Representations: position, momentum, wave functions, and function spaces

S. Rajasekar, R. Velusamy

This chapter discusses the momentum eigenfunctions and momentum probability distribution for some simple time-independent quantum systems. The position-space distribution is useful when we perform measurements of position of a particle. Photo-cells and photographic plates are useful for position measurements for photons. The momentum representation is helpful in the case of measurement of momentum of a particle. The momentum representation is completely equivalent to the coordinate representation. In spite of their complete equivalence, the momentum representation could throw much insight on certain features that may remain hidden in the coordinate representation. A method using delta function interactions has been introduced for constructing momentum wave function for hydrogen atom and hydrogen molecule ion. Momentum space is shown to be better suited for exploring electronic wave functions of quasiperiodic systems.

Agus Budiyono

It is shown that a novel ``epistemically restricted phase-space distribution,'' directly derivable from the wave function, can be operationally determined by a weak measurement of momentum followed by postselection on position. The epistemically restricted phase-space representation thereby reveals deep links between a specific epistemic restriction manifesting transparently quantum uncertainty in classical phase space, weak momentum value, and complex wave function.

Aleksei G. Sitenko

Jun Lu

The exact solutions of the stationary Schrödinger equations for the diatomic system with an empirical potential function in one-dimensional nanomaterials are solved within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The wave functions in position and momentum representations can be obtained through the Fourier-like projection transformation from the phase-space wave functions.

Andrei Yu. Khrennikov