Position and Momentum Representations

Pavel Cherkashin

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.

James M. Feagin

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.

Spiros Konstantogiannis

Making an intuitive assumption, and using the completeness of the position and momentum eigenstates, along with the postulates of quantum mechanics, we provide a geometric presentation of the position and momentum representations in quantum mechanics, in the hope of offering a perspective complementary to those given in standard textbooks.

James M. Feagin