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Zeitschrift für Naturforschung A

On the Field Theoretic Functional Calculus for the Anharmonic Oscillator I

Dieter Maison, Harald Stumpf

Published Nov 1, 1966• 4 citations
Publisher

Abstract

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.

DOI
10.1515/zna-1966-1103

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Physical Review A•Oct 1, 1988•9 citations
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.
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 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 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.
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.
For an anharmonic oscillator, it follows from the number of zeros of the exact wave functions as functions of the quantum number nnn that the matrix elements $〈n|x|{n}^{\ensuremath{'}}〉$ and $〈n|p|{n}^{\ensuremath{'}}〉$ should be rapidly decreasing functions of $|n\ensuremath{-}{n}^{\ensuremath{'}}|$. Matrix elements of polynomials in xxx and ppp should therefore be well approximated by a finite number of terms in their sum-rule decomposition. From the matrix elements of the equations of motion for xxx and for ppp and of the commutator ∣x,p∣|x,p|∣x,p∣, one thereby obtains closed sets of nonlinear algebraic equations to characterize subspaces of the Hilbert space of exact eigenfunctions. The approximations are also derived from a novel variational principle, and numerous variant approximation schemes are suggested. Essentially exact numerical results are obtained and compared with previous work. The broad applicability of the techniques is emphasized.
The time evolution of a quantum single-quartic anharmonic oscillator is considered. The study is carried on in operational form by use of the raising and lowering operators of the oscillator. The equation of motion is solved by application of a new integration method based on iteration techniques, and the rigorous solutions that describe the time development of the displacement and momentum operators of the oscillator are obtained. These operators are presented as a Laplace transform and a subsequent inverse Laplace transform of suitable functionals. Finally, the results are employed to describe the time evolution of a quasiclassical anharmonic oscillator.