Wavelet transform in the context of quantum mechanics and new orthogonal property of mother wavelets in parameter space

Abstract

Based on the quantum mechanical version of wavelet transform (WT) W _ψ f(μ,s) = ⟨ψ|U(μ,s)| f ⟩ [Fan, H.Y.; Lu, H.L. Opt. Lett. 2006, 31, 407] we further prove Parseval theorem and inversion formula for WT in the context of quantum mechanics. The concise proof not only manifestly shows the merit of Dirac's representation theory, but also leads to a new orthogonal property of mother wavelets in parameter space.

Keywords:

• wavelet transform
• quantum mechanics
• Parseval theorem

Acknowledgements

We sincerely thank the referee for his useful suggestion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10775097 and 10475056).