Coordinate and Momentum Representations of Quantum States, Fourier Transforms

Chisheng Li, Luo Han-Wen

Using the intermediate coordinate–momentum representation | x ⟩ s,r , we introduce a new Hadamard transform. It is found that the operator U corresponding to this transform can be considered as a combination of the Fresnel operator F ( r, s ) and the Fourier transform operator F by decomposing U. We also find that the matrix element s,r ⟨| x | U | f ⟩ just corresponds to an optical scaled Fresnel–Fourier transform.

L. S. Kuz’menkov, Сергей Геннадьевич Максимов

Zhang Bing-yun

Three non-symmetric projective operators were made by coordinate,momentum eigenstates and coherent states.By IWOP technology,three non-symmetric projective operators were the same Fourier transform operator.This operator could realize the Fourier transform between the coordinate and momentum eigenstates,and the unitary transformation between the coordinate and momentum operators.

Marco O. Lanzagorta, Jeffrey K. Uhlmann

FANHong-Yi

We find quantum mechanical Fourier-Hankel representation transform for an electron moving in a uniform magnetic field. The physical meaning of Fourier decomposition states of electron's coordinate eigenstate and the momentum eigenstate are revealed.

Martin Rötteler, Thomas Beth