Quantum states can be represented mathematically in different ways, depending on the chosen basis states. For example, position-based states have a definite position but an indefinite momentum, while momentum-based states have a definite momentum but an indefinite position.The Fourier transform is a mathematical operation that converts a quantum state from its position-space representation to its momentum-space representation, and vice versa. This transformation allows a function to be expressed in a different domain.This paper examines the concept and characteristics of the momentum-space representation of a quantum state and compares it to the position-space representation. Additionally, we investigate the applications and benefits of using the momentum-space representation to solve certain issues in quantum mechanics.
We construct here families of coherent states for the full Poincare group, for representations corresponding to mass
m>0 and arbitrary integral or half-integral spin. Each family of coherent states is defined by an affine section in the group and constitutes a frame. The sections, in their turn, are determined by particular velocity vector fields, the latter always appearing in dual pairs. We discretize the coherent states of Poincare group in 1-space and 1-time dimensions and show that they form a discrete frame, develop a transform, similar to a windowed Fourier transform, which we call the relativistic windowed Fourier transform. We also obtain a reconstruction formula. Finally, we perform numerical computations. We evaluate the discrete frame operator numerically and present it graphically for different sections and windows. We also reconstruct some functions, compare reconstructed functions with the original ones graphically. We compare the reconstruction scheme of the relativistic windowed Fourier transform with that of the standard windowed Fourier transform.
Drawing upon some earlier work on coherent states of the Poincare' group in l-space and l-time dimensions, we use these states to define a relativistic windowed Fourier transform, as relativistic extension of the usual windowed Fourier transform. We discretize the resulting transform and obtain conditions under which the discretized transform can be used to reconstruct arbitrary square integrable functions. We present some numerical and graphical exercises to illustrate the theory, as well as to compare the relativistic windowed Fourier transform with the orthodox windowed Fourier transform.