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Progress of Theoretical Physics Supplement

Phase Quantization of Chaos and the Role of the Semiclassical Amplitude Factor(Quantum Mechanics and Chaos)

Takefumi Yamashita, Kazuo Takatsuka

Published May 31, 2007• 0 citations
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Phase Quantization of Chaos and the Role of the Semiclassical Amplitude Factor

Takefumi Yamashita, Kazuo Takatsuka

In a recent paper [Takatsuka, Takahashi, Koh and Yamashita, J. Chem. Phys. 126 (2007), 021104], it was shown that the semiclassical quantization of chaos can be accurately achieved only with the phases (the action and Maslov phases) without use of the amplitude (preexponential) factor, which is known to diverge exponentially in chaos. The aim of this paper is to analyze the role of the semiclassical amplitude in energy quantization from various points of view, and we actually show that it is indeed quite limited. Therefore even chaos is quantized mainly by the phase constructive and destructive interferences.

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A geometrical method to calculate the Maslov index, which is an important part of the quantum phase, is proposed. This method is particularly useful for the recently proposed amplitude-free quasicorrelation function approach for the quantization of chaos [K. Hotta and K. Takatsuka, J. Phys. A 36, 4785 (2003)]. In this theory the Maslov index is involved with no need to calculate the amplitude factor, which is usually obtained through integration of the stability matrix. Since this matrix constitutes the major origin of difficulty in semiclassical quantization of chaos and since its integration is time consuming, the present method, which avoids the stability matrix, should assist in opening a gate for practical semiclassical quantization of chaos in a large molecular system.

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In a recent paper [Takatsuka, Takahashi, Koh and Yamashita, J. Chem. Phys. 126 (2007), 021104], it was shown that the semiclassical quantization of chaos can be accurately achieved only with the phases (the action and Maslov phases) without use of the amplitude (preexponential) factor, which is known to diverge exponentially in chaos. The aim of this paper is to analyze the role of the semiclassical amplitude in energy quantization from various points of view, and we actually show that it is indeed quite limited. Therefore even chaos is quantized mainly by the phase constructive and destructive interferences.
A geometrical method to calculate the Maslov index, which is an important part of the quantum phase, is proposed. This method is particularly useful for the recently proposed amplitude-free quasicorrelation function approach for the quantization of chaos [K. Hotta and K. Takatsuka, J. Phys. A 36, 4785 (2003)]. In this theory the Maslov index is involved with no need to calculate the amplitude factor, which is usually obtained through integration of the stability matrix. Since this matrix constitutes the major origin of difficulty in semiclassical quantization of chaos and since its integration is time consuming, the present method, which avoids the stability matrix, should assist in opening a gate for practical semiclassical quantization of chaos in a large molecular system.