# Weak values in collision theory

## TL;DR

This paper extends the concept of weak measurements in quantum mechanics by incorporating kinetic energy and three-dimensional motion, re-deriving weak values within collision theory and introducing weak vectors for a more complete Stern-Gerlach description.

## Contribution

It introduces two approaches to derive weak values considering kinetic energy and motion, demonstrating their compatibility and generalizing weak values to weak vectors.

## Key findings

- Weak values are compatible with collision theory.
- Two approaches to derive weak values are consistent.
- Weak vectors generalize weak values for better Stern-Gerlach modeling.

## Abstract

Weak measurements have an increasing number of applications in contemporary quantum mechanics. They were originally described as a weak interaction that slightly entangled the translational degrees of freedom of a particle to its spin, yielding surprising results after post-selection. That description often ignores the kinetic energy of the particle and its movement in three dimensions. Here, we include these elements and re-obtain the weak values within the context of collision theory by two different approaches, and prove that the results are compatible with each other and with the results from the traditional approach. To provide a more complete description of the Stern-Gerlach apparatus, we use weak vectors, a generalization of the weak values.