Momentum Representation

T.T. TAYLOR

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.

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.

H. N. Núñez‐Yépez, E. Guillaumı́n-España, R. P. Martı́nez-y-Romero +1 more

The momentum representation is seldom used in quantum mechanics courses. Some students are thence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In this work, we give an introduction to quantum mechanics in momentum space, where the Schrödinger equation becomes an integral equation. To this end we discuss standard problems, namely, the free particle, the quantum motion under a constant potential, a particle interacting with a potential step, and the motion of a particle under a harmonic potential. What is not so standard is that they are all conceived from momentum space and hence they, with the exception of the free particle, are not equivalent to the coordinate space ones with the same names. All the problems are solved within the momentum representation making no reference to the systems they correspond to in the coordinate representation.