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Generalization of Bandlimited Functions and Applications to Quantum Probability Distributions.

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Entropy (Basel, Switzerland)
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This study generalizes bandlimited functions to arbitrary representations, yielding new inequalities. These findings provide bounds for quantum mechanical wave functions and probability distributions in various scenarios.

Keywords:
bandlimited functionsinequalitiesquantum mechanicstransformswave functions

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Area of Science:

  • Mathematical Physics
  • Quantum Mechanics
  • Signal Processing

Background:

  • Bandlimited functions are crucial in signal processing, defined by their Fourier transform's finite frequency band.
  • Generalizing this concept beyond the Fourier transform is essential for broader applications.

Purpose of the Study:

  • To generalize the concept of bandlimited functions to arbitrary representations.
  • To derive novel inequalities applicable in various mathematical and physical contexts.
  • To explore applications in quantum mechanics, particularly for wave functions and probability distributions.

Main Methods:

  • Generalization of bandlimitedness to arbitrary representations.
  • Derivation of inequalities for frequency, dilation, and chirplet transforms.
  • Application of derived inequalities to quantum mechanical wave functions.

Main Results:

  • Established a framework for inequalities in arbitrary representations.
  • Derived explicit bounds for position and momentum wave functions and their probability distributions.
  • Investigated cases including bounded momentum/position and sums of energy eigenfunctions.

Conclusions:

  • The generalization of bandlimitedness provides powerful tools for bounding quantum mechanical properties.
  • The derived inequalities offer new insights into the behavior of wave functions and probability distributions.
  • Specific examples like truncated Gaussian and quantum bump wave functions illustrate the practical utility of the results.