The Combined Transformation And Convolution Properties Of Fractional Fourier Transform Studied By Dirac Notation Method

Dirac’s notation unified the statements of Heisenberg and Schrodinger in a wonderful and concise way,and Dirac had been looking forward to further develop the Dirac notation in quantum field.It was not until the birth of the technique of integration within an ordered product of operators(IWOP)that the Dirac notation method was finally applied in a deeper level.A large number of new representations and unitary operators can be derived by the IWOP technique,which makes the theory of quantum mechanical representations and optical transformation a more self-consistent system.In this paper,based on the study of fractional Fourier transform,several new combinatorial optical transformations are proposed using the Dirac symbol and representation theory,and the transformation results corresponding to different quantum states or different state vectors of the mother wavelet are obtained by the IWOP technique.The characteristics of these transformation results are of certain importance in the analysis and identification of quantum states or the mother wavelets.Finally,two new convolution theorems of fractional Fourier transform are proved by Dirac notation,and the convolution of three-mode fractional Fourier transform and three-mode fractional squeezing transform are proposed and proved by Dirac notation.The specific contents are as follows:Firstly,based on the three-mode wavelet transform and the three-mode fractional Fourier transform expressed by Dirac notation,a new combined optical transform,the three-mode wavelet-fractional Fourier transform,is obtained by recombining the integral kernel of the two optical transforms.The specific expression of the core operator of the optical transformation is obtained by using IWOP technique,and different state vectors and quantum states are substituted into the new combined optical transformation to obtain the corresponding optical transformation results.Several transformation results are numerically simulated to obtain corresponding optical transformation wave maps,and their characteristics are discussed.These analysis results can identify different mother wavelets and quantum states.Quantum state is the information carrier in quantum communication.Whether the quantum state can be recognized successfully is very important for the reliable transmission of quantum information.Secondly,considering the relationship between fractional squeezing transform and fractional Fourier transform,we extend the above study to fractional squeezing transform.Based on the Dirac notation of fractional squeezing transform,the wavelet-fractional squeezing transform is obtained and we also extended it to two-mode and three-mode entangled state case.Similarly,the IWOP technique is used to obtain the specific forms of these optical transformation core operators.And then the optical transformation results of different quantum states are calculated.By virtue of the MATLAB,we simulated the transformation results to get several different types of wave spectrogram,and comparatively analysed the wave spectrogram and the characteristics of these optical transformations.Finally,two new convolution theorems of fractional Fourier transform are proved by using Dirac notation and representation theory.The convolution theorem of fractional Fourier transform and fractional compression transform is also extended to the three-mode entangled state case.