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International Journal of Theoretical Physics

Comparison of Corrected Wave Functions Associated to Two Different Approaches for the Time-Dependent Hamiltonian Systems Involving $(1/\hat{x})\hat{p}+\hat{p}(1/\hat{x})$ Term

Jeong Ryeol Choi, Jun‐Young Oh

Published Jun 4, 2007• 2 citations
Publisher
DOI
10.1007/s10773-007-9373-7

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Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and (1/x^)p^+p^(1/x^)(1/\hat x)\hat p + \hat p(1/\hat x)(1/x^)p^​+p^​(1/x^) Terms
Equations are derived for the time evolution of time-dependent vibrational coupled cluster (TDVCC) wave functions covering both the TDVCC ket state and the associated so-called Λ bra state. The equations are implemented in the special case of both the Hamiltonian and the cluster operator containing at most two-mode coupling terms. The nontrivial behavior of the evolution of norm, energy, and expectation values due to the nonunitary time-evolution of the nonvariational TDVCC theory is analyzed theoretically and confirmed in numerical experiments that also include time-dependent Hamiltonians. In the spirit of time-independent size-consistency analysis, the separability of both the coupled cluster and Λ states for noninteracting systems is studied. While the coupled cluster state clearly has the correct behavior, the behavior of the Λ state is more intricate, and the consequence for different properties is shown theoretically and numerically. Overall, the numerical experiments show that TDVCC in incomplete expansions gives higher accuracy than a standard linear variational wave function parameterization with the same number of independent parameters, while equivalent results are obtained for complete expansions. The efficiency of the methodology is illustrated in computations on polycyclic aromatic hydrocarbons with up to 156 modes.
We explore how to apply perturbation theory to complicated time-dependent Hamiltonian systems that involve complex potentials. To do this, we introduce a generalized time-dependent oscillator to which the complex potentials are connected through a weak coupling strength. We regard the complex potentials in the Hamiltonian as the perturbed terms. Quantum characteristics of the system, such as wave functions and expectation values of the Hamiltonian, are investigated on the basis of the perturbation theory. We apply our theory to particular systems with explicit choices of time-dependent parameters. Through such applications, the time behavior of the quantum wave packets and the spectrum of expectation values of the Hamiltonian are analyzed in detail. We confirm that the imaginary parts of expectation values of the Hamiltonian are not zero but very small, while the real parts deviate slightly from those of the unperturbed system.