Letters in Mathematical Physics

, Volume 71, Issue 1, pp 1–11 | Cite as

On Decaying Rate of Quantum States

  • Shunlong Luo
  • Zhengmin Zhang


The survival amplitude of a quantum state (wave function) under the Schrödinger evolution can be expressed as the Fourier transform of the probability density induced by the wave function in the energy representation. In particular, the first zero of the survival amplitude is a fundamental quantity in characterizing the decaying rate of the quantum state. A basic problem in quantum mechanics is to study how fast the survival amplitude can fall. We present a general estimation of the decaying rate of a quantum state in terms of a moment of any order. The result is established by integrating an inequality which involves controlling trigonometric sums by power functions. This inequality is of independent interest in estimating exponential sums.


quantum state survival amplitude Fourier transform moment time--energy uncertainty relations 


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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Academy of Mathematics and Systems SciencesChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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