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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Exact solution to a quantum-forced anharmonic oscillator

Jialun Zhang, Raphaël Khan, Sheng Ding, Wenda Shen

Published Oct 28, 1992• 2 citations
Publisher

Abstract

The exact solution to a quantum forced anharmonic oscillator is derived. The equation of motion is solved by using the integral operators and iteration method. As a result, the time development of the displacement and momentum operators of the forced anharmonic oscillator is obtained. These operators are presented as a Laplace and a subsequent inverse Laplace transform of suitable functionals.

DOI
10.1117/12.130400

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The exact solution to a quantum forced anharmonic oscillator is derived. The equation of motion is solved by using the integral operators and iteration method. As a result, the time development of the displacement and momentum operators of the forced anharmonic oscillator is obtained. These operators are presented as a Laplace and a subsequent inverse Laplace transform of suitable functionals.
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.
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.
The solution of the quantum dynamical equation $\frac{i\ensuremath{\hbar}\mathrm{dT}}{\mathrm{dt}}=HT$, for the time-displacement operator TTT, is given, when the Hamiltonian HHH is a polynomial of the second degree in canonically conjugate variables, with arbitrary time-dependent coefficients. Heisenberg's equations of motion are then solved, and the general integral of Schr\"odinger's equation in coordinate space is expressed by the Green's function corresponding to TTT. An example is given.