the functional equations for such expansions are discussed in Sect. 3. In the next Sect. 4 the unsymmetrical DYSON representation is investigated and the explicit representation of the smeared out functional equation by an infinite system of equations is derived. Then in Sect. 5 and 6 the system of equations is truncated for N = 3 and the corresponding eigenvalue equation is considered. The same is done in Sect. 7 and 8 for the HERWITTE representation. In the following Sect. 9 the original functional equation in a not smeared out form is treated in the DYSON representation and the corresponding system of unsymmetrized equations is given. Furthermore in Sect. 10 the N = 3 approximation together with other possibilities is investigated again. Finally the numerical results of our calculations for eigenvalues are stated and discussed. In the appendices technical details are derived.

DOI
10.1515/zna-1968-1201

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The theory of solution for quantum field functional equations as developped in II and III for a suitable test problem of quantum mechanics is investigated in low approximations. In Sect. 1 the functional formulation of the anharmonic oscillator is once more given and in Sect. 2 general translational equivalent functional equations. The expansion of the physical state functional into series of unsymmetrical and symmetrical base functionals and the representation of the functional equations for such expansions are discussed in Sect. 3. In the next Sect. 4 the unsymmetrical DYSON representation is investigated and the explicit representation of the smeared out functional equation by an infinite system of equations is derived. Then in Sect. 5 and 6 the system of equations is truncated for N = 3 and the corresponding eigenvalue equation is considered. The same is done in Sect. 7 and 8 for the HERWITTE representation. In the following Sect. 9 the original functional equation in a not smeared out form is treated in the DYSON representation and the corresponding system of unsymmetrized equations is given. Furthermore in Sect. 10 the N = 3 approximation together with other possibilities is investigated again. Finally the numerical results of our calculations for eigenvalues are stated and discussed. In the appendices technical details are derived.
The theory of solution for quantum field functional equations is developped for a suitable testproblem of quantum mechanics. In Sect. 1 the functional formulation of the anharmonic oscillator in its spinorial representation is given, and in Sect. 2 translational equivalent functional equations are discussed. The expansion of the physical state functionals into series of basefunctionals and the symmetrical representation of the functional equations for such an expansion is discussed in Sect. 3. In the following Sect. 4 the special symmetric orthogonal Hermitean functionals are used and the explicit representation is derived. In Sect. 5 the functionals are approximated by expansions with only a finite number of terms and the resulting equations are prepared for integration and in Sect. 6 a necessary condition of stationarity is considered. In Sect. 7 the simplest equation for N=1 is discussed in detail and the lowest eigenvalue is obtained. In the appendices technical details are derived.
The theory of solution for quantum field functional equations is developed for a suitable testproblem of quantum mechanics. In Sect. 1 the anharmonic oscillator is described in a field theoretic fashion. In Sect. 2 its functional equations are derived and in Sect. 3 these equations are symmetrized due to physical conditions. In Sect. 4, 5 the expansion of the physical functionals into series of base functionals is discussed and a convenient notation for the operator representation is introduced. In Sect. 6 the representation of the functional equation for an expansion into Dyson base functionals is given. In Sect. 7 and 8 functionals are approximated by expansions with only a finite number of terms and the resulting equations are prepared for integration. In Sect. 9, 10 the integration of the resulting equations for N = 2 and N = 4 is discussed in detail so that one finally obtains eigenvalue equations which contain only integrals to be solved. In the appendices technical details are derived.
In sections 1, 2, 3, 4 the many-time and one-time functional calculus is developed for the anharmonic oscilator in analogy to the requirements of nonlinear spinor theory. In section 5 the N.T.D.-method is discussed for the eigenvalue functional equation. It is shown that the N.T.D.-method admits different representations, namely a symmetric one and an unsymmetric one. The proof of convergence is given for the symmetric one in section 6. In section 7 the connection between the unsymmetric and the symmetric representation is discussed and in section 8 follow numerical values in comparison with SCHRÖDINGER theory.
As in the usual quantum field theory, the states, and therefore also the eigenvalue spectrum of an anharmonic oscillator can be characterized by means of the so-called τ-functions, that is the matrix element of the type 〈0 | qn | j〉. For the calculation of these matrix elements, the equation of motion of the anharmonic oscillator can be used to obtain an infinite set of equations, which define an eigenvalue problem. To solve it a new set of functions, the so-called φ-functions, are introduced by means of a transformation, whose matrix corresponds formally to the WICK rule. An analysis of this infinite system of φ-equations shows that a convergent secular polynomial can be obtained, which exists as a limiting value of the polynomials for the truncated N φ -equationsystems in the limit N → ∞ . It is therefore permissible to calculate the eigenvalues of the infinite system in an approximate way from the truncated systems. Such an approximation procedure is the essential content of the so-called TAMM-DANCOFF method. The above mentioned convergence of the determinants therefore provides its justification. The convergence of the eigenvalues of the truncated systems to the exact oscillator values is numerically examined up to N= 20. The results are satisfactory.
Matrix mechanic methods are used to find approximate equations and solutions for quantum anharmonic oscillator problems. A series of hypotheses are introduced that truncate and partially decouple the infinite set of coupled equations that specify the problem in the matrix mechanics formulation. The dependent variables or unknowns in these equations are the matrix elements of the coordinate and momentum operators. The independent variables are the matrix indices and coupling strengths. The equations themselves specify that the off diagonal matrix elements of the Hamiltonian and the commutators expressed in terms of the unknowns vanish and that the diagonal commutator matrix elements vanish except for canonical pairs in which case they are equal to −ih/. The truncation and decoupling hypotheses offer an orderly procedure for dealing approximately with the vast array of equations of the exact problems. Only the leading behavior of the coordinate and momentum operator matrix elements is found in terms of the matrix indices and coupling parameters. Although general techniques are presented to find the equations, the solutions discussed and the applications are brief extensions of problems that have already been treated.
We develop a general formalism for calculating the large-order behavior of perturbation theory for quantized systems of unequal-mass coupled anharmonic oscillators. Our technique is based on a generalization of the semiclassical approximation which was used to study equal-mass oscillators in the first paper of this series. The unequal-mass problem is much more difficult because the path which minimizes the classical action is not a straight line. Assuming that this tunneling path is known, we derive a general expression for the physical-optics approximation to the wave function of a tunneling particle. This derivation rests on the construction of a WKB approximation in curved space. We thus completely reduce the general quantum problem to the much simpler classical one of determining the path. Then we present a perturbation scheme for finding the classical path for systems of oscillators whose masses only differ by a small amount. Finally, we illustrate our techniques by solving a two-mode unequal-mass oscillator and comparing these results with a computer calculation. Our theoretical predictions and numerical calculations agree.